raw-angle from scale-rotated-ellipse

Percentage Accurate: 14.4% → 46.3%

Time: 33.9s

Alternatives: 29

Speedup: 49.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) x-scale) x-scale)))
   (*
    180.0
    (/
     (atan
      (/ (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0)))) t_3))
     PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.cos(t_0)
	t_2 = math.sin(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale
	return 180.0 * (math.atan((((t_4 - t_5) - math.sqrt((math.pow((t_5 - t_4), 2.0) + math.pow(t_3, 2.0)))) / t_3)) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	return Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Initial Program: 14.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) x-scale) x-scale)))
   (*
    180.0
    (/
     (atan
      (/ (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0)))) t_3))
     PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.cos(t_0)
	t_2 = math.sin(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale
	return 180.0 * (math.atan((((t_4 - t_5) - math.sqrt((math.pow((t_5 - t_4), 2.0) + math.pow(t_3, 2.0)))) / t_3)) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	return Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Alternative 1: 46.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ t_3 := -1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale}\\ t_4 := \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\\ t_5 := \sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;b\_m \leq 1.75 \cdot 10^{-191}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{t\_3}{\frac{\cos \left(t\_0 - t\_4\right) - \cos \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_4\right)\right)}{2}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.65 \cdot 10^{+143}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{t\_3}{\frac{\sin \left(t\_0 - \left(-t\_0\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, t\_0\right)\right)}{2}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_5}^{4}} + {t\_5}^{2}\right)}{x-scale}}{t\_5 \cdot \sin t\_1}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1))
        (t_3
         (*
          -1.0
          (/ (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0))) x-scale)))
        (t_4 (fma (* PI angle) 0.005555555555555556 (/ PI 2.0)))
        (t_5 (sin (fma (fabs (* PI angle)) 0.005555555555555556 (/ PI 2.0)))))
   (if (<= b_m 1.75e-191)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          t_3
          (/
           (-
            (cos (- t_0 t_4))
            (cos (fma (* 0.005555555555555556 angle) PI t_4)))
           2.0))))
       PI))
     (if (<= b_m 1.65e+143)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            t_3
            (/
             (+
              (sin (- t_0 (- t_0)))
              (sin (fma (* PI angle) -0.005555555555555556 t_0)))
             2.0))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_5 4.0)) (pow t_5 2.0))) x-scale))
            (* t_5 (sin t_1)))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = -1.0 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / x_45_scale);
	double t_4 = fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0));
	double t_5 = sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double tmp;
	if (b_m <= 1.75e-191) {
		tmp = 180.0 * (atan((0.5 * (t_3 / ((cos((t_0 - t_4)) - cos(fma((0.005555555555555556 * angle), ((double) M_PI), t_4))) / 2.0)))) / ((double) M_PI));
	} else if (b_m <= 1.65e+143) {
		tmp = 180.0 * (atan((0.5 * (t_3 / ((sin((t_0 - -t_0)) + sin(fma((((double) M_PI) * angle), -0.005555555555555556, t_0))) / 2.0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_5, 4.0)) + pow(t_5, 2.0))) / x_45_scale)) / (t_5 * sin(t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / x_45_scale))
	t_4 = fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0))
	t_5 = sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi / 2.0)))
	tmp = 0.0
	if (b_m <= 1.75e-191)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(t_3 / Float64(Float64(cos(Float64(t_0 - t_4)) - cos(fma(Float64(0.005555555555555556 * angle), pi, t_4))) / 2.0)))) / pi));
	elseif (b_m <= 1.65e+143)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(t_3 / Float64(Float64(sin(Float64(t_0 - Float64(-t_0))) + sin(fma(Float64(pi * angle), -0.005555555555555556, t_0))) / 2.0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_5 ^ 4.0)) + (t_5 ^ 2.0))) / x_45_scale)) / Float64(t_5 * sin(t_1))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1.75e-191], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(t$95$3 / N[(N[(N[Cos[N[(t$95$0 - t$95$4), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.65e+143], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(t$95$3 / N[(N[(N[Sin[N[(t$95$0 - (-t$95$0)), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$5, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.75000000000000003e-191

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 34.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) - \cos \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   if 1.75000000000000003e-191 < b < 1.65e143

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       3. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       7. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}{2}}\right)}{\pi} \]

       9. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)}{2}}\right)}{\pi} \]

       10. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)}{2}}\right)}{\pi} \]

       11. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       12. associate-*l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}{2}}\right)}{\pi} \]

       13. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}{2}}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}{2}}\right)}{\pi} \]

       15. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       17. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \mathsf{neg}\left(\frac{1}{180}\right), \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       18. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \mathsf{neg}\left(\frac{1}{180}\right), \left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       19. metadata-eval 45.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   11. Applied rewrites 45.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 1.65e143 < b

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. cos-fabs-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right|\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. fabs-mul N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|angle \cdot \pi\right| \cdot \left|\frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left|angle \cdot \pi\right| \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|angle \cdot \pi\right|, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. cos-fabs-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right|\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. fabs-mul N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|angle \cdot \pi\right| \cdot \left|\frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left|angle \cdot \pi\right| \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|angle \cdot \pi\right|, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. cos-fabs-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right|\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       13. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. fabs-mul N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|angle \cdot \pi\right| \cdot \left|\frac{1}{180}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left|angle \cdot \pi\right| \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\left|angle \cdot \pi\right|, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 45.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 46.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)\\ t_3 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;y-scale \leq -1 \cdot 10^{+232}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale}}{t\_1 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq -3 \cdot 10^{-66}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}{x-scale}}{t\_0 \cdot t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale}}{t\_2 \cdot t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* PI angle) -0.005555555555555556 (* PI 0.5))))
        (t_1 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))))
        (t_2 (sin (fma (* PI angle) 0.005555555555555556 (* PI 0.5))))
        (t_3 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= y-scale -1e+232)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/ (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0))) x-scale))
          (* t_1 (sin (* (* 0.005555555555555556 angle) PI))))))
       PI))
     (if (<= y-scale -3e-66)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))) x-scale))
            (* t_0 t_3))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0))) x-scale))
            (* t_2 t_3))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((((double) M_PI) * angle), -0.005555555555555556, (((double) M_PI) * 0.5)));
	double t_1 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double t_2 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) * 0.5)));
	double t_3 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (y_45_scale <= -1e+232) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / x_45_scale)) / (t_1 * sin(((0.005555555555555556 * angle) * ((double) M_PI))))))) / ((double) M_PI));
	} else if (y_45_scale <= -3e-66) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0))) / x_45_scale)) / (t_0 * t_3)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / x_45_scale)) / (t_2 * t_3)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(pi * angle), -0.005555555555555556, Float64(pi * 0.5)))
	t_1 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	t_2 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi * 0.5)))
	t_3 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (y_45_scale <= -1e+232)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / x_45_scale)) / Float64(t_1 * sin(Float64(Float64(0.005555555555555556 * angle) * pi)))))) / pi));
	elseif (y_45_scale <= -3e-66)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))) / x_45_scale)) / Float64(t_0 * t_3)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / x_45_scale)) / Float64(t_2 * t_3)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$45$scale, -1e+232], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -3e-66], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.00000000000000006e232

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)}\right)}{\pi} \]

       5. lower-*.f64 46.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}{\pi} \]

