raw-angle from scale-rotated-ellipse

Percentage Accurate: 14.4% → 47.2%

Time: 29.4s

Alternatives: 15

Speedup: 29.2×

Specification

Math

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

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

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

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: 47.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-168}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale}}{t\_2 \cdot t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}{x-scale}}{t\_0 \cdot t\_1}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))
        (t_1 (sin (* 0.005555555555555556 (* angle PI))))
        (t_2 (sin (fma (fabs (* PI angle)) 0.005555555555555556 (/ PI 2.0)))))
   (if (<= (fabs b) 4.8e-168)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (if (<= (fabs b) 1.4e+17)
       (*
        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_1))))
         PI))
       (*
        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_1))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double tmp;
	if (fabs(b) <= 4.8e-168) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.4e+17) {
		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_1)))) / ((double) M_PI));
	} else {
		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_1)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi / 2.0)))
	tmp = 0.0
	if (abs(b) <= 4.8e-168)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	elseif (abs(b) <= 1.4e+17)
		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_1)))) / pi));
	else
		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_1)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-168], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e+17], 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$1), $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$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$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.7999999999999999e-168

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.7999999999999999e-168 < b < 1.4e17

   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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. cos-fabs-rev N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      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. fabs-mul N/A

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

      16. metadata-eval N/A

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

      17. 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 44.8%

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

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. cos-fabs-rev N/A

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

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

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       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. fabs-mul N/A

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

       16. metadata-eval N/A

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

       17. 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 44.8%

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

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. cos-fabs-rev N/A

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

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

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-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. fabs-mul N/A

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

       16. metadata-eval N/A

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

       17. 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 45.4%

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

   if 1.4e17 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 44.8%

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

   9. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lower-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.8%

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

   11. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lower-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.9%

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

   13. Applied rewrites 44.9%

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

Alternative 2: 47.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right), 0.5, 0.5\right)\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\\ t_2 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-168}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 2 \cdot 10^{+17}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_3}^{4}} + t\_0\right)}{x-scale}}{t\_3 \cdot t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + t\_0\right)}{x-scale}}{t\_1 \cdot t\_2}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma (cos (* (* (* 0.005555555555555556 angle) PI) 2.0)) 0.5 0.5))
        (t_1 (sin (fma (* 0.005555555555555556 angle) PI (* PI 0.5))))
        (t_2 (sin (* 0.005555555555555556 (* angle PI))))
        (t_3 (sin (fma (fabs (* PI angle)) 0.005555555555555556 (* PI 0.5)))))
   (if (<= (fabs b) 4.8e-168)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (if (<= (fabs b) 2e+17)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -1.0 (/ (* y-scale (+ (sqrt (pow t_3 4.0)) t_0)) x-scale))
            (* t_3 t_2))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -1.0 (/ (* y-scale (+ (sqrt (pow t_1 4.0)) t_0)) x-scale))
            (* t_1 t_2))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(cos((((0.005555555555555556 * angle) * ((double) M_PI)) * 2.0)), 0.5, 0.5);
	double t_1 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) * 0.5)));
	double t_2 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_3 = sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) * 0.5)));
	double tmp;
	if (fabs(b) <= 4.8e-168) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else if (fabs(b) <= 2e+17) {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_3, 4.0)) + t_0)) / x_45_scale)) / (t_3 * t_2)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + t_0)) / x_45_scale)) / (t_1 * t_2)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = fma(cos(Float64(Float64(Float64(0.005555555555555556 * angle) * pi) * 2.0)), 0.5, 0.5)
	t_1 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi * 0.5)))
	t_2 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_3 = sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi * 0.5)))
	tmp = 0.0
	if (abs(b) <= 4.8e-168)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	elseif (abs(b) <= 2e+17)
		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_0)) / x_45_scale)) / Float64(t_3 * t_2)))) / pi));
	else
		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_0)) / x_45_scale)) / Float64(t_1 * t_2)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-168], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 2e+17], 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] + t$95$0), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$2), $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$1, 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.7999999999999999e-168

