raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.9% → 46.9%

Time: 25.9s

Alternatives: 16

Speedup: 49.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 46.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := {t\_1}^{2}\\ t_3 := {t\_1}^{4}\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_5 := \cos t\_4\\ t_6 := \sin t\_4\\ t_7 := \sin \left(0.5 \cdot \pi\right)\\ t_8 := t\_6 \cdot t\_1\\ t_9 := {\left(a \cdot t\_6\right)}^{2} + {\left(b\_m \cdot t\_5\right)}^{2}\\ \mathbf{if}\;b\_m \leq 1.3 \cdot 10^{-255}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{t\_3}{{x-scale}^{4}}} + \frac{t\_2}{x-scale \cdot x-scale}\right)\right)}{\frac{\cos \left(t\_4 - t\_0\right) - \cos \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, t\_0\right)\right)}{2}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.2 \cdot 10^{-208}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{t\_6}^{4}}{{x-scale}^{4}}} + \frac{{t\_6}^{2}}{x-scale \cdot x-scale}\right)\right)}{t\_8}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{t\_7}^{4}}{{x-scale}^{4}}} + \frac{{t\_7}^{2}}{x-scale \cdot x-scale}\right)\right)}{t\_6 \cdot t\_7}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 6.6 \cdot 10^{+79}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\frac{y-scale \cdot \left(\sqrt{{t\_9}^{2}} + t\_9\right)}{x-scale}}{t\_5 \cdot \left(t\_6 \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{t\_3} + t\_2\right)}{x-scale \cdot t\_8}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma 0.005555555555555556 (* angle PI) (* 0.5 PI)))
        (t_1 (sin t_0))
        (t_2 (pow t_1 2.0))
        (t_3 (pow t_1 4.0))
        (t_4 (* 0.005555555555555556 (* angle PI)))
        (t_5 (cos t_4))
        (t_6 (sin t_4))
        (t_7 (sin (* 0.5 PI)))
        (t_8 (* t_6 t_1))
        (t_9 (+ (pow (* a t_6) 2.0) (pow (* b_m t_5) 2.0))))
   (if (<= b_m 1.3e-255)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ t_3 (pow x-scale 4.0))) (/ t_2 (* x-scale x-scale)))))
          (/
           (-
            (cos (- t_4 t_0))
            (cos (fma 0.005555555555555556 (* angle PI) t_0)))
           2.0))))
       PI))
     (if (<= b_m 2.2e-208)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             x-scale
             (*
              y-scale
              (+
               (sqrt (/ (pow t_6 4.0) (pow x-scale 4.0)))
               (/ (pow t_6 2.0) (* x-scale x-scale)))))
            t_8)))
         PI))
       (if (<= b_m 2.6e-72)
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/
              (*
               x-scale
               (*
                y-scale
                (+
                 (sqrt (/ (pow t_7 4.0) (pow x-scale 4.0)))
                 (/ (pow t_7 2.0) (* x-scale x-scale)))))
              (* t_6 t_7))))
           PI))
         (if (<= b_m 6.6e+79)
           (*
            180.0
            (/
             (atan
              (*
               -0.5
               (/
                (/ (* y-scale (+ (sqrt (pow t_9 2.0)) t_9)) x-scale)
                (* t_5 (* t_6 (- (* b_m b_m) (* a a)))))))
             PI))
           (*
            180.0
            (/
             (atan (* -0.5 (/ (* y-scale (+ (sqrt t_3) t_2)) (* x-scale t_8))))
             PI))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)));
	double t_1 = sin(t_0);
	double t_2 = pow(t_1, 2.0);
	double t_3 = pow(t_1, 4.0);
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_5 = cos(t_4);
	double t_6 = sin(t_4);
	double t_7 = sin((0.5 * ((double) M_PI)));
	double t_8 = t_6 * t_1;
	double t_9 = pow((a * t_6), 2.0) + pow((b_m * t_5), 2.0);
	double tmp;
	if (b_m <= 1.3e-255) {
		tmp = 180.0 * (atan((-0.5 * ((x_45_scale * (y_45_scale * (sqrt((t_3 / pow(x_45_scale, 4.0))) + (t_2 / (x_45_scale * x_45_scale))))) / ((cos((t_4 - t_0)) - cos(fma(0.005555555555555556, (angle * ((double) M_PI)), t_0))) / 2.0)))) / ((double) M_PI));
	} else if (b_m <= 2.2e-208) {
		tmp = 180.0 * (atan((0.5 * ((x_45_scale * (y_45_scale * (sqrt((pow(t_6, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_6, 2.0) / (x_45_scale * x_45_scale))))) / t_8))) / ((double) M_PI));
	} else if (b_m <= 2.6e-72) {
		tmp = 180.0 * (atan((-0.5 * ((x_45_scale * (y_45_scale * (sqrt((pow(t_7, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_7, 2.0) / (x_45_scale * x_45_scale))))) / (t_6 * t_7)))) / ((double) M_PI));
	} else if (b_m <= 6.6e+79) {
		tmp = 180.0 * (atan((-0.5 * (((y_45_scale * (sqrt(pow(t_9, 2.0)) + t_9)) / x_45_scale) / (t_5 * (t_6 * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(t_3) + t_2)) / (x_45_scale * t_8)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))
	t_1 = sin(t_0)
	t_2 = t_1 ^ 2.0
	t_3 = t_1 ^ 4.0
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_5 = cos(t_4)
	t_6 = sin(t_4)
	t_7 = sin(Float64(0.5 * pi))
	t_8 = Float64(t_6 * t_1)
	t_9 = Float64((Float64(a * t_6) ^ 2.0) + (Float64(b_m * t_5) ^ 2.0))
	tmp = 0.0
	if (b_m <= 1.3e-255)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(t_3 / (x_45_scale ^ 4.0))) + Float64(t_2 / Float64(x_45_scale * x_45_scale))))) / Float64(Float64(cos(Float64(t_4 - t_0)) - cos(fma(0.005555555555555556, Float64(angle * pi), t_0))) / 2.0)))) / pi));
	elseif (b_m <= 2.2e-208)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64((t_6 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_6 ^ 2.0) / Float64(x_45_scale * x_45_scale))))) / t_8))) / pi));
	elseif (b_m <= 2.6e-72)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64((t_7 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_7 ^ 2.0) / Float64(x_45_scale * x_45_scale))))) / Float64(t_6 * t_7)))) / pi));
	elseif (b_m <= 6.6e+79)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(y_45_scale * Float64(sqrt((t_9 ^ 2.0)) + t_9)) / x_45_scale) / Float64(t_5 * Float64(t_6 * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt(t_3) + t_2)) / Float64(x_45_scale * t_8)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 4.0], $MachinePrecision]}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 * t$95$1), $MachinePrecision]}, Block[{t$95$9 = N[(N[Power[N[(a * t$95$6), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.3e-255], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(t$95$3 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(t$95$4 - t$95$0), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.2e-208], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[Power[t$95$6, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$6, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.6e-72], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[Power[t$95$7, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$7, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.6e+79], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$9, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(t$95$5 * N[(t$95$6 * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[t$95$3], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 1.3000000000000001e-255