   15. Applied rewrites 46.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}{\pi} \]

   if -1.00000000000000006e232 < y-scale < -3.0000000000000002e-66

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if -3.0000000000000002e-66 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 46.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)\\ t_3 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\\ \mathbf{if}\;y-scale \leq -1 \cdot 10^{+232}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_3}^{4}} + {t\_3}^{2}\right)}{x-scale}}{t\_3 \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq -3 \cdot 10^{-66}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale}}{t\_1 \cdot t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale}}{t\_2 \cdot t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (fma (* PI angle) -0.005555555555555556 (* PI 0.5))))
        (t_2 (sin (fma (* PI angle) 0.005555555555555556 (* PI 0.5))))
        (t_3 (sin (fma (* 0.005555555555555556 angle) PI (* PI 0.5)))))
   (if (<= y-scale -1e+232)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/ (* y-scale (+ (sqrt (pow t_3 4.0)) (pow t_3 2.0))) x-scale))
          (* t_3 t_0))))
       PI))
     (if (<= y-scale -3e-66)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0))) x-scale))
            (* t_1 t_0))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0))) x-scale))
            (* t_2 t_0))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin(fma((((double) M_PI) * angle), -0.005555555555555556, (((double) M_PI) * 0.5)));
	double t_2 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) * 0.5)));
	double t_3 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) * 0.5)));
	double tmp;
	if (y_45_scale <= -1e+232) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_3, 4.0)) + pow(t_3, 2.0))) / x_45_scale)) / (t_3 * t_0)))) / ((double) M_PI));
	} else if (y_45_scale <= -3e-66) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / x_45_scale)) / (t_1 * t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / x_45_scale)) / (t_2 * t_0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(fma(Float64(pi * angle), -0.005555555555555556, Float64(pi * 0.5)))
	t_2 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi * 0.5)))
	t_3 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi * 0.5)))
	tmp = 0.0
	if (y_45_scale <= -1e+232)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0))) / x_45_scale)) / Float64(t_3 * t_0)))) / pi));
	elseif (y_45_scale <= -3e-66)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / x_45_scale)) / Float64(t_1 * t_0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / x_45_scale)) / Float64(t_2 * t_0)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$45$scale, -1e+232], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -3e-66], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.00000000000000006e232

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 46.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right) \cdot \sin \left(\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if -1.00000000000000006e232 < y-scale < -3.0000000000000002e-66

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if -3.0000000000000002e-66 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. metadata-eval N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 46.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ t_3 := \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\\ t_4 := \sin t\_3\\ \mathbf{if}\;y-scale \leq -1 \cdot 10^{+232}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_4}^{4}} + {t\_4}^{2}\right)}{x-scale}}{t\_4 \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq -3 \cdot 10^{-66}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale}}{t\_1 \cdot t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_3\right)\right)\right)}{x-scale}}{t\_2 \cdot t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (fma (* PI angle) -0.005555555555555556 (* PI 0.5))))
        (t_2 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))))
        (t_3 (fma (* 0.005555555555555556 angle) PI (* PI 0.5)))
        (t_4 (sin t_3)))
   (if (<= y-scale -1e+232)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/ (* y-scale (+ (sqrt (pow t_4 4.0)) (pow t_4 2.0))) x-scale))
          (* t_4 t_0))))
       PI))
     (if (<= y-scale -3e-66)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/ (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0))) x-scale))
            (* t_1 t_0))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (*
               y-scale
               (+ (sqrt (pow t_2 4.0)) (- 0.5 (* 0.5 (cos (* 2.0 t_3))))))
              x-scale))
            (* t_2 t_0))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin(fma((((double) M_PI) * angle), -0.005555555555555556, (((double) M_PI) * 0.5)));
	double t_2 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double t_3 = fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) * 0.5));
	double t_4 = sin(t_3);
	double tmp;
	if (y_45_scale <= -1e+232) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_4, 4.0)) + pow(t_4, 2.0))) / x_45_scale)) / (t_4 * t_0)))) / ((double) M_PI));
	} else if (y_45_scale <= -3e-66) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / x_45_scale)) / (t_1 * t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + (0.5 - (0.5 * cos((2.0 * t_3)))))) / x_45_scale)) / (t_2 * t_0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(fma(Float64(pi * angle), -0.005555555555555556, Float64(pi * 0.5)))
	t_2 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	t_3 = fma(Float64(0.005555555555555556 * angle), pi, Float64(pi * 0.5))
	t_4 = sin(t_3)
	tmp = 0.0
	if (y_45_scale <= -1e+232)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_4 ^ 4.0)) + (t_4 ^ 2.0))) / x_45_scale)) / Float64(t_4 * t_0)))) / pi));
	elseif (y_45_scale <= -3e-66)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / x_45_scale)) / Float64(t_1 * t_0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_3)))))) / x_45_scale)) / Float64(t_2 * t_0)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[y$45$scale, -1e+232], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -3e-66], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.00000000000000006e232

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-fma.f64 46.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. mult-flip N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. metadata-eval N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-*.f64 46.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 46.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right) \cdot \sin \left(\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if -1.00000000000000006e232 < y-scale < -3.0000000000000002e-66

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   16. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   18. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       7. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       8. lift-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       10. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lift-neg.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. distribute-rgt-neg-in N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) + \frac{\pi}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{-1}{180}, \pi \cdot \frac{1}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \mathsf{neg}\left(\frac{1}{180}\right), \frac{\pi}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, -0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if -3.0000000000000002e-66 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
   14. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. lift-cos.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-/.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lift-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lift-cos.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 46.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos t\_0\\ t_2 := -t\_0\\ t_3 := \frac{\sin \left(t\_0 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, \pi \cdot angle, \pi \cdot 0.5\right)\right) + 1, 0.5, \sqrt{{t\_1}^{4}}\right)}{x-scale}}{\sin t\_0 \cdot t\_1} \cdot 0.5\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (cos t_0))
        (t_2 (- t_0))
        (t_3
         (/
          (+
           (sin (- t_0 t_2))
           (sin (fma (* 0.005555555555555556 angle) PI t_2)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (/
      (*
       180.0
       (atan
        (*
         (/
          (-
           (*
            y-scale
            (/
             (fma
              (+ (sin (fma 0.011111111111111112 (* PI angle) (* PI 0.5))) 1.0)
              0.5
              (sqrt (pow t_1 4.0)))
             x-scale)))
          (* (sin t_0) t_1))
         0.5)))
      PI)
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = cos(t_0);
	double t_2 = -t_0;
	double t_3 = (sin((t_0 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = (180.0 * atan(((-(y_45_scale * (fma((sin(fma(0.011111111111111112, (((double) M_PI) * angle), (((double) M_PI) * 0.5))) + 1.0), 0.5, sqrt(pow(t_1, 4.0))) / x_45_scale)) / (sin(t_0) * t_1)) * 0.5))) / ((double) M_PI);
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = cos(t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(sin(Float64(t_0 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(fma(Float64(sin(fma(0.011111111111111112, Float64(pi * angle), Float64(pi * 0.5))) + 1.0), 0.5, sqrt((t_1 ^ 4.0))) / x_45_scale))) / Float64(sin(t_0) * t_1)) * 0.5))) / pi);
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(N[(180.0 * N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       2. sin-+PI/2-rev N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90} + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       3. lower-sin.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90} + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90} + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       5. *-commutative N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       7. *-commutative N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       9. lift-PI.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\pi}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       10. lift-/.f64 N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\pi}{2}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       11. lower-fma.f64 45.7