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.7999999999999999e-168 < b < 2e17

   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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. cos-neg-rev N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-PI.f64 N/A

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

   9. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. cos-fabs-rev N/A

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

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

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. fabs-mul N/A

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

       16. metadata-eval N/A

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

       17. lift-PI.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lower-fma.f64 N/A

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

   11. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. cos-fabs-rev N/A

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

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

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. fabs-mul N/A

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

       16. metadata-eval N/A

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

       17. lift-PI.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. lower-fma.f64 N/A

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

   13. Applied rewrites 45.4%

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

   if 2e17 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. cos-neg-rev N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-PI.f64 N/A

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

   9. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-fma.f64 N/A

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

       12. lower-sin.f64 44.8%

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

       13. lift-/.f64 N/A

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

       14. mult-flip N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 44.8%

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

   11. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-fma.f64 N/A

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

       12. lower-sin.f64 44.9%

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

       13. lift-/.f64 N/A

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

       14. mult-flip N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 44.9%

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

   13. Applied rewrites 44.9%

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

Alternative 3: 47.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 4.9 \cdot 10^{-145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}{x-scale}}{t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0)))))
   (if (<= (fabs b) 4.9e-145)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/ (* y-scale (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))) x-scale))
          (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if (fabs(b) <= 4.9e-145) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		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 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))
	tmp = 0.0
	if (abs(b) <= 4.9e-145)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		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 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.9e-145], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $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$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $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. Split input into 2 regimes

2. if b < 4.8999999999999997e-145

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.8999999999999997e-145 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 44.8%

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

   9. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lower-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.8%

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

   11. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lower-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.9%

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

   13. Applied rewrites 44.9%

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

Alternative 4: 47.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 4.9 \cdot 10^{-145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_0}^{4}} + \mathsf{fma}\left(\cos \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right), 0.5, 0.5\right)\right)}{x-scale}}{t\_0 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* 0.005555555555555556 angle) PI (* PI 0.5)))))
   (if (<= (fabs b) 4.9e-145)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/
            (*
             y-scale
             (+
              (sqrt (pow t_0 4.0))
              (fma
               (cos (* (* (* 0.005555555555555556 angle) PI) 2.0))
               0.5
               0.5)))
            x-scale))
          (* t_0 (sin (* 0.005555555555555556 (* angle PI)))))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) * 0.5)));
	double tmp;
	if (fabs(b) <= 4.9e-145) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_0, 4.0)) + fma(cos((((0.005555555555555556 * angle) * ((double) M_PI)) * 2.0)), 0.5, 0.5))) / x_45_scale)) / (t_0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi * 0.5)))
	tmp = 0.0
	if (abs(b) <= 4.9e-145)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_0 ^ 4.0)) + fma(cos(Float64(Float64(Float64(0.005555555555555556 * angle) * pi) * 2.0)), 0.5, 0.5))) / x_45_scale)) / Float64(t_0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.9e-145], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $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$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $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. Split input into 2 regimes

2. if b < 4.8999999999999997e-145

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.8999999999999997e-145 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. cos-neg-rev N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-PI.f64 N/A