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. sin-mult N/A

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

   15. Applied rewrites 31.5%

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

   if 1.3000000000000001e-255 < b < 2.2e-208

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 32.5%

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

   if 2.2e-208 < b < 2.59999999999999996e-72

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 42.3

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

   16. Applied rewrites 42.3%

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

   17. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 42.3

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

   19. Applied rewrites 42.3%

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

   20. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 41.9

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

   22. Applied rewrites 41.9%

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

   if 2.59999999999999996e-72 < b < 6.6000000000000003e79

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 26.7%

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

   if 6.6000000000000003e79 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 46.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := {t\_1}^{4}\\ t_3 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_4 := \sin t\_3\\ t_5 := t\_4 \cdot t\_1\\ t_6 := {t\_1}^{2}\\ t_7 := \cos t\_3\\ t_8 := \sin \left(0.5 \cdot \pi\right)\\ t_9 := {\left(a \cdot t\_4\right)}^{2} + {\left(b\_m \cdot t\_7\right)}^{2}\\ \mathbf{if}\;b\_m \leq 1.3 \cdot 10^{-255}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{t\_2}{{x-scale}^{4}}} + \frac{t\_6}{x-scale \cdot x-scale}\right)\right)}{\frac{\cos \left(t\_3 - t\_0\right) - \cos \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, t\_0\right)\right)}{2}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.2 \cdot 10^{-208}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{t\_4}^{4}}{{x-scale}^{4}}} + \frac{{t\_4}^{2}}{x-scale \cdot x-scale}\right)\right)}{t\_5}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{t\_8}^{4}}{{x-scale}^{4}}} + \frac{{t\_8}^{2}}{x-scale \cdot x-scale}\right)\right)}{t\_4 \cdot t\_8}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 6 \cdot 10^{+79}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_9}^{2}} + t\_9\right)}{x-scale \cdot \left(t\_7 \cdot \left(t\_4 \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{t\_2} + t\_6\right)}{x-scale \cdot t\_5}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma 0.005555555555555556 (* angle PI) (* 0.5 PI)))
        (t_1 (sin t_0))
        (t_2 (pow t_1 4.0))
        (t_3 (* 0.005555555555555556 (* angle PI)))
        (t_4 (sin t_3))
        (t_5 (* t_4 t_1))
        (t_6 (pow t_1 2.0))
        (t_7 (cos t_3))
        (t_8 (sin (* 0.5 PI)))
        (t_9 (+ (pow (* a t_4) 2.0) (pow (* b_m t_7) 2.0))))
   (if (<= b_m 1.3e-255)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ t_2 (pow x-scale 4.0))) (/ t_6 (* x-scale x-scale)))))
          (/
           (-
            (cos (- t_3 t_0))
            (cos (fma 0.005555555555555556 (* angle PI) t_0)))
           2.0))))
       PI))
     (if (<= b_m 2.2e-208)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             x-scale
             (*
              y-scale
              (+
               (sqrt (/ (pow t_4 4.0) (pow x-scale 4.0)))
               (/ (pow t_4 2.0) (* x-scale x-scale)))))
            t_5)))
         PI))
       (if (<= b_m 2.6e-72)
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/
              (*
               x-scale
               (*
                y-scale
                (+
                 (sqrt (/ (pow t_8 4.0) (pow x-scale 4.0)))
                 (/ (pow t_8 2.0) (* x-scale x-scale)))))
              (* t_4 t_8))))
           PI))
         (if (<= b_m 6e+79)
           (*
            180.0
            (/
             (atan
              (*
               -0.5
               (/
                (* y-scale (+ (sqrt (pow t_9 2.0)) t_9))
                (* x-scale (* t_7 (* t_4 (- (* b_m b_m) (* a a))))))))
             PI))
           (*
            180.0
            (/
             (atan (* -0.5 (/ (* y-scale (+ (sqrt t_2) t_6)) (* x-scale t_5))))
             PI))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)));
	double t_1 = sin(t_0);
	double t_2 = pow(t_1, 4.0);
	double t_3 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_4 = sin(t_3);
	double t_5 = t_4 * t_1;
	double t_6 = pow(t_1, 2.0);
	double t_7 = cos(t_3);
	double t_8 = sin((0.5 * ((double) M_PI)));
	double t_9 = pow((a * t_4), 2.0) + pow((b_m * t_7), 2.0);
	double tmp;
	if (b_m <= 1.3e-255) {
		tmp = 180.0 * (atan((-0.5 * ((x_45_scale * (y_45_scale * (sqrt((t_2 / pow(x_45_scale, 4.0))) + (t_6 / (x_45_scale * x_45_scale))))) / ((cos((t_3 - t_0)) - cos(fma(0.005555555555555556, (angle * ((double) M_PI)), t_0))) / 2.0)))) / ((double) M_PI));
	} else if (b_m <= 2.2e-208) {
		tmp = 180.0 * (atan((0.5 * ((x_45_scale * (y_45_scale * (sqrt((pow(t_4, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_4, 2.0) / (x_45_scale * x_45_scale))))) / t_5))) / ((double) M_PI));
	} else if (b_m <= 2.6e-72) {
		tmp = 180.0 * (atan((-0.5 * ((x_45_scale * (y_45_scale * (sqrt((pow(t_8, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_8, 2.0) / (x_45_scale * x_45_scale))))) / (t_4 * t_8)))) / ((double) M_PI));
	} else if (b_m <= 6e+79) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_9, 2.0)) + t_9)) / (x_45_scale * (t_7 * (t_4 * ((b_m * b_m) - (a * a)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(t_2) + t_6)) / (x_45_scale * t_5)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))
	t_1 = sin(t_0)
	t_2 = t_1 ^ 4.0
	t_3 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_4 = sin(t_3)
	t_5 = Float64(t_4 * t_1)
	t_6 = t_1 ^ 2.0
	t_7 = cos(t_3)
	t_8 = sin(Float64(0.5 * pi))
	t_9 = Float64((Float64(a * t_4) ^ 2.0) + (Float64(b_m * t_7) ^ 2.0))
	tmp = 0.0
	if (b_m <= 1.3e-255)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(t_2 / (x_45_scale ^ 4.0))) + Float64(t_6 / Float64(x_45_scale * x_45_scale))))) / Float64(Float64(cos(Float64(t_3 - t_0)) - cos(fma(0.005555555555555556, Float64(angle * pi), t_0))) / 2.0)))) / pi));
	elseif (b_m <= 2.2e-208)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64((t_4 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_4 ^ 2.0) / Float64(x_45_scale * x_45_scale))))) / t_5))) / pi));
	elseif (b_m <= 2.6e-72)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64((t_8 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_8 ^ 2.0) / Float64(x_45_scale * x_45_scale))))) / Float64(t_4 * t_8)))) / pi));
	elseif (b_m <= 6e+79)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_9 ^ 2.0)) + t_9)) / Float64(x_45_scale * Float64(t_7 * Float64(t_4 * Float64(Float64(b_m * b_m) - Float64(a * a)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt(t_2) + t_6)) / Float64(x_45_scale * t_5)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 4.0], $MachinePrecision]}, Block[{t$95$3 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$7 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$8 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$9 = N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$7), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.3e-255], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(t$95$2 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$6 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(t$95$3 - t$95$0), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.2e-208], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.6e-72], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[Power[t$95$8, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$8, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6e+79], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$9, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$7 * N[(t$95$4 * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[t$95$2], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 1.3000000000000001e-255