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \frac{\pi}{2}\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{90}, angle \cdot \pi, \frac{\pi}{2}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{90}, \pi \cdot angle, \frac{\pi}{2}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       14. lift-*.f64 45.7

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, \pi \cdot angle, \frac{\pi}{2}\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

       15. lift-/.f64 N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{90}, \pi \cdot angle, \frac{\pi}{2}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       16. mult-flip N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{90}, \pi \cdot angle, \pi \cdot \frac{1}{2}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       17. metadata-eval N/A

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\frac{1}{90}, \pi \cdot angle, \pi \cdot \frac{1}{2}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       18. lower-*.f64 45.7

           \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, \pi \cdot angle, \pi \cdot 0.5\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   11. Applied rewrites 45.7%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, \pi \cdot angle, \pi \cdot 0.5\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 46.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{1}{\frac{\sin t\_0 \cdot t\_2}{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{t\_2}^{4}}\right)}{x-scale}}}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 2.8 \cdot 10^{-78}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* 0.005555555555555556 angle) PI t_1)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          1.0
          (/
           (* (sin t_0) t_2)
           (-
            (*
             y-scale
             (/
              (fma
               (+ (cos (* (* PI angle) 0.011111111111111112)) 1.0)
               0.5
               (sqrt (pow t_2 4.0)))
              x-scale)))))))
       PI))
     (if (<= x-scale 2.8e-78)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = cos(t_0);
	double t_3 = (sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = 180.0 * (atan((0.5 * (1.0 / ((sin(t_0) * t_2) / -(y_45_scale * (fma((cos(((((double) M_PI) * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt(pow(t_2, 4.0))) / x_45_scale)))))) / ((double) M_PI));
	} else if (x_45_scale <= 2.8e-78) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(1.0 / Float64(Float64(sin(t_0) * t_2) / Float64(-Float64(y_45_scale * Float64(fma(Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt((t_2 ^ 4.0))) / x_45_scale))))))) / pi));
	elseif (x_45_scale <= 2.8e-78)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(1.0 / N[(N[(N[Sin[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision] / (-N[(y$45$scale * N[(N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.8e-78], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{1}{\color{blue}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}}}\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 2.80000000000000024e-78

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 2.80000000000000024e-78 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos t\_0\\ t_2 := -t\_0\\ t_3 := \frac{\sin \left(t\_0 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{1}{\frac{\pi}{\tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{t\_1}^{4}}\right)}{x-scale}}{\sin t\_0 \cdot t\_1} \cdot 0.5\right)}}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (cos t_0))
        (t_2 (- t_0))
        (t_3
         (/
          (+
           (sin (- t_0 t_2))
           (sin (fma (* 0.005555555555555556 angle) PI t_2)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (*
      180.0
      (/
       1.0
       (/
        PI
        (atan
         (*
          (/
           (-
            (*
             y-scale
             (/
              (fma
               (+ (cos (* (* PI angle) 0.011111111111111112)) 1.0)
               0.5
               (sqrt (pow t_1 4.0)))
              x-scale)))
           (* (sin t_0) t_1))
          0.5)))))
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = cos(t_0);
	double t_2 = -t_0;
	double t_3 = (sin((t_0 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = 180.0 * (1.0 / (((double) M_PI) / atan(((-(y_45_scale * (fma((cos(((((double) M_PI) * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt(pow(t_1, 4.0))) / x_45_scale)) / (sin(t_0) * t_1)) * 0.5))));
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = cos(t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(sin(Float64(t_0 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(180.0 * Float64(1.0 / Float64(pi / atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(fma(Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt((t_1 ^ 4.0))) / x_45_scale))) / Float64(sin(t_0) * t_1)) * 0.5)))));
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(180.0 * N[(1.0 / N[(Pi / N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \color{blue}{\frac{1}{\frac{\pi}{\tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}}} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 45.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := -t\_0\\ t_3 := \frac{\sin \left(t\_0 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{\frac{-1 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos t\_0}^{4}}\right) \cdot y-scale\right)}{x-scale}}{\cos t\_1 \cdot \sin t\_1}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (- t_0))
        (t_3
         (/
          (+
           (sin (- t_0 t_2))
           (sin (fma (* 0.005555555555555556 angle) PI t_2)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (/
           (*
            -1.0
            (*
             (fma
              (+ (cos (* (* PI angle) 0.011111111111111112)) 1.0)
              0.5
              (sqrt (pow (cos t_0) 4.0)))
             y-scale))
           x-scale)
          (* (cos t_1) (sin t_1)))))
       PI))
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = -t_0;
	double t_3 = (sin((t_0 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = 180.0 * (atan((0.5 * (((-1.0 * (fma((cos(((((double) M_PI) * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt(pow(cos(t_0), 4.0))) * y_45_scale)) / x_45_scale) / (cos(t_1) * sin(t_1))))) / ((double) M_PI));
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(sin(Float64(t_0 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(Float64(-1.0 * Float64(fma(Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt((cos(t_0) ^ 4.0))) * y_45_scale)) / x_45_scale) / Float64(cos(t_1) * sin(t_1))))) / pi));
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(N[(-1.0 * N[(N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{\frac{-1 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot y-scale\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos t\_0\\ t_2 := -t\_0\\ t_3 := \frac{\sin \left(t\_0 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.5}{\sin t\_0} \cdot \frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{t\_1}^{4}}\right)}{x-scale}}{t\_1}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (cos t_0))
        (t_2 (- t_0))
        (t_3
         (/
          (+
           (sin (- t_0 t_2))
           (sin (fma (* 0.005555555555555556 angle) PI t_2)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (*
      180.0
      (/
       (atan
        (*
         (/ 0.5 (sin t_0))
         (/
          (-
           (*
            y-scale
            (/
             (fma
              (+ (cos (* (* PI angle) 0.011111111111111112)) 1.0)
              0.5
              (sqrt (pow t_1 4.0)))
             x-scale)))
          t_1)))
       PI))
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = cos(t_0);
	double t_2 = -t_0;
	double t_3 = (sin((t_0 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = 180.0 * (atan(((0.5 / sin(t_0)) * (-(y_45_scale * (fma((cos(((((double) M_PI) * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt(pow(t_1, 4.0))) / x_45_scale)) / t_1))) / ((double) M_PI));
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = cos(t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(sin(Float64(t_0 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.5 / sin(t_0)) * Float64(Float64(-Float64(y_45_scale * Float64(fma(Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt((t_1 ^ 4.0))) / x_45_scale))) / t_1))) / pi));
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(180.0 * N[(N[ArcTan[N[(N[(0.5 / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[((-N[(y$45$scale * N[(N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \color{blue}{\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}}\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}\\ t_3 := \cos t\_0\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1, 0.5, \sqrt{{t\_3}^{4}}\right)}{x-scale}}{\sin t\_0 \cdot t\_3} \cdot 0.5\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_2}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* 0.005555555555555556 angle) PI t_1)))
          2.0))
        (t_3 (cos t_0)))
   (if (<= x-scale -3.05e-38)
     (/
      (*
       180.0
       (atan
        (*
         (/
          (-
           (*
            y-scale
            (/
             (fma
              (+ (cos (* PI (* angle 0.011111111111111112))) 1.0)
              0.5
              (sqrt (pow t_3 4.0)))
             x-scale)))
          (* (sin t_0) t_3))
         0.5)))
      PI)
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_2)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_2))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = (sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0;
	double t_3 = cos(t_0);
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = (180.0 * atan(((-(y_45_scale * (fma((cos((((double) M_PI) * (angle * 0.011111111111111112))) + 1.0), 0.5, sqrt(pow(t_3, 4.0))) / x_45_scale)) / (sin(t_0) * t_3)) * 0.5))) / ((double) M_PI);
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_2))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_2))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0)
	t_3 = cos(t_0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(fma(Float64(cos(Float64(pi * Float64(angle * 0.011111111111111112))) + 1.0), 0.5, sqrt((t_3 ^ 4.0))) / x_45_scale))) / Float64(sin(t_0) * t_3)) * 0.5))) / pi);
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_2))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_2))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(N[(180.0 * N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       3. associate-*l* N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot \frac{1}{90}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot \frac{1}{90}\right)\right) + 1, \frac{1}{2}, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       5. lower-*.f64 45.6