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

   9. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-fma.f64 N/A

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

       12. lower-sin.f64 44.8%

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

       13. lift-/.f64 N/A

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

       14. mult-flip N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 44.8%

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

   11. Applied rewrites 44.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

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

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-fma.f64 N/A

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

       12. lower-sin.f64 44.9%

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

       13. lift-/.f64 N/A

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

       14. mult-flip N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 44.9%

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

   13. Applied rewrites 44.9%

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

Alternative 5: 46.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 5.5 \cdot 10^{-139}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + \left(\sin \left(0.5 \cdot \pi\right) + 0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \cos \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs b) 5.5e-139)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/
            (*
             y-scale
             (+
              1.0
              (+
               (sin (* 0.5 PI))
               (* 0.005555555555555556 (* angle (* PI (cos (* 0.5 PI))))))))
            x-scale))
          (* (cos t_0) (sin t_0)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 5.5e-139) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + (sin((0.5 * ((double) M_PI))) + (0.005555555555555556 * (angle * (((double) M_PI) * cos((0.5 * ((double) M_PI))))))))) / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(b) <= 5.5e-139) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * Math.log(Math.pow(Math.exp(Math.PI), x_45_scale)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + (Math.sin((0.5 * Math.PI)) + (0.005555555555555556 * (angle * (Math.PI * Math.cos((0.5 * Math.PI)))))))) / x_45_scale)) / (Math.cos(t_0) * Math.sin(t_0))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(b) <= 5.5e-139:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * math.log(math.pow(math.exp(math.pi), x_45_scale)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + (math.sin((0.5 * math.pi)) + (0.005555555555555556 * (angle * (math.pi * math.cos((0.5 * math.pi)))))))) / x_45_scale)) / (math.cos(t_0) * math.sin(t_0))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 5.5e-139)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + Float64(sin(Float64(0.5 * pi)) + Float64(0.005555555555555556 * Float64(angle * Float64(pi * cos(Float64(0.5 * pi)))))))) / x_45_scale)) / Float64(cos(t_0) * sin(t_0))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(b) <= 5.5e-139)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log((exp(pi) ^ x_45_scale)))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + (sin((0.5 * pi)) + (0.005555555555555556 * (angle * (pi * cos((0.5 * pi)))))))) / x_45_scale)) / (cos(t_0) * sin(t_0))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.5e-139], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $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[(1.0 + N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + N[(0.005555555555555556 * N[(angle * N[(Pi * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Split input into 2 regimes

2. if b < 5.4999999999999997e-139

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 5.4999999999999997e-139 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. cos-neg-rev N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-PI.f64 N/A

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

   9. Applied rewrites 44.9%

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

   10. Taylor expanded in angle around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-PI.f64 N/A

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

       10. lower-cos.f64 N/A

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

   12. Applied rewrites 40.5%

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

Alternative 6: 46.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;\left|b\right| \leq 4.9 \cdot 10^{-145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {t\_1}^{2}\right)}{x-scale}}{t\_1 \cdot \sin t\_0}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (cos t_0)))
   (if (<= (fabs b) 4.9e-145)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* -1.0 (/ (* y-scale (+ 1.0 (pow t_1 2.0))) x-scale))
          (* t_1 (sin t_0)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double tmp;
	if (fabs(b) <= 4.9e-145) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + pow(t_1, 2.0))) / x_45_scale)) / (t_1 * sin(t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double tmp;
	if (Math.abs(b) <= 4.9e-145) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * Math.log(Math.pow(Math.exp(Math.PI), x_45_scale)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + Math.pow(t_1, 2.0))) / x_45_scale)) / (t_1 * Math.sin(t_0))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	tmp = 0
	if math.fabs(b) <= 4.9e-145:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * math.log(math.pow(math.exp(math.pi), x_45_scale)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + math.pow(t_1, 2.0))) / x_45_scale)) / (t_1 * math.sin(t_0))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	tmp = 0.0
	if (abs(b) <= 4.9e-145)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + (t_1 ^ 2.0))) / x_45_scale)) / Float64(t_1 * sin(t_0))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	tmp = 0.0;
	if (abs(b) <= 4.9e-145)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log((exp(pi) ^ x_45_scale)))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + (t_1 ^ 2.0))) / x_45_scale)) / (t_1 * sin(t_0))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.9e-145], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $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[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8999999999999997e-145

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.8999999999999997e-145 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.5%

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

Alternative 7: 46.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 4.9 \cdot 10^{-145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + \mathsf{fma}\left(\cos \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right), 0.5, 0.5\right)\right)}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs b) 4.9e-145)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -1.0
           (/
            (*
             y-scale
             (+
              1.0
              (fma
               (cos (* (* (* 0.005555555555555556 angle) PI) 2.0))
               0.5
               0.5)))
            x-scale))
          (* (cos t_0) (sin t_0)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 4.9e-145) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + fma(cos((((0.005555555555555556 * angle) * ((double) M_PI)) * 2.0)), 0.5, 0.5))) / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 4.9e-145)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + fma(cos(Float64(Float64(Float64(0.005555555555555556 * angle) * pi) * 2.0)), 0.5, 0.5))) / x_45_scale)) / Float64(cos(t_0) * sin(t_0))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.9e-145], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $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[(1.0 + N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Split input into 2 regimes