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. sin-mult N/A

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

   15. Applied rewrites 31.5%

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

   if 1.3000000000000001e-255 < b < 2.2e-208

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 32.5%

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

   if 2.2e-208 < b < 2.59999999999999996e-72

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 42.3

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

   16. Applied rewrites 42.3%

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

   17. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 42.3

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

   19. Applied rewrites 42.3%

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

   20. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 41.9

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

   22. Applied rewrites 41.9%

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

   if 2.59999999999999996e-72 < b < 5.99999999999999948e79

   1. Initial program 13.9%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.7%

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

   if 5.99999999999999948e79 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 46.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_4 := y-scale \cdot \left(\sqrt{{t\_3}^{4}} + {t\_3}^{2}\right)\\ t_5 := t\_1 \cdot t\_3\\ \mathbf{if}\;b\_m \leq 8.2 \cdot 10^{-47}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\frac{t\_4}{x-scale}}{t\_5}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1050:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale \cdot \left(\sqrt{{b\_m}^{4}} + b\_m \cdot b\_m\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(1 + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_2 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{t\_4}{x-scale \cdot t\_5}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_4 (* y-scale (+ (sqrt (pow t_3 4.0)) (pow t_3 2.0))))
        (t_5 (* t_1 t_3)))
   (if (<= b_m 8.2e-47)
     (* 180.0 (/ (atan (* -0.5 (/ (/ t_4 x-scale) t_5))) PI))
     (if (<= b_m 1050.0)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (* y-scale (+ (sqrt (pow b_m 4.0)) (* b_m b_m)))
            (* angle (* x-scale (* PI (- (* b_m b_m) (* a a))))))))
         PI))
       (if (<= b_m 5.8e+102)
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/ (* y-scale (+ 1.0 (pow t_2 2.0))) (* x-scale (* t_2 t_1)))))
           PI))
         (* 180.0 (/ (atan (* -0.5 (/ t_4 (* x-scale t_5)))) PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_4 = y_45_scale * (sqrt(pow(t_3, 4.0)) + pow(t_3, 2.0));
	double t_5 = t_1 * t_3;
	double tmp;
	if (b_m <= 8.2e-47) {
		tmp = 180.0 * (atan((-0.5 * ((t_4 / x_45_scale) / t_5))) / ((double) M_PI));
	} else if (b_m <= 1050.0) {
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt(pow(b_m, 4.0)) + (b_m * b_m))) / (angle * (x_45_scale * (((double) M_PI) * ((b_m * b_m) - (a * a)))))))) / ((double) M_PI));
	} else if (b_m <= 5.8e+102) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + pow(t_2, 2.0))) / (x_45_scale * (t_2 * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (t_4 / (x_45_scale * t_5)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_4 = Float64(y_45_scale * Float64(sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0)))
	t_5 = Float64(t_1 * t_3)
	tmp = 0.0
	if (b_m <= 8.2e-47)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(t_4 / x_45_scale) / t_5))) / pi));
	elseif (b_m <= 1050.0)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(y_45_scale * Float64(sqrt((b_m ^ 4.0)) + Float64(b_m * b_m))) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a)))))))) / pi));
	elseif (b_m <= 5.8e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(1.0 + (t_2 ^ 2.0))) / Float64(x_45_scale * Float64(t_2 * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(t_4 / Float64(x_45_scale * t_5)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * t$95$3), $MachinePrecision]}, If[LessEqual[b$95$m, 8.2e-47], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(t$95$4 / x$45$scale), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1050.0], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.8e+102], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(t$95$4 / N[(x$45$scale * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 8.20000000000000003e-47