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 45.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}\\ t_3 := \cos t\_0\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{t\_3}^{4}}\right)}{x-scale}}{\sin t\_0 \cdot t\_3} \cdot 0.5\right)}{\pi} \cdot 180\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_2}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* 0.005555555555555556 angle) PI t_1)))
          2.0))
        (t_3 (cos t_0)))
   (if (<= x-scale -3.05e-38)
     (*
      (/
       (atan
        (*
         (/
          (-
           (*
            y-scale
            (/
             (fma
              (+ (cos (* (* PI angle) 0.011111111111111112)) 1.0)
              0.5
              (sqrt (pow t_3 4.0)))
             x-scale)))
          (* (sin t_0) t_3))
         0.5))
       PI)
      180.0)
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_2)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_2))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = (sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0;
	double t_3 = cos(t_0);
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = (atan(((-(y_45_scale * (fma((cos(((((double) M_PI) * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt(pow(t_3, 4.0))) / x_45_scale)) / (sin(t_0) * t_3)) * 0.5)) / ((double) M_PI)) * 180.0;
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_2))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_2))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0)
	t_3 = cos(t_0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(Float64(atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(fma(Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) + 1.0), 0.5, sqrt((t_3 ^ 4.0))) / x_45_scale))) / Float64(sin(t_0) * t_3)) * 0.5)) / pi) * 180.0);
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_2))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_2))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(N[(N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \cdot 180} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := -t\_1\\ t_3 := \frac{\sin \left(t\_1 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}\\ \mathbf{if}\;x-scale \leq -3.05 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}{x-scale}}{0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 9.2 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI))))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (- t_1))
        (t_3
         (/
          (+
           (sin (- t_1 t_2))
           (sin (fma (* 0.005555555555555556 angle) PI t_2)))
          2.0)))
   (if (<= x-scale -3.05e-38)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/ (* y-scale (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))) x-scale))
          (* 0.5 (sin (* 0.011111111111111112 (* angle PI)))))))
       PI))
     (if (<= x-scale 9.2e-61)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_3)))
         PI))
       (*
        180.0
        (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_3))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = -t_1;
	double t_3 = (sin((t_1 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0;
	double tmp;
	if (x_45_scale <= -3.05e-38) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0))) / x_45_scale)) / (0.5 * sin((0.011111111111111112 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
	} else if (x_45_scale <= 9.2e-61) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(-t_1)
	t_3 = Float64(Float64(sin(Float64(t_1 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)
	tmp = 0.0
	if (x_45_scale <= -3.05e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))) / x_45_scale)) / Float64(0.5 * sin(Float64(0.011111111111111112 * Float64(angle * pi))))))) / pi));
	elseif (x_45_scale <= 9.2e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_3))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(N[(N[Sin[N[(t$95$1 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.05e-38], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9.2e-61], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.04999999999999986e-38

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{1}{2} \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

       2. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

       5. lower-PI.f64 45.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{0.5 \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{0.5 \cdot \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if -3.04999999999999986e-38 < x-scale < 9.19999999999999967e-61

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if 9.19999999999999967e-61 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 45.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{t\_2}\right)}{\pi}\\ \mathbf{if}\;x-scale \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x-scale \leq 2.8 \cdot 10^{-78}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* 0.005555555555555556 angle) PI t_1)))
          2.0))
        (t_3
         (*
          180.0
          (/ (atan (* 0.5 (/ (* -1.0 (/ (* 2.0 y-scale) x-scale)) t_2))) PI))))
   (if (<= x-scale -8.2e-21)
     t_3
     (if (<= x-scale 2.8e-78)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (fma
               -6.17283950617284e-5
               (* (pow angle 2.0) (* y-scale (pow PI 2.0)))
               (* 2.0 y-scale))
              x-scale))
            t_2)))
         PI))
       t_3))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = (sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0;
	double t_3 = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / t_2))) / ((double) M_PI));
	double tmp;
	if (x_45_scale <= -8.2e-21) {
		tmp = t_3;
	} else if (x_45_scale <= 2.8e-78) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (fma(-6.17283950617284e-5, (pow(angle, 2.0) * (y_45_scale * pow(((double) M_PI), 2.0))), (2.0 * y_45_scale)) / x_45_scale)) / t_2))) / ((double) M_PI));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0)
	t_3 = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / t_2))) / pi))
	tmp = 0.0
	if (x_45_scale <= -8.2e-21)
		tmp = t_3;
	elseif (x_45_scale <= 2.8e-78)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(fma(-6.17283950617284e-5, Float64((angle ^ 2.0) * Float64(y_45_scale * (pi ^ 2.0))), Float64(2.0 * y_45_scale)) / x_45_scale)) / t_2))) / pi));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -8.2e-21], t$95$3, If[LessEqual[x$45$scale, 2.8e-78], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[(y$45$scale * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -8.19999999999999988e-21 or 2.80000000000000024e-78 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   if -8.19999999999999988e-21 < x-scale < 2.80000000000000024e-78

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{-1}{16200} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       3. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       6. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16200}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]

       7. lower-*.f64 40.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 40.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, {angle}^{2} \cdot \left(y-scale \cdot {\pi}^{2}\right), 2 \cdot y-scale\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;y-scale \leq 7.5 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {t\_2}^{2}\right)}{x-scale}}{t\_2 \cdot t\_1}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq 2.35 \cdot 10^{+57}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin t\_3 \cdot \cos t\_3} \cdot 0.5\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3 (* (* PI angle) 0.005555555555555556)))
   (if (<= y-scale 7.5e-63)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* -1.0 (/ (* y-scale (+ 1.0 (pow t_2 2.0))) x-scale))
          (* t_2 t_1))))
       PI))
     (if (<= y-scale 2.35e+57)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (*
              x-scale
              (*
               y-scale
               (+
                (sqrt (/ 1.0 (pow x-scale 4.0)))
                (/ 1.0 (pow x-scale 2.0))))))
            (* (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))) t_1))))
         PI))
       (/
        (*
         180.0
         (atan
          (*
           (/
            (-
             (*
              y-scale
              (/
               (+
                2.0
                (* -6.17283950617284e-5 (* (pow angle 2.0) (pow PI 2.0))))
               x-scale)))
            (* (sin t_3) (cos t_3)))
           0.5)))
        PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (y_45_scale <= 7.5e-63) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + pow(t_2, 2.0))) / x_45_scale)) / (t_2 * t_1)))) / ((double) M_PI));
	} else if (y_45_scale <= 2.35e+57) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0)))))) / (sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0))) * t_1)))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * (pow(angle, 2.0) * pow(((double) M_PI), 2.0)))) / x_45_scale)) / (sin(t_3) * cos(t_3))) * 0.5))) / ((double) M_PI);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (y_45_scale <= 7.5e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + (t_2 ^ 2.0))) / x_45_scale)) / Float64(t_2 * t_1)))) / pi));
	elseif (y_45_scale <= 2.35e+57)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0)))))) / Float64(sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0))) * t_1)))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(Float64(2.0 + Float64(-6.17283950617284e-5 * Float64((angle ^ 2.0) * (pi ^ 2.0)))) / x_45_scale))) / Float64(sin(t_3) * cos(t_3))) * 0.5))) / pi);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[y$45$scale, 7.5e-63], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 2.35e+57], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(2.0 + N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$3], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 7.5000000000000003e-63