2. if b < 4.8999999999999997e-145

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.8999999999999997e-145 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. cos-neg-rev N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-PI.f64 N/A

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

   9. Applied rewrites 44.8%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 44.5%

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

Alternative 8: 43.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 4.9 \cdot 10^{-145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_0 \cdot \sin t\_0}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs b) 4.9e-145)
     (*
      180.0
      (/
       (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
       PI))
     (*
      180.0
      (/
       (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_0) (sin t_0)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 4.9e-145) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(b) <= 4.9e-145) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * Math.log(Math.pow(Math.exp(Math.PI), x_45_scale)))))) / Math.PI);
	} else {
		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);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(b) <= 4.9e-145:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * math.log(math.pow(math.exp(math.pi), x_45_scale)))))) / math.pi)
	else:
		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)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 4.9e-145)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		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));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(b) <= 4.9e-145)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log((exp(pi) ^ x_45_scale)))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_0) * sin(t_0))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.9e-145], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 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. Split input into 2 regimes

2. if b < 4.8999999999999997e-145

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 4.8999999999999997e-145 < 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

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale}}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.8%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   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 44.5%

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

   10. Applied rewrites 44.5%

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

Alternative 9: 42.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.5 \cdot 10^{-140}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 1.5e-140)
   (*
    180.0
    (/
     (atan (* -180.0 (/ y-scale (* angle (log (pow (exp PI) x-scale))))))
     PI))
   (* 180.0 (/ (atan (* (/ -180.0 angle) (/ y-scale (* PI x-scale)))) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 1.5e-140) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log(pow(exp(((double) M_PI)), x_45_scale)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 1.5e-140) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * Math.log(Math.pow(Math.exp(Math.PI), x_45_scale)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((-180.0 / angle) * (y_45_scale / (Math.PI * x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 1.5e-140:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * math.log(math.pow(math.exp(math.pi), x_45_scale)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((-180.0 / angle) * (y_45_scale / (math.pi * x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 1.5e-140)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * log((exp(pi) ^ x_45_scale)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / angle) * Float64(y_45_scale / Float64(pi * x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 1.5e-140)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * log((exp(pi) ^ x_45_scale)))))) / pi);
	else
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (pi * x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.5e-140], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[Log[N[Power[N[Exp[Pi], $MachinePrecision], x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / angle), $MachinePrecision] * N[(y$45$scale / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5000000000000001e-140

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       2. lift-PI.f64 N/A

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

       3. add-log-exp N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\pi} \]

       4. log-pow-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       5. lower-log.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       6. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{x-scale}\right)}\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

       8. lower-exp.f64 35.3%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   12. Applied rewrites 35.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \log \left({\left(e^{\pi}\right)}^{x-scale}\right)}\right)}{\pi} \]

   if 1.5000000000000001e-140 < 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 angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       8. lower-/.f64 39.2%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       10. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