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 44.9%

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

   if 8.20000000000000003e-47 < b < 1050

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   if 1050 < b < 5.8000000000000005e102

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 43.3%

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

   if 5.8000000000000005e102 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

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

Alternative 4: 45.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_1 \cdot t\_2\right)}\right)}{\pi}\\ t_4 := \cos t\_0\\ \mathbf{if}\;b\_m \leq 8.2 \cdot 10^{-47}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;b\_m \leq 1050:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale \cdot \left(\sqrt{{b\_m}^{4}} + b\_m \cdot b\_m\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(1 + {t\_4}^{2}\right)}{x-scale \cdot \left(t\_4 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_3
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/
              (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
              (* x-scale (* t_1 t_2)))))
           PI)))
        (t_4 (cos t_0)))
   (if (<= b_m 8.2e-47)
     t_3
     (if (<= b_m 1050.0)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (* y-scale (+ (sqrt (pow b_m 4.0)) (* b_m b_m)))
            (* angle (* x-scale (* PI (- (* b_m b_m) (* a a))))))))
         PI))
       (if (<= b_m 5.8e+102)
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/ (* y-scale (+ 1.0 (pow t_4 2.0))) (* x-scale (* t_4 t_1)))))
           PI))
         t_3)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_3 = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / (x_45_scale * (t_1 * t_2))))) / ((double) M_PI));
	double t_4 = cos(t_0);
	double tmp;
	if (b_m <= 8.2e-47) {
		tmp = t_3;
	} else if (b_m <= 1050.0) {
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt(pow(b_m, 4.0)) + (b_m * b_m))) / (angle * (x_45_scale * (((double) M_PI) * ((b_m * b_m) - (a * a)))))))) / ((double) M_PI));
	} else if (b_m <= 5.8e+102) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + pow(t_4, 2.0))) / (x_45_scale * (t_4 * t_1))))) / ((double) M_PI));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_3 = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / Float64(x_45_scale * Float64(t_1 * t_2))))) / pi))
	t_4 = cos(t_0)
	tmp = 0.0
	if (b_m <= 8.2e-47)
		tmp = t_3;
	elseif (b_m <= 1050.0)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(y_45_scale * Float64(sqrt((b_m ^ 4.0)) + Float64(b_m * b_m))) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a)))))))) / pi));
	elseif (b_m <= 5.8e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(1.0 + (t_4 ^ 2.0))) / Float64(x_45_scale * Float64(t_4 * t_1))))) / pi));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(180.0 * N[(N[ArcTan[N[(-0.5 * 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] / N[(x$45$scale * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 8.2e-47], t$95$3, If[LessEqual[b$95$m, 1050.0], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[b$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.8e+102], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.20000000000000003e-47 or 5.8000000000000005e102 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

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

   if 8.20000000000000003e-47 < b < 1050

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 22.5%

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

   if 1050 < b < 5.8000000000000005e102

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 43.3%

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

Alternative 5: 44.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{t\_2}^{4}}{{x-scale}^{4}}} + \frac{{t\_2}^{2}}{x-scale \cdot x-scale}\right)\right)}{t\_3}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(1 + {t\_1}^{2}\right)}{x-scale \cdot t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_4}^{4}} + {t\_4}^{2}\right)}{x-scale \cdot \left(t\_2 \cdot t\_4\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (* t_1 t_2))
        (t_4 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI)))))
   (if (<= b_m 4.2e-109)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           x-scale
           (*
            y-scale
            (+
             (sqrt (/ (pow t_2 4.0) (pow x-scale 4.0)))
             (/ (pow t_2 2.0) (* x-scale x-scale)))))
          t_3)))
       PI))
     (if (<= b_m 5.8e+102)
       (*
        180.0
        (/
         (atan (* -0.5 (/ (* y-scale (+ 1.0 (pow t_1 2.0))) (* x-scale t_3))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow t_4 4.0)) (pow t_4 2.0)))
            (* x-scale (* t_2 t_4)))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = t_1 * t_2;
	double t_4 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double tmp;
	if (b_m <= 4.2e-109) {
		tmp = 180.0 * (atan((0.5 * ((x_45_scale * (y_45_scale * (sqrt((pow(t_2, 4.0) / pow(x_45_scale, 4.0))) + (pow(t_2, 2.0) / (x_45_scale * x_45_scale))))) / t_3))) / ((double) M_PI));
	} else if (b_m <= 5.8e+102) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + pow(t_1, 2.0))) / (x_45_scale * t_3)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_4, 4.0)) + pow(t_4, 2.0))) / (x_45_scale * (t_2 * t_4))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(t_1 * t_2)
	t_4 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	tmp = 0.0
	if (b_m <= 4.2e-109)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64((t_2 ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((t_2 ^ 2.0) / Float64(x_45_scale * x_45_scale))))) / t_3))) / pi));
	elseif (b_m <= 5.8e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(1.0 + (t_1 ^ 2.0))) / Float64(x_45_scale * t_3)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_4 ^ 4.0)) + (t_4 ^ 2.0))) / Float64(x_45_scale * Float64(t_2 * t_4))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e-109], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[Power[t$95$2, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.8e+102], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999992e-109