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 7.5000000000000003e-63 < y-scale < 2.3500000000000001e57

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   14. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. Applied rewrites 42.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 2.3500000000000001e57 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       6. lower-PI.f64 40.1

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   12. Applied rewrites 40.1%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;y-scale \leq 7.5 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_0 \cdot t\_1}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq 2.35 \cdot 10^{+57}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin t\_2 \cdot \cos t\_2} \cdot 0.5\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= y-scale 7.5e-63)
     (*
      180.0
      (/ (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_0) t_1)))) PI))
     (if (<= y-scale 2.35e+57)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (*
              x-scale
              (*
               y-scale
               (+
                (sqrt (/ 1.0 (pow x-scale 4.0)))
                (/ 1.0 (pow x-scale 2.0))))))
            (* (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))) t_1))))
         PI))
       (/
        (*
         180.0
         (atan
          (*
           (/
            (-
             (*
              y-scale
              (/
               (+
                2.0
                (* -6.17283950617284e-5 (* (pow angle 2.0) (pow PI 2.0))))
               x-scale)))
            (* (sin t_2) (cos t_2)))
           0.5)))
        PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (y_45_scale <= 7.5e-63) {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * t_1)))) / ((double) M_PI));
	} else if (y_45_scale <= 2.35e+57) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * (x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0)))))) / (sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0))) * t_1)))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * (pow(angle, 2.0) * pow(((double) M_PI), 2.0)))) / x_45_scale)) / (sin(t_2) * cos(t_2))) * 0.5))) / ((double) M_PI);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (y_45_scale <= 7.5e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(cos(t_0) * t_1)))) / pi));
	elseif (y_45_scale <= 2.35e+57)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0)))))) / Float64(sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0))) * t_1)))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(Float64(2.0 + Float64(-6.17283950617284e-5 * Float64((angle ^ 2.0) * (pi ^ 2.0)))) / x_45_scale))) / Float64(sin(t_2) * cos(t_2))) * 0.5))) / pi);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[y$45$scale, 7.5e-63], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 2.35e+57], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(2.0 + N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$2], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 7.5000000000000003e-63

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 45.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 7.5000000000000003e-63 < y-scale < 2.3500000000000001e57

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      4. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      16. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      18. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      19. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      20. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      21. lower-/.f64 45.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 45.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 45.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       11. lower-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

       12. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       13. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       15. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       16. lower-fma.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       17. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       18. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       19. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       20. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       21. lower-/.f64 46.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   14. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. Applied rewrites 42.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 2.3500000000000001e57 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       6. lower-PI.f64 40.1

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   12. Applied rewrites 40.1%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := -t\_1\\ \mathbf{if}\;x-scale \leq -3.1 \cdot 10^{-170}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{-42}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(t\_1 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (- t_1)))
   (if (<= x-scale -3.1e-170)
     (*
      180.0
      (/
       (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_0) (sin t_0)))))
       PI))
     (if (<= x-scale 5e-42)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (*
             x-scale
             (*
              y-scale
              (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
            (* angle PI))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -1.0 (/ (* 2.0 y-scale) x-scale))
            (/
             (+
              (sin (- t_1 t_2))
              (sin (fma (* 0.005555555555555556 angle) PI t_2)))
             2.0))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = -t_1;
	double tmp;
	if (x_45_scale <= -3.1e-170) {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
	} else if (x_45_scale <= 5e-42) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((2.0 * y_45_scale) / x_45_scale)) / ((sin((t_1 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(-t_1)
	tmp = 0.0
	if (x_45_scale <= -3.1e-170)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(cos(t_0) * sin(t_0))))) / pi));
	elseif (x_45_scale <= 5e-42)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(2.0 * y_45_scale) / x_45_scale)) / Float64(Float64(sin(Float64(t_1 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, If[LessEqual[x$45$scale, -3.1e-170], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 5e-42], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(2.0 * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$1 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.09999999999999986e-170

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 45.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if -3.09999999999999986e-170 < x-scale < 5.00000000000000003e-42

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

   if 5.00000000000000003e-42 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      3. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      5. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      6. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      8. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      11. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      12. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      14. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      15. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\pi} \]

      16. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      17. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      18. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}\right)}{\pi} \]

      19. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

      20. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}{\pi} \]

   9. Applied rewrites 45.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}}}\right)}{\pi} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(-\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{2}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-*.f64 45.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]

   12. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{2 \cdot y-scale}{x-scale}}{\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 - \left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{2}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\right)\right)}{x-scale}}{t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0)))))
   (*
    180.0
    (/
     (atan
      (*
       0.5
       (/
        (*
         -1.0
         (/
          (*
           y-scale
           (+
            (sqrt (pow t_0 4.0))
            (-
             0.5
             (*
              0.5
              (cos
               (* 2.0 (fma (* 0.005555555555555556 angle) PI (* PI 0.5))))))))
          x-scale))
        (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	return 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_0, 4.0)) + (0.5 - (0.5 * cos((2.0 * fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) * 0.5)))))))) / x_45_scale)) / (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	return Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_0 ^ 4.0)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(Float64(0.005555555555555556 * angle), pi, Float64(pi * 0.5)))))))) / x_45_scale)) / Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 23.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

7. Applied rewrites 45.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   3. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   4. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   5. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   6. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. sin-+PI/2-rev N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. lower-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   14. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. lower-fma.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   18. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   20. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   21. lower-/.f64 45.7

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

9. Applied rewrites 45.7%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
10. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 45.6

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

11. Applied rewrites 45.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
12. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 46.3

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

13. Applied rewrites 46.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
14. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. lift-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-fma.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. sin-+PI/2-rev N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. unpow2 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. lift-cos.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    11. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-/.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lift-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lift-cos.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

15. Applied rewrites 46.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
16. Add Preprocessing

Alternative 18: 45.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}{x-scale}}{t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0)))))
   (*
    180.0
    (/
     (atan
      (*
       0.5
       (/
        (*
         -1.0
         (/
          (*
           y-scale
           (+
            (sqrt (pow t_0 4.0))
            (+
             0.5
             (* 0.5 (cos (* 2.0 (* (* PI angle) 0.005555555555555556)))))))
          x-scale))
        (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	return 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_0, 4.0)) + (0.5 + (0.5 * cos((2.0 * ((((double) M_PI) * angle) * 0.005555555555555556))))))) / x_45_scale)) / (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	return Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_0 ^ 4.0)) + Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * angle) * 0.005555555555555556))))))) / x_45_scale)) / Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 23.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