       11. lower-*.f64 39.2%

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

   12. Applied rewrites 39.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot \color{blue}{x-scale}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 40.4% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 10^{-47}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \left(x-scale \cdot \frac{\left(\frac{1}{x-scale \cdot x-scale} + \sqrt{\frac{1}{\left(x-scale \cdot x-scale\right) \cdot \left(x-scale \cdot x-scale\right)}}\right) \cdot y-scale}{\pi \cdot angle}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1e-47)
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (*
        x-scale
        (/
         (*
          (+
           (/ 1.0 (* x-scale x-scale))
           (sqrt (/ 1.0 (* (* x-scale x-scale) (* x-scale x-scale)))))
          y-scale)
         (* PI angle)))))
     PI))
   (* 180.0 (/ (atan (* (/ -180.0 angle) (/ y-scale (* PI x-scale)))) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1e-47) {
		tmp = 180.0 * (atan((-90.0 * (x_45_scale * ((((1.0 / (x_45_scale * x_45_scale)) + sqrt((1.0 / ((x_45_scale * x_45_scale) * (x_45_scale * x_45_scale))))) * y_45_scale) / (((double) M_PI) * angle))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1e-47) {
		tmp = 180.0 * (Math.atan((-90.0 * (x_45_scale * ((((1.0 / (x_45_scale * x_45_scale)) + Math.sqrt((1.0 / ((x_45_scale * x_45_scale) * (x_45_scale * x_45_scale))))) * y_45_scale) / (Math.PI * angle))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((-180.0 / angle) * (y_45_scale / (Math.PI * x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1e-47:
		tmp = 180.0 * (math.atan((-90.0 * (x_45_scale * ((((1.0 / (x_45_scale * x_45_scale)) + math.sqrt((1.0 / ((x_45_scale * x_45_scale) * (x_45_scale * x_45_scale))))) * y_45_scale) / (math.pi * angle))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((-180.0 / angle) * (y_45_scale / (math.pi * x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1e-47)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(x_45_scale * Float64(Float64(Float64(Float64(1.0 / Float64(x_45_scale * x_45_scale)) + sqrt(Float64(1.0 / Float64(Float64(x_45_scale * x_45_scale) * Float64(x_45_scale * x_45_scale))))) * y_45_scale) / Float64(pi * angle))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / angle) * Float64(y_45_scale / Float64(pi * x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1e-47)
		tmp = 180.0 * (atan((-90.0 * (x_45_scale * ((((1.0 / (x_45_scale * x_45_scale)) + sqrt((1.0 / ((x_45_scale * x_45_scale) * (x_45_scale * x_45_scale))))) * y_45_scale) / (pi * angle))))) / pi);
	else
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (pi * x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1e-47], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(x$45$scale * N[(N[(N[(N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(1.0 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / angle), $MachinePrecision] * N[(y$45$scale / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.9999999999999997e-48

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\pi}}\right)}{\pi} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi} \]

      3. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \left(x-scale \cdot \frac{y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)}{\color{blue}{angle \cdot \pi}}\right)\right)}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \left(x-scale \cdot \frac{y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)}{\color{blue}{angle \cdot \pi}}\right)\right)}{\pi} \]

      5. lower-/.f64 40.1%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \left(x-scale \cdot \frac{y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)}{angle \cdot \color{blue}{\pi}}\right)\right)}{\pi} \]

   9. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \left(x-scale \cdot \frac{\left(\frac{1}{x-scale \cdot x-scale} + \sqrt{\frac{1}{\left(x-scale \cdot x-scale\right) \cdot \left(x-scale \cdot x-scale\right)}}\right) \cdot y-scale}{\color{blue}{\pi \cdot angle}}\right)\right)}{\pi} \]

   if 9.9999999999999997e-48 < 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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       8. lower-/.f64 39.2%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       10. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

       11. lower-*.f64 39.2%

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

   12. Applied rewrites 39.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot \color{blue}{x-scale}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 10^{-47}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{2}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1e-47)
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/ (* x-scale (* y-scale (/ 2.0 (pow x-scale 2.0)))) (* angle PI))))
     PI))
   (* 180.0 (/ (atan (* (/ -180.0 angle) (/ y-scale (* PI x-scale)))) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1e-47) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (2.0 / pow(x_45_scale, 2.0)))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1e-47) {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (2.0 / Math.pow(x_45_scale, 2.0)))) / (angle * Math.PI)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((-180.0 / angle) * (y_45_scale / (Math.PI * x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1e-47:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (2.0 / math.pow(x_45_scale, 2.0)))) / (angle * math.pi)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((-180.0 / angle) * (y_45_scale / (math.pi * x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1e-47)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(2.0 / (x_45_scale ^ 2.0)))) / Float64(angle * pi)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / angle) * Float64(y_45_scale / Float64(pi * x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1e-47)
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (2.0 / (x_45_scale ^ 2.0)))) / (angle * pi)))) / pi);
	else
		tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (pi * x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1e-47], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(2.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / angle), $MachinePrecision] * N[(y$45$scale / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.9999999999999997e-48