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.3%

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

   if 4.19999999999999992e-109 < b < 5.8000000000000005e102

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 43.3%

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

   if 5.8000000000000005e102 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

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

Alternative 6: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_2 := \sin t\_0\\ t_3 := \cos t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_3}{x-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_6 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_3\right)}^{2}}{x-scale}}{x-scale}\\ t_7 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_8 := \frac{{\left(a \cdot t\_7\right)}^{2} + {\left(b\_m \cdot 1\right)}^{2}}{x-scale \cdot x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_5 - t\_6\right) - \sqrt{{\left(t\_6 - t\_5\right)}^{2} + {t\_4}^{2}}}{t\_4}\right)}{\pi} \leq 100:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{t\_8}^{2}} + t\_8\right)\right)}{1 \cdot \left(t\_7 \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale \cdot \left(t\_7 \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_2 (sin t_0))
        (t_3 (cos t_0))
        (t_4
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_2) t_3) x-scale)
          y-scale))
        (t_5
         (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b_m t_2) 2.0)) y-scale) y-scale))
        (t_6
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_3) 2.0)) x-scale) x-scale))
        (t_7 (sin (* 0.005555555555555556 (* angle PI))))
        (t_8
         (/
          (+ (pow (* a t_7) 2.0) (pow (* b_m 1.0) 2.0))
          (* x-scale x-scale))))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_5 t_6) (sqrt (+ (pow (- t_6 t_5) 2.0) (pow t_4 2.0))))
            t_4))
          PI))
        100.0)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (* x-scale (* y-scale (+ (sqrt (pow t_8 2.0)) t_8)))
          (* 1.0 (* t_7 (- (* b_m b_m) (* a a)))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
          (* x-scale (* t_7 t_1)))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_2 = sin(t_0);
	double t_3 = cos(t_0);
	double t_4 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_3), 2.0) + pow((b_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_6 = ((pow((a * t_2), 2.0) + pow((b_m * t_3), 2.0)) / x_45_scale) / x_45_scale;
	double t_7 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_8 = (pow((a * t_7), 2.0) + pow((b_m * 1.0), 2.0)) / (x_45_scale * x_45_scale);
	double tmp;
	if ((180.0 * (atan((((t_5 - t_6) - sqrt((pow((t_6 - t_5), 2.0) + pow(t_4, 2.0)))) / t_4)) / ((double) M_PI))) <= 100.0) {
		tmp = 180.0 * (atan((-0.5 * ((x_45_scale * (y_45_scale * (sqrt(pow(t_8, 2.0)) + t_8))) / (1.0 * (t_7 * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / (x_45_scale * (t_7 * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_2 = sin(t_0)
	t_3 = cos(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_6 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b_m * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_7 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_8 = Float64(Float64((Float64(a * t_7) ^ 2.0) + (Float64(b_m * 1.0) ^ 2.0)) / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_5 - t_6) - sqrt(Float64((Float64(t_6 - t_5) ^ 2.0) + (t_4 ^ 2.0)))) / t_4)) / pi)) <= 100.0)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt((t_8 ^ 2.0)) + t_8))) / Float64(1.0 * Float64(t_7 * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / Float64(x_45_scale * Float64(t_7 * t_1))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$7 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[Power[N[(a * t$95$7), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$5 - t$95$6), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$6 - t$95$5), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$8, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(t$95$7 * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$7 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64))) < 100

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 26.5%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 26.0%

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

   if 100 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64)))

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   6. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 26.4

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

   10. Applied rewrites 26.4%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 42.4%

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

   14. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   16. Applied rewrites 43.4%

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

Alternative 7: 43.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := {t\_1}^{2}\\ t_3 := x-scale \cdot \left(t\_1 \cdot \sin t\_0\right)\\ \mathbf{if}\;angle \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(1 + t\_2\right)}{t\_3}\right)}{\pi}\\ \mathbf{elif}\;angle \leq 10^{+118}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + t\_2\right)}{t\_3}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (pow t_1 2.0))
        (t_3 (* x-scale (* t_1 (sin t_0)))))
   (if (<= angle 4.5e+29)
     (* 180.0 (/ (atan (* -0.5 (/ (* y-scale (+ 1.0 t_2)) t_3))) PI))
     (if (<= angle 1e+118)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (*
             y-scale
             (+ 2.0 (* -6.17283950617284e-5 (* (* angle angle) (* PI PI)))))
            t_3)))
         PI))
       (*
        180.0
        (/
         (atan (* -0.5 (/ (* y-scale (+ (sqrt (pow t_1 4.0)) t_2)) t_3)))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = pow(t_1, 2.0);
	double t_3 = x_45_scale * (t_1 * sin(t_0));
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + t_2)) / t_3))) / ((double) M_PI));
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI)))))) / t_3))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + t_2)) / t_3))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.pow(t_1, 2.0);
	double t_3 = x_45_scale * (t_1 * Math.sin(t_0));
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (1.0 + t_2)) / t_3))) / Math.PI);
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (Math.PI * Math.PI))))) / t_3))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (Math.sqrt(Math.pow(t_1, 4.0)) + t_2)) / t_3))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	t_2 = math.pow(t_1, 2.0)
	t_3 = x_45_scale * (t_1 * math.sin(t_0))
	tmp = 0
	if angle <= 4.5e+29:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (1.0 + t_2)) / t_3))) / math.pi)
	elif angle <= 1e+118:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (math.pi * math.pi))))) / t_3))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (math.sqrt(math.pow(t_1, 4.0)) + t_2)) / t_3))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = t_1 ^ 2.0
	t_3 = Float64(x_45_scale * Float64(t_1 * sin(t_0)))
	tmp = 0.0
	if (angle <= 4.5e+29)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(1.0 + t_2)) / t_3))) / pi));
	elseif (angle <= 1e+118)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(2.0 + Float64(-6.17283950617284e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))) / t_3))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + t_2)) / t_3))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	t_2 = t_1 ^ 2.0;
	t_3 = x_45_scale * (t_1 * sin(t_0));
	tmp = 0.0;
	if (angle <= 4.5e+29)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + t_2)) / t_3))) / pi);
	elseif (angle <= 1e+118)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (pi * pi))))) / t_3))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt((t_1 ^ 4.0)) + t_2)) / t_3))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(x$45$scale * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.5e+29], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1e+118], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(2.0 + N[(-6.17283950617284e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 4.5000000000000002e29