7. Applied rewrites 45.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   3. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   4. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   5. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   6. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. sin-+PI/2-rev N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. lower-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   14. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. lower-fma.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   18. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   20. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   21. lower-/.f64 45.7

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

9. Applied rewrites 45.7%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
10. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 45.6

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

11. Applied rewrites 45.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
12. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 46.3

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

13. Applied rewrites 46.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
14. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. lift-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-fma.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\pi}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. sin-+PI/2-rev N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. unpow2 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. lift-cos.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    11. lift-cos.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    12. sqr-cos-a N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. lower-+.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-cos.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-*.f64 46.3

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

15. Applied rewrites 46.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
16. Add Preprocessing

Alternative 19: 45.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot -0.005555555555555556\\ 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \sqrt{{\cos t\_0}^{4}}}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) -0.005555555555555556)))
   (*
    180.0
    (/
     (atan
      (*
       0.5
       (/
        (-
         (*
          y-scale
          (/
           (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (sqrt (pow (cos t_0) 4.0)))
           x-scale)))
        (*
         (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0)))
         (sin (* 0.005555555555555556 (* angle PI)))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * -0.005555555555555556;
	return 180.0 * (atan((0.5 * (-(y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_0)))) + sqrt(pow(cos(t_0), 4.0))) / x_45_scale)) / (sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0))) * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * -0.005555555555555556)
	return Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) + sqrt((cos(t_0) ^ 4.0))) / x_45_scale))) / Float64(sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0))) * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(0.5 * N[((-N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 23.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

7. Applied rewrites 45.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   3. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   4. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   5. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   6. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. sin-+PI/2-rev N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   11. lower-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   13. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   14. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   15. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. lower-fma.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   17. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   18. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   19. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   20. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   21. lower-/.f64 45.7

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

9. Applied rewrites 45.7%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
10. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 45.6

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

11. Applied rewrites 45.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
12. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    3. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    4. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    5. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    6. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    7. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    8. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    11. lower-sin.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

    12. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    13. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    14. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    15. lower-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    16. lower-fma.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    17. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    18. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    19. lift-*.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    20. lift-PI.f64 N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    21. lower-/.f64 46.3

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

13. Applied rewrites 46.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)}{x-scale}}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

15. Applied rewrites 46.2%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)\right)\right) + \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)}^{4}}}{x-scale}}{\color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
16. Add Preprocessing

Alternative 20: 45.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;y-scale \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin t\_1 \cdot \cos t\_1} \cdot 0.5\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (* PI angle) 0.005555555555555556)))
   (if (<= y-scale 1.15e+56)
     (*
      180.0
      (/
       (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_0) (sin t_0)))))
       PI))
     (/
      (*
       180.0
       (atan
        (*
         (/
          (-
           (*
            y-scale
            (/
             (+ 2.0 (* -6.17283950617284e-5 (* (pow angle 2.0) (pow PI 2.0))))
             x-scale)))
          (* (sin t_1) (cos t_1)))
         0.5)))
      PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (y_45_scale <= 1.15e+56) {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * (pow(angle, 2.0) * pow(((double) M_PI), 2.0)))) / x_45_scale)) / (sin(t_1) * cos(t_1))) * 0.5))) / ((double) M_PI);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (y_45_scale <= 1.15e+56) {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (Math.cos(t_0) * Math.sin(t_0))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * (Math.pow(angle, 2.0) * Math.pow(Math.PI, 2.0)))) / x_45_scale)) / (Math.sin(t_1) * Math.cos(t_1))) * 0.5))) / Math.PI;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if y_45_scale <= 1.15e+56:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (math.cos(t_0) * math.sin(t_0))))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * (math.pow(angle, 2.0) * math.pow(math.pi, 2.0)))) / x_45_scale)) / (math.sin(t_1) * math.cos(t_1))) * 0.5))) / math.pi
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (y_45_scale <= 1.15e+56)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(cos(t_0) * sin(t_0))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-Float64(y_45_scale * Float64(Float64(2.0 + Float64(-6.17283950617284e-5 * Float64((angle ^ 2.0) * (pi ^ 2.0)))) / x_45_scale))) / Float64(sin(t_1) * cos(t_1))) * 0.5))) / pi);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (y_45_scale <= 1.15e+56)
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / pi);
	else
		tmp = (180.0 * atan(((-(y_45_scale * ((2.0 + (-6.17283950617284e-5 * ((angle ^ 2.0) * (pi ^ 2.0)))) / x_45_scale)) / (sin(t_1) * cos(t_1))) * 0.5))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[y$45$scale, 1.15e+56], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[((-N[(y$45$scale * N[(N[(2.0 + N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.15000000000000007e56

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 45.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   10. Applied rewrites 45.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 1.15000000000000007e56 < y-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 23.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 45.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

       6. lower-PI.f64 40.1

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

   12. Applied rewrites 40.1%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 45.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \frac{180 \cdot \tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin t\_0 \cdot \cos t\_0} \cdot 0.5\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (/
    (*
     180.0
     (atan (* (/ (* -2.0 (/ y-scale x-scale)) (* (sin t_0) (cos t_0))) 0.5)))
    PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	return (180.0 * atan((((-2.0 * (y_45_scale / x_45_scale)) / (sin(t_0) * cos(t_0))) * 0.5))) / ((double) M_PI);
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	return (180.0 * Math.atan((((-2.0 * (y_45_scale / x_45_scale)) / (Math.sin(t_0) * Math.cos(t_0))) * 0.5))) / Math.PI;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (math.pi * angle) * 0.005555555555555556
	return (180.0 * math.atan((((-2.0 * (y_45_scale / x_45_scale)) / (math.sin(t_0) * math.cos(t_0))) * 0.5))) / math.pi

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	return Float64(Float64(180.0 * atan(Float64(Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(sin(t_0) * cos(t_0))) * 0.5))) / pi)
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = (180.0 * atan((((-2.0 * (y_45_scale / x_45_scale)) / (sin(t_0) * cos(t_0))) * 0.5))) / pi;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[(180.0 * N[ArcTan[N[(N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 23.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

7. Applied rewrites 45.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

9. Applied rewrites 45.6%

   \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{-y-scale \cdot \frac{\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 1, 0.5, \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi}} \]

10. Taylor expanded in angle around 0

    \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{\frac{-4}{2} \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{\frac{-4}{2} \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

    3. metadata-eval N/A

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)} \cdot \frac{1}{2}\right)}{\pi} \]

    4. lower-/.f64 45.3

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]

12. Applied rewrites 45.3%

    \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-2 \cdot \frac{y-scale}{x-scale}}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 0.5\right)}{\pi} \]
13. Add Preprocessing

Alternative 22: 44.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (*
    180.0
    (/
     (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_0) (sin t_0)))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	return 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	return 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (Math.cos(t_0) * Math.sin(t_0))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	return 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (math.cos(t_0) * math.sin(t_0))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	return Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(cos(t_0) * sin(t_0))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 23.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} - \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