   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{2}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{2}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi} \]

      2. lower-pow.f64 39.9%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{2}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi} \]

   10. Applied rewrites 39.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \frac{2}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi} \]

   if 9.9999999999999997e-48 < 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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
   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

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
   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. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.6%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

   10. Applied rewrites 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       8. lower-/.f64 39.2%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

       10. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

       11. lower-*.f64 39.2%

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

   12. Applied rewrites 39.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot \color{blue}{x-scale}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.2% accurate, 28.4× speedup

Math

\[180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* (/ -180.0 angle) (/ y-scale (* PI x-scale)))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(((-180.0 / angle) * (y_45_scale / (Math.PI * x_45_scale)))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(((-180.0 / angle) * (y_45_scale / (math.pi * x_45_scale)))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / angle) * Float64(y_45_scale / Float64(pi * x_45_scale)))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(((-180.0 / angle) * (y_45_scale / (pi * x_45_scale)))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / angle), $MachinePrecision] * N[(y$45$scale / N[(Pi * x$45$scale), $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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
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. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 37.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

10. Applied rewrites 37.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

    5. times-frac N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

    8. lower-/.f64 39.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

    9. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{x-scale \cdot \pi}\right)}{\pi} \]

    10. *-commutative N/A

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

    11. lower-*.f64 39.2%

        \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{\pi} \]

12. Applied rewrites 39.2%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{angle} \cdot \frac{y-scale}{\pi \cdot \color{blue}{x-scale}}\right)}{\pi} \]
13. Add Preprocessing

Alternative 13: 37.6% accurate, 29.2× speedup

Math

\[\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -180\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/ (* 180.0 (atan (* (/ y-scale (* (* PI x-scale) angle)) -180.0))) PI))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (180.0 * atan(((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -180.0))) / ((double) M_PI);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (180.0 * Math.atan(((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -180.0))) / Math.PI;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (180.0 * math.atan(((y_45_scale / ((math.pi * x_45_scale) * angle)) * -180.0))) / math.pi

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -180.0))) / pi)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (180.0 * atan(((y_45_scale / ((pi * x_45_scale) * angle)) * -180.0))) / pi;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -180.0), $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 angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
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. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 37.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

10. Applied rewrites 37.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
11. Step-by-step derivation

12. Applied rewrites 37.6%

    \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -180\right)}{\pi}} \]
13. Add Preprocessing

Alternative 14: 37.6% accurate, 29.2× speedup

Math

\[180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(angle \cdot x-scale\right) \cdot \pi}\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ y-scale (* (* angle x-scale) PI)))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-180.0 * (y_45_scale / ((angle * x_45_scale) * ((double) M_PI))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-180.0 * (y_45_scale / ((angle * x_45_scale) * Math.PI)))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-180.0 * (y_45_scale / ((angle * x_45_scale) * math.pi)))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(Float64(angle * x_45_scale) * pi)))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-180.0 * (y_45_scale / ((angle * x_45_scale) * pi)))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(N[(angle * x$45$scale), $MachinePrecision] * 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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
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. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 37.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

10. Applied rewrites 37.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

    2. lift-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

    3. associate-*r* N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(angle \cdot x-scale\right) \cdot \pi}\right)}{\pi} \]

    4. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(angle \cdot x-scale\right) \cdot \pi}\right)}{\pi} \]

    5. lower-*.f64 37.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(angle \cdot x-scale\right) \cdot \pi}\right)}{\pi} \]

12. Applied rewrites 37.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\left(angle \cdot x-scale\right) \cdot \pi}\right)}{\pi} \]
13. Add Preprocessing

Alternative 15: 37.6% accurate, 29.2× speedup

Math

\[180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $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 angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]
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

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
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. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 37.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \]

10. Applied rewrites 37.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(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)))