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 43.3%

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

   if 4.5000000000000002e29 < angle < 9.99999999999999967e117

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lift-PI.f64 37.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 37.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]

   if 9.99999999999999967e117 < angle

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

Alternative 8: 42.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := x-scale \cdot \left(t\_1 \cdot \sin t\_0\right)\\ t_3 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(1 + {t\_1}^{2}\right)}{t\_2}\right)}{\pi}\\ \mathbf{if}\;angle \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;angle \leq 10^{+118}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (* x-scale (* t_1 (sin t_0))))
        (t_3
         (*
          180.0
          (/ (atan (* -0.5 (/ (* y-scale (+ 1.0 (pow t_1 2.0))) t_2))) PI))))
   (if (<= angle 4.5e+29)
     t_3
     (if (<= angle 1e+118)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (*
             y-scale
             (+ 2.0 (* -6.17283950617284e-5 (* (* angle angle) (* PI PI)))))
            t_2)))
         PI))
       t_3))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = x_45_scale * (t_1 * sin(t_0));
	double t_3 = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + pow(t_1, 2.0))) / t_2))) / ((double) M_PI));
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = t_3;
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI)))))) / t_2))) / ((double) M_PI));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = x_45_scale * (t_1 * Math.sin(t_0));
	double t_3 = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (1.0 + Math.pow(t_1, 2.0))) / t_2))) / Math.PI);
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = t_3;
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (Math.PI * Math.PI))))) / t_2))) / Math.PI);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	t_2 = x_45_scale * (t_1 * math.sin(t_0))
	t_3 = 180.0 * (math.atan((-0.5 * ((y_45_scale * (1.0 + math.pow(t_1, 2.0))) / t_2))) / math.pi)
	tmp = 0
	if angle <= 4.5e+29:
		tmp = t_3
	elif angle <= 1e+118:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (math.pi * math.pi))))) / t_2))) / math.pi)
	else:
		tmp = t_3
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = Float64(x_45_scale * Float64(t_1 * sin(t_0)))
	t_3 = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(1.0 + (t_1 ^ 2.0))) / t_2))) / pi))
	tmp = 0.0
	if (angle <= 4.5e+29)
		tmp = t_3;
	elseif (angle <= 1e+118)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(2.0 + Float64(-6.17283950617284e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))) / t_2))) / pi));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	t_2 = x_45_scale * (t_1 * sin(t_0));
	t_3 = 180.0 * (atan((-0.5 * ((y_45_scale * (1.0 + (t_1 ^ 2.0))) / t_2))) / pi);
	tmp = 0.0;
	if (angle <= 4.5e+29)
		tmp = t_3;
	elseif (angle <= 1e+118)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (pi * pi))))) / t_2))) / pi);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(x$45$scale * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.5e+29], t$95$3, If[LessEqual[angle, 1e+118], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(2.0 + N[(-6.17283950617284e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.5000000000000002e29 or 9.99999999999999967e117 < angle

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 43.3%

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

   if 4.5000000000000002e29 < angle < 9.99999999999999967e117

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lift-PI.f64 37.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 37.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{t\_1}\right)}{\pi}\\ \mathbf{if}\;angle \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;angle \leq 10^{+118}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* x-scale (* (cos t_0) (sin t_0))))
        (t_2 (* 180.0 (/ (atan (* -0.5 (/ (* y-scale 2.0) t_1))) PI))))
   (if (<= angle 4.5e+29)
     t_2
     (if (<= angle 1e+118)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (*
             y-scale
             (+ 2.0 (* -6.17283950617284e-5 (* (* angle angle) (* PI PI)))))
            t_1)))
         PI))
       t_2))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	double t_2 = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / ((double) M_PI));
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = t_2;
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI)))))) / t_1))) / ((double) M_PI));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = x_45_scale * (Math.cos(t_0) * Math.sin(t_0));
	double t_2 = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / Math.PI);
	double tmp;
	if (angle <= 4.5e+29) {
		tmp = t_2;
	} else if (angle <= 1e+118) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (Math.PI * Math.PI))))) / t_1))) / Math.PI);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = x_45_scale * (math.cos(t_0) * math.sin(t_0))
	t_2 = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / math.pi)
	tmp = 0
	if angle <= 4.5e+29:
		tmp = t_2
	elif angle <= 1e+118:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (math.pi * math.pi))))) / t_1))) / math.pi)
	else:
		tmp = t_2
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))
	t_2 = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / t_1))) / pi))
	tmp = 0.0
	if (angle <= 4.5e+29)
		tmp = t_2;
	elseif (angle <= 1e+118)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(2.0 + Float64(-6.17283950617284e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))) / t_1))) / pi));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	t_2 = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / pi);
	tmp = 0.0;
	if (angle <= 4.5e+29)
		tmp = t_2;
	elseif (angle <= 1e+118)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle * angle) * (pi * pi))))) / t_1))) / pi);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.5e+29], t$95$2, If[LessEqual[angle, 1e+118], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(2.0 + N[(-6.17283950617284e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.5000000000000002e29 or 9.99999999999999967e117 < angle