7. Applied rewrites 45.6%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

8. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   2. lower-/.f64 45.3

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

10. Applied rewrites 45.3%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
11. Add Preprocessing

Alternative 23: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := {b\_m}^{2} - {a}^{2}\\ t_3 := \sin t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot t\_2\right) \cdot t\_3\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ t_6 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_3\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{if}\;\frac{\left(t\_6 - t\_5\right) - \sqrt{{\left(t\_5 - t\_6\right)}^{2} + {t\_4}^{2}}}{t\_4} \leq 10^{+274}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\mathsf{fma}\left(b\_m, b\_m, \sqrt{{b\_m}^{4}}\right)}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot t\_2\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (- (pow b_m 2.0) (pow a 2.0)))
        (t_3 (sin t_0))
        (t_4 (/ (/ (* (* (* 2.0 t_2) t_3) t_1) x-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b_m t_1) 2.0)) x-scale) x-scale))
        (t_6
         (/
          (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_3) 2.0)) y-scale)
          y-scale)))
   (if (<=
        (/ (- (- t_6 t_5) (sqrt (+ (pow (- t_5 t_6) 2.0) (pow t_4 2.0)))) t_4)
        1e+274)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (*
             (/ -1.0 x-scale)
             (/ (fma b_m b_m (sqrt (pow b_m 4.0))) x-scale))))
          (* angle (* PI t_2)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
          (* angle PI))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = pow(b_m, 2.0) - pow(a, 2.0);
	double t_3 = sin(t_0);
	double t_4 = ((((2.0 * t_2) * t_3) * t_1) / x_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_3), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double t_6 = ((pow((a * t_1), 2.0) + pow((b_m * t_3), 2.0)) / y_45_scale) / y_45_scale;
	double tmp;
	if ((((t_6 - t_5) - sqrt((pow((t_5 - t_6), 2.0) + pow(t_4, 2.0)))) / t_4) <= 1e+274) {
		tmp = 180.0 * (atan((90.0 * ((x_45_scale * (y_45_scale * ((-1.0 / x_45_scale) * (fma(b_m, b_m, sqrt(pow(b_m, 4.0))) / x_45_scale)))) / (angle * (((double) M_PI) * t_2))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	t_3 = sin(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * t_2) * t_3) * t_1) / x_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_6 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_3) ^ 2.0)) / y_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(Float64(t_6 - t_5) - sqrt(Float64((Float64(t_5 - t_6) ^ 2.0) + (t_4 ^ 2.0)))) / t_4) <= 1e+274)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(Float64(-1.0 / x_45_scale) * Float64(fma(b_m, b_m, sqrt((b_m ^ 4.0))) / x_45_scale)))) / Float64(angle * Float64(pi * t_2))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$6 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$6), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], 1e+274], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[(-1.0 / x$45$scale), $MachinePrecision] * N[(N[(b$95$m * b$95$m + N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(Pi * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) < 9.99999999999999921e273

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      3. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      5. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      6. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      7. lower-pow.f64 22.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 22.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      3. lift-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      4. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{-1 \cdot \left(\sqrt{{b}^{4}} + {b}^{2}\right)}{{x-scale}^{2}}\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      5. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{-1 \cdot \left(\sqrt{{b}^{4}} + {b}^{2}\right)}{x-scale \cdot x-scale}\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      6. times-frac N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      8. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      9. lower-/.f64 22.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      10. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      11. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{{b}^{2} + \sqrt{{b}^{4}}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      12. lift-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{{b}^{2} + \sqrt{{b}^{4}}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      13. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{b \cdot b + \sqrt{{b}^{4}}}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      14. lower-fma.f64 22.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\mathsf{fma}\left(b, b, \sqrt{{b}^{4}}\right)}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 22.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{-1}{x-scale} \cdot \frac{\mathsf{fma}\left(b, b, \sqrt{{b}^{4}}\right)}{x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   if 9.99999999999999921e273 < (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ t_6 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ \mathbf{if}\;\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3} \leq 10^{+274}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{\frac{\left(\left(t\_6 - \mathsf{fma}\left(\frac{b\_m}{x-scale}, \frac{b\_m}{x-scale}, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_6\right|\right)\right) \cdot y-scale\right) \cdot x-scale}{angle}}{\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_1) 2.0)) x-scale) x-scale))
        (t_6 (/ (* a a) (* y-scale y-scale))))
   (if (<=
        (/ (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0)))) t_3)
        1e+274)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (/
           (*
            (*
             (-
              t_6
              (fma
               (/ b_m x-scale)
               (/ b_m x-scale)
               (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) t_6))))
             y-scale)
            x-scale)
           angle)
          (* (* (- b_m a) (+ b_m a)) PI))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
          (* angle PI))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double t_6 = (a * a) / (y_45_scale * y_45_scale);
	double tmp;
	if ((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3) <= 1e+274) {
		tmp = 180.0 * (atan((90.0 * (((((t_6 - fma((b_m / x_45_scale), (b_m / x_45_scale), fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - t_6)))) * y_45_scale) * x_45_scale) / angle) / (((b_m - a) * (b_m + a)) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_6 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3) <= 1e+274)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(Float64(Float64(Float64(t_6 - fma(Float64(b_m / x_45_scale), Float64(b_m / x_45_scale), abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - t_6)))) * y_45_scale) * x_45_scale) / angle) / Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * pi)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], 1e+274], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(N[(N[(N[(t$95$6 - N[(N[(b$95$m / x$45$scale), $MachinePrecision] * N[(b$95$m / x$45$scale), $MachinePrecision] + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / angle), $MachinePrecision] / N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) < 9.99999999999999921e273

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   6. Applied rewrites 15.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{\frac{\left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} - \mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right) \cdot y-scale\right) \cdot x-scale}{angle}}{\color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \pi}}\right)}{\pi} \]

   if 9.99999999999999921e273 < (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 40.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ t_3 := \sin t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_3\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ t_6 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_3\right)}^{2}}{y-scale}}{y-scale}\\ \mathbf{if}\;\frac{\left(t\_6 - t\_5\right) - \sqrt{{\left(t\_5 - t\_6\right)}^{2} + {t\_4}^{2}}}{t\_4} \leq 10^{+274}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{90 \cdot \left(\left(\left(t\_2 - \mathsf{fma}\left(\frac{b\_m}{x-scale}, \frac{b\_m}{x-scale}, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_2\right|\right)\right) \cdot y-scale\right) \cdot x-scale\right)}{\left(\pi \cdot angle\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (/ (* a a) (* y-scale y-scale)))
        (t_3 (sin t_0))
        (t_4
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_3) t_1) x-scale)
          y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b_m t_1) 2.0)) x-scale) x-scale))
        (t_6
         (/
          (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_3) 2.0)) y-scale)
          y-scale)))
   (if (<=
        (/ (- (- t_6 t_5) (sqrt (+ (pow (- t_5 t_6) 2.0) (pow t_4 2.0)))) t_4)
        1e+274)
     (*
      (/
       (atan
        (/
         (*
          90.0
          (*
           (*
            (-
             t_2
             (fma
              (/ b_m x-scale)
              (/ b_m x-scale)
              (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) t_2))))
            y-scale)
           x-scale))
         (* (* PI angle) (* (- b_m a) (+ b_m a)))))
       PI)
      180.0)
     (*
      180.0
      (/
       (atan
        (*
         -90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
          (* angle PI))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = (a * a) / (y_45_scale * y_45_scale);
	double t_3 = sin(t_0);
	double t_4 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_3) * t_1) / x_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_3), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double t_6 = ((pow((a * t_1), 2.0) + pow((b_m * t_3), 2.0)) / y_45_scale) / y_45_scale;
	double tmp;
	if ((((t_6 - t_5) - sqrt((pow((t_5 - t_6), 2.0) + pow(t_4, 2.0)))) / t_4) <= 1e+274) {
		tmp = (atan(((90.0 * (((t_2 - fma((b_m / x_45_scale), (b_m / x_45_scale), fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - t_2)))) * y_45_scale) * x_45_scale)) / ((((double) M_PI) * angle) * ((b_m - a) * (b_m + a))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	t_3 = sin(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_3) * t_1) / x_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_6 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_3) ^ 2.0)) / y_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(Float64(t_6 - t_5) - sqrt(Float64((Float64(t_5 - t_6) ^ 2.0) + (t_4 ^ 2.0)))) / t_4) <= 1e+274)
		tmp = Float64(Float64(atan(Float64(Float64(90.0 * Float64(Float64(Float64(t_2 - fma(Float64(b_m / x_45_scale), Float64(b_m / x_45_scale), abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - t_2)))) * y_45_scale) * x_45_scale)) / Float64(Float64(pi * angle) * Float64(Float64(b_m - a) * Float64(b_m + a))))) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$6 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$6), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], 1e+274], N[(N[(N[ArcTan[N[(N[(90.0 * N[(N[(N[(t$95$2 - N[(N[(b$95$m / x$45$scale), $MachinePrecision] * N[(b$95$m / x$45$scale), $MachinePrecision] + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * angle), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) < 9.99999999999999921e273

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   6. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{90 \cdot \left(\left(\left(\frac{a \cdot a}{y-scale \cdot y-scale} - \mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right) \cdot y-scale\right) \cdot x-scale\right)}{\left(\pi \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi} \cdot 180} \]

   if 9.99999999999999921e273 < (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 40.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/
   (atan
    (*
     -90.0
     (/
      (*
       x-scale
       (*
        y-scale
        (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
      (* angle PI))))
   PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
3. Step-by-step derivation

4. Applied rewrites 12.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

7. Applied rewrites 40.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
8. Add Preprocessing

Alternative 27: 28.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \mathbf{if}\;x-scale \leq -2 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale \leq 2500000000000:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90}{\left(b\_m \cdot b\_m - a \cdot a\right) \cdot \pi} \cdot \frac{\left(\left(-\frac{\mathsf{fma}\left(b\_m, b\_m, \sqrt{{b\_m}^{4}}\right)}{x-scale \cdot x-scale}\right) \cdot y-scale\right) \cdot x-scale}{angle}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ (atan 0.0) PI))))
   (if (<= x-scale -2e-19)
     t_0
     (if (<= x-scale 2500000000000.0)
       (*
        180.0
        (/
         (atan
          (*
           (/ 90.0 (* (- (* b_m b_m) (* a a)) PI))
           (/
            (*
             (*
              (- (/ (fma b_m b_m (sqrt (pow b_m 4.0))) (* x-scale x-scale)))
              y-scale)
             x-scale)
            angle)))
         PI))
       t_0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (atan(0.0) / ((double) M_PI));
	double tmp;
	if (x_45_scale <= -2e-19) {
		tmp = t_0;
	} else if (x_45_scale <= 2500000000000.0) {
		tmp = 180.0 * (atan(((90.0 / (((b_m * b_m) - (a * a)) * ((double) M_PI))) * (((-(fma(b_m, b_m, sqrt(pow(b_m, 4.0))) / (x_45_scale * x_45_scale)) * y_45_scale) * x_45_scale) / angle))) / ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(180.0 * Float64(atan(0.0) / pi))
	tmp = 0.0
	if (x_45_scale <= -2e-19)
		tmp = t_0;
	elseif (x_45_scale <= 2500000000000.0)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 / Float64(Float64(Float64(b_m * b_m) - Float64(a * a)) * pi)) * Float64(Float64(Float64(Float64(-Float64(fma(b_m, b_m, sqrt((b_m ^ 4.0))) / Float64(x_45_scale * x_45_scale))) * y_45_scale) * x_45_scale) / angle))) / pi));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2e-19], t$95$0, If[LessEqual[x$45$scale, 2500000000000.0], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 / N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(N[((-N[(N[(b$95$m * b$95$m + N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]) * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -2e-19 or 2.5e12 < x-scale

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 7.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]

   if -2e-19 < x-scale < 2.5e12

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      3. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      5. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      6. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      7. lower-pow.f64 22.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 22.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}}\right)}{\pi} \]

      2. lift-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}}\right)}{\pi} \]

      3. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 24.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90}{\left(b \cdot b - a \cdot a\right) \cdot \pi} \cdot \color{blue}{\frac{\left(\left(-\frac{\mathsf{fma}\left(b, b, \sqrt{{b}^{4}}\right)}{x-scale \cdot x-scale}\right) \cdot y-scale\right) \cdot x-scale}{angle}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 27.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \leq 100:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(\left(-\frac{\mathsf{fma}\left(b\_m, b\_m, \sqrt{{b\_m}^{4}}\right)}{x-scale \cdot x-scale}\right) \cdot y-scale\right) \cdot x-scale}{\left(\pi \cdot angle\right) \cdot \left(b\_m \cdot b\_m - a \cdot a\right)} \cdot 90\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/
          (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_1) 2.0)) x-scale)
          x-scale)))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0))))
            t_3))
          PI))
        100.0)
     (*
      (/
       (atan
        (*
         (/
          (*
           (*
            (- (/ (fma b_m b_m (sqrt (pow b_m 4.0))) (* x-scale x-scale)))
            y-scale)
           x-scale)
          (* (* PI angle) (- (* b_m b_m) (* a a))))
         90.0))
       PI)
      180.0)
     (* 180.0 (/ (atan 0.0) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI))) <= 100.0) {
		tmp = (atan(((((-(fma(b_m, b_m, sqrt(pow(b_m, 4.0))) / (x_45_scale * x_45_scale)) * y_45_scale) * x_45_scale) / ((((double) M_PI) * angle) * ((b_m * b_m) - (a * a)))) * 90.0)) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan(0.0) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = Float64(Float64(atan(Float64(Float64(Float64(Float64(Float64(-Float64(fma(b_m, b_m, sqrt((b_m ^ 4.0))) / Float64(x_45_scale * x_45_scale))) * y_45_scale) * x_45_scale) / Float64(Float64(pi * angle) * Float64(Float64(b_m * b_m) - Float64(a * a)))) * 90.0)) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(N[(N[ArcTan[N[(N[(N[(N[((-N[(N[(b$95$m * b$95$m + N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]) * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(N[(Pi * angle), $MachinePrecision] * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64))) < 100

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      3. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      5. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      6. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

      7. lower-pow.f64 22.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 22.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}{\pi} \]

   9. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(\left(-\frac{\mathsf{fma}\left(b, b, \sqrt{{b}^{4}}\right)}{x-scale \cdot x-scale}\right) \cdot y-scale\right) \cdot x-scale}{\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)} \cdot 90\right)}{\pi} \cdot 180} \]

   if 100 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64)))

   1. Initial program 14.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 7.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 18.7% accurate, 49.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} 0}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan 0.0) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(0.0) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(0.0) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(0.0) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(0.0) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(0.0) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
3. Step-by-step derivation

4. Applied rewrites 12.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

7. Applied rewrites 7.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

8. Taylor expanded in y-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 18.7%

    \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (a b angle x-scale y-scale)
  :name "raw-angle from scale-rotated-ellipse"
  :precision binary64
  (* 180.0 (/ (atan (/ (- (- (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale) (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0)))) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale))) PI)))