   1. Initial program 13.9%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 26.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 42.5%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

   13. Applied rewrites 43.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]

   if 4.5000000000000002e29 < angle < 9.99999999999999967e117

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lift-PI.f64 37.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   13. Applied rewrites 37.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)}\right)}{\pi} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (*
    180.0
    (/
     (atan (* -0.5 (/ (* y-scale 2.0) (* x-scale (* (cos t_0) (sin t_0))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	return 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	return 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	return 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * math.sin(t_0)))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 13.9%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in y-scale around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 26.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

7. Applied rewrites 42.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

8. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

10. Applied rewrites 43.4%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

11. Taylor expanded in angle around 0

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
12. Step-by-step derivation

13. Applied rewrites 43.2%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}\right)}{\pi} \]
14. Add Preprocessing

Alternative 11: 39.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ \mathbf{if}\;angle \leq -6 \cdot 10^{+97}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\mathsf{fma}\left(360, \frac{y-scale}{x-scale \cdot \pi}, \left(angle \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot \pi}{x-scale} - 64800 \cdot \frac{y-scale \cdot \mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, t\_0, -2.8577960676726107 \cdot 10^{-8} \cdot t\_0\right)}{x-scale \cdot \left(\pi \cdot \pi\right)}\right)\right)}{angle}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI PI) PI)))
   (if (<= angle -6e+97)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (fma
           360.0
           (/ y-scale (* x-scale PI))
           (*
            (* angle angle)
            (-
             (* -0.011111111111111112 (/ (* y-scale PI) x-scale))
             (*
              64800.0
              (/
               (*
                y-scale
                (fma -8.573388203017833e-8 t_0 (* -2.8577960676726107e-8 t_0)))
               (* x-scale (* PI PI)))))))
          angle)))
       PI))
     (*
      180.0
      (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double tmp;
	if (angle <= -6e+97) {
		tmp = 180.0 * (atan((-0.5 * (fma(360.0, (y_45_scale / (x_45_scale * ((double) M_PI))), ((angle * angle) * ((-0.011111111111111112 * ((y_45_scale * ((double) M_PI)) / x_45_scale)) - (64800.0 * ((y_45_scale * fma(-8.573388203017833e-8, t_0, (-2.8577960676726107e-8 * t_0))) / (x_45_scale * (((double) M_PI) * ((double) M_PI)))))))) / angle))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * pi) * pi)
	tmp = 0.0
	if (angle <= -6e+97)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(fma(360.0, Float64(y_45_scale / Float64(x_45_scale * pi)), Float64(Float64(angle * angle) * Float64(Float64(-0.011111111111111112 * Float64(Float64(y_45_scale * pi) / x_45_scale)) - Float64(64800.0 * Float64(Float64(y_45_scale * fma(-8.573388203017833e-8, t_0, Float64(-2.8577960676726107e-8 * t_0))) / Float64(x_45_scale * Float64(pi * pi))))))) / angle))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[angle, -6e+97], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(360.0 * N[(y$45$scale / N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(angle * angle), $MachinePrecision] * N[(N[(-0.011111111111111112 * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(64800.0 * N[(N[(y$45$scale * N[(-8.573388203017833e-8 * t$95$0 + N[(-2.8577960676726107e-8 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < -5.9999999999999997e97

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{360 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)} + {angle}^{2} \cdot \left(\frac{-1}{90} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - 64800 \cdot \frac{y-scale \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{x-scale \cdot {\mathsf{PI}\left(\right)}^{2}}\right)}{angle}\right)}{\pi} \]

   13. Applied rewrites 33.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\mathsf{fma}\left(360, \frac{y-scale}{x-scale \cdot \pi}, \left(angle \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \frac{y-scale \cdot \pi}{x-scale} - 64800 \cdot \frac{y-scale \cdot \mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\pi \cdot \pi\right) \cdot \pi, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{x-scale \cdot \left(\pi \cdot \pi\right)}\right)\right)}{angle}\right)}{\pi} \]

   if -5.9999999999999997e97 < angle

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       5. lift-PI.f64 37.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   13. Applied rewrites 37.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \leq 100:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/
          (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_1) 2.0)) x-scale)
          x-scale)))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0))))
            t_3))
          PI))
        100.0)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (*
             -1.0
             (*
              (* b_m b_m)
              (+ (sqrt (pow x-scale -4.0)) (/ 1.0 (* x-scale x-scale)))))))
          (* angle (* PI (- (* b_m b_m) (* a a)))))))
       PI))
     (*
      180.0
      (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI))) <= 100.0) {
		tmp = 180.0 * (atan((90.0 * ((x_45_scale * (y_45_scale * (-1.0 * ((b_m * b_m) * (sqrt(pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))))) / (angle * (((double) M_PI) * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (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_m, 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_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((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)) <= 100.0) {
		tmp = 180.0 * (Math.atan((90.0 * ((x_45_scale * (y_45_scale * (-1.0 * ((b_m * b_m) * (Math.sqrt(Math.pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))))) / (angle * (Math.PI * ((b_m * b_m) - (a * a))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, 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_m, 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_m * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a * t_2), 2.0) + math.pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (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)) <= 100.0:
		tmp = 180.0 * (math.atan((90.0 * ((x_45_scale * (y_45_scale * (-1.0 * ((b_m * b_m) * (math.sqrt(math.pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))))) / (angle * (math.pi * ((b_m * b_m) - (a * a))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(-1.0 * Float64(Float64(b_m * b_m) * Float64(sqrt((x_45_scale ^ -4.0)) + Float64(1.0 / Float64(x_45_scale * x_45_scale))))))) / Float64(angle * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	t_3 = ((((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a * t_2) ^ 2.0) + ((b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = 180.0 * (atan((90.0 * ((x_45_scale * (y_45_scale * (-1.0 * ((b_m * b_m) * (sqrt((x_45_scale ^ -4.0)) + (1.0 / (x_45_scale * x_45_scale))))))) / (angle * (pi * ((b_m * b_m) - (a * a))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(-1.0 * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(N[Sqrt[N[Power[x$45$scale, -4.0], $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64))) < 100

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 12.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \left(\sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}} + \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left({b}^{2} \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left({b}^{2} \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left({b}^{2} \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      3. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      5. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      7. pow-flip N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{\left(\mathsf{neg}\left(4\right)\right)}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      8. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{\left(\mathsf{neg}\left(4\right)\right)}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      9. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      10. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{{x-scale}^{2}}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      11. pow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      12. lift-*.f64 23.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 23.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \left(\left(b \cdot b\right) \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   if 100 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64)))

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       5. lift-PI.f64 37.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   13. Applied rewrites 37.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 38.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 2 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 2e-38)
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/
        (*
         x-scale
         (* y-scale (+ (sqrt (pow x-scale -4.0)) (/ 1.0 (* x-scale x-scale)))))
        (* angle PI))))
     PI))
   (*
    180.0
    (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2e-38) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt(pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2e-38) {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt(Math.pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))) / (angle * Math.PI)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 2e-38:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt(math.pow(x_45_scale, -4.0)) + (1.0 / (x_45_scale * x_45_scale))))) / (angle * math.pi)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 2e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt((x_45_scale ^ -4.0)) + Float64(1.0 / Float64(x_45_scale * x_45_scale))))) / Float64(angle * pi)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 2e-38)
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((x_45_scale ^ -4.0)) + (1.0 / (x_45_scale * x_45_scale))))) / (angle * pi)))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 2e-38], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[Power[x$45$scale, -4.0], $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.9999999999999999e-38

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 12.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \left(\sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}} + \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

   7. Applied rewrites 40.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{x-scale}^{-4}} + \frac{1}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

   if 1.9999999999999999e-38 < x-scale

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       5. lift-PI.f64 37.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   13. Applied rewrites 37.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 37.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b\_m \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \leq 100:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b\_m \cdot b\_m}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b_m t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/
          (/ (+ (pow (* a t_2) 2.0) (pow (* b_m t_1) 2.0)) x-scale)
          x-scale)))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0))))
            t_3))
          PI))
        100.0)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (* x-scale (* y-scale (* -2.0 (/ (* b_m b_m) (* x-scale x-scale)))))
          (* angle (* PI (- (* b_m b_m) (* a a)))))))
       PI))
     (*
      180.0
      (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a * t_2), 2.0) + pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI))) <= 100.0) {
		tmp = 180.0 * (atan((90.0 * ((x_45_scale * (y_45_scale * (-2.0 * ((b_m * b_m) / (x_45_scale * x_45_scale))))) / (angle * (((double) M_PI) * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (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_m, 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_m * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((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)) <= 100.0) {
		tmp = 180.0 * (Math.atan((90.0 * ((x_45_scale * (y_45_scale * (-2.0 * ((b_m * b_m) / (x_45_scale * x_45_scale))))) / (angle * (Math.PI * ((b_m * b_m) - (a * a))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, 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_m, 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_m * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a * t_2), 2.0) + math.pow((b_m * t_1), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (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)) <= 100.0:
		tmp = 180.0 * (math.atan((90.0 * ((x_45_scale * (y_45_scale * (-2.0 * ((b_m * b_m) / (x_45_scale * x_45_scale))))) / (angle * (math.pi * ((b_m * b_m) - (a * a))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(-2.0 * Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale))))) / Float64(angle * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	t_3 = ((((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b_m * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a * t_2) ^ 2.0) + ((b_m * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = 180.0 * (atan((90.0 * ((x_45_scale * (y_45_scale * (-2.0 * ((b_m * b_m) / (x_45_scale * x_45_scale))))) / (angle * (pi * ((b_m * b_m) - (a * a))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(-2.0 * N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64))) < 100

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 12.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \left(\sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}} + \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      3. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      5. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + {b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      6. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + b \cdot b}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + b \cdot b}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      8. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      9. lift-*.f64 21.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   7. Applied rewrites 21.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-1 \cdot \frac{\sqrt{{b}^{4}} + b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   8. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      3. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b \cdot b}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b \cdot b}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      5. pow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

      6. lift-*.f64 23.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   10. Applied rewrites 23.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(-2 \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}{\pi} \]

   if 100 < (*.f64 #s(literal 180 binary64) (/.f64 (atan.f64 (/.f64 (-.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64))))) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale))) (PI.f64)))

   1. Initial program 13.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

   2. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

   4. Applied rewrites 26.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

   5. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

   7. Applied rewrites 42.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   8. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 43.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       5. lift-PI.f64 37.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   13. Applied rewrites 37.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.2% accurate, 25.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.9%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in y-scale around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 26.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{{\left(\frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}^{2}} + \frac{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\pi} \]

7. Applied rewrites 42.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}}} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right)\right)}{\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

8. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)}{\pi} \]

10. Applied rewrites 43.4%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]

11. Taylor expanded in angle around 0

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

    2. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

    3. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

    4. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

    5. lift-PI.f64 37.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

13. Applied rewrites 37.8%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
14. Add Preprocessing

Alternative 16: 19.0% accurate, 49.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} 0}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan 0.0) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(0.0) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(0.0) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(0.0) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(0.0) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(0.0) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.9%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]

2. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\pi} \]

4. Applied rewrites 12.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \left(\sqrt{{\left(\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right)}^{2}} + \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)}}{\pi} \]

5. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

7. Applied rewrites 7.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{y-scale \cdot y-scale} - \sqrt{{y-scale}^{-4}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

8. Taylor expanded in y-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 19.0%

    \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))