raw-angle from scale-rotated-ellipse

Percentage Accurate: 14.3% → 50.6%

Time: 26.4s

Alternatives: 13

Speedup: 49.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 14.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 50.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale \cdot \left(\cos t\_0 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_2 \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2
         (sin
          (fma (fabs (* PI angle)) 0.005555555555555556 1.5707963267948966))))
   (if (<= (fabs b) 7.2e-86)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
          (* x-scale (* (cos t_0) t_1)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
          (* x-scale (* t_2 t_1)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, 1.5707963267948966));
	double tmp;
	if (fabs(b) <= 7.2e-86) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / (x_45_scale * (cos(t_0) * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / (x_45_scale * (t_2 * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, 1.5707963267948966))
	tmp = 0.0
	if (abs(b) <= 7.2e-86)
		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(cos(t_0) * t_1))))) / pi));
	else
		tmp = 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_2 * t_1))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + 1.5707963267948966), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7.2e-86], 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[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $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$2, 4.0], $MachinePrecision]], $MachinePrecision] + 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]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.19999999999999932e-86

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 37.6%

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

   if 7.19999999999999932e-86 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fabs-mul N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 43.5%

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

   9. Applied rewrites 43.5%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.4%

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

   11. Applied rewrites 43.4%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   14. Evaluated real constant 43.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right)}^{2}\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   15. Evaluated real constant 43.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)}^{2}\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   16. Evaluated real constant 43.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)}^{4}} + {\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right)}^{2}\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, 1.5707963267948966\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 50.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale \cdot \left(\cos t\_0 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + \mathsf{fma}\left(\cos \left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right), 0.5, 0.5\right)\right)}{x-scale \cdot \left(t\_2 \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (sin (fma (fabs (* PI angle)) 0.005555555555555556 (* PI 0.5)))))
   (if (<= (fabs b) 7.2e-86)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
          (* x-scale (* (cos t_0) t_1)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           y-scale
           (+
            (sqrt (pow t_2 4.0))
            (fma (cos (* (* (* PI angle) 0.005555555555555556) 2.0)) 0.5 0.5)))
          (* x-scale (* t_2 t_1)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) * 0.5)));
	double tmp;
	if (fabs(b) <= 7.2e-86) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / (x_45_scale * (cos(t_0) * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + fma(cos((((((double) M_PI) * angle) * 0.005555555555555556) * 2.0)), 0.5, 0.5))) / (x_45_scale * (t_2 * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi * 0.5)))
	tmp = 0.0
	if (abs(b) <= 7.2e-86)
		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(cos(t_0) * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + fma(cos(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * 2.0)), 0.5, 0.5))) / Float64(x_45_scale * Float64(t_2 * t_1))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7.2e-86], 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[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $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$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.19999999999999932e-86

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 37.6%

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

   if 7.19999999999999932e-86 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fabs-mul N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 43.5%

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

   9. Applied rewrites 43.5%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.4%

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

   11. Applied rewrites 43.4%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   15. Applied rewrites 43.7%

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

Alternative 3: 50.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|b\right| \leq 7.2 \cdot 10^{-86}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale \cdot \left(\cos t\_0 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\left(\mathsf{fma}\left(\cos \left(t\_2 \cdot 2\right), 0.5, 0.5\right) + \sqrt{{\cos t\_2}^{4}}\right) \cdot y-scale}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs b) 7.2e-86)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
          (* x-scale (* (cos t_0) t_1)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           (+ (fma (cos (* t_2 2.0)) 0.5 0.5) (sqrt (pow (cos t_2) 4.0)))
           y-scale)
          (*
           x-scale
           (*
            (sin (fma (fabs (* PI angle)) 0.005555555555555556 (* PI 0.5)))
            t_1)))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 7.2e-86) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / (x_45_scale * (cos(t_0) * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (((fma(cos((t_2 * 2.0)), 0.5, 0.5) + sqrt(pow(cos(t_2), 4.0))) * y_45_scale) / (x_45_scale * (sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) * 0.5))) * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 7.2e-86)
		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(cos(t_0) * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(fma(cos(Float64(t_2 * 2.0)), 0.5, 0.5) + sqrt((cos(t_2) ^ 4.0))) * y_45_scale) / Float64(x_45_scale * Float64(sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi * 0.5))) * t_1))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7.2e-86], 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[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(N[(N[Cos[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$2], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(x$45$scale * N[(N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.19999999999999932e-86

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 37.6%

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

   if 7.19999999999999932e-86 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fabs-mul N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 43.5%

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

   9. Applied rewrites 43.5%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.4%

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

   11. Applied rewrites 43.4%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   15. Applied rewrites 43.6%

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

Alternative 4: 44.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|b\right| \leq 1.32 \cdot 10^{-139}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 5.2 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{x-scale \cdot \left(\cos t\_1 \cdot \left(\sin t\_1 \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{\left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_2\right)\right) + \sqrt{{\cos t\_2}^{4}}\right) \cdot y-scale}{x-scale \cdot \sin t\_2} \cdot -0.5\right)}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs b) 1.32e-139)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           angle
           (*
            y-scale
            (+
             (sqrt (* 9.525986892242036e-10 (* (pow a 4.0) (pow PI 4.0))))
             (* 3.08641975308642e-5 (* (pow a 2.0) (pow PI 2.0))))))
          (* (pow a 2.0) (* x-scale PI)))))
       PI))
     (if (<= (fabs b) 5.2e+72)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* x-scale (* (cos t_1) (* (sin t_1) (- t_0 (pow a 2.0))))))))
         PI))
       (/
        1.0
        (/
         PI
         (*
          180.0
          (atan
           (*
            (/
             (*
              (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_2)))) (sqrt (pow (cos t_2) 4.0)))
              y-scale)
             (* x-scale (sin t_2)))
            -0.5)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 1.32e-139) {
		tmp = 180.0 * (atan((90.0 * ((angle * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(a, 4.0) * pow(((double) M_PI), 4.0)))) + (3.08641975308642e-5 * (pow(a, 2.0) * pow(((double) M_PI), 2.0)))))) / (pow(a, 2.0) * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else if (fabs(b) <= 5.2e+72) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (x_45_scale * (cos(t_1) * (sin(t_1) * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan((((((0.5 + (0.5 * cos((2.0 * t_2)))) + sqrt(pow(cos(t_2), 4.0))) * y_45_scale) / (x_45_scale * sin(t_2))) * -0.5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(b), 2.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (Math.abs(b) <= 1.32e-139) {
		tmp = 180.0 * (Math.atan((90.0 * ((angle * (y_45_scale * (Math.sqrt((9.525986892242036e-10 * (Math.pow(a, 4.0) * Math.pow(Math.PI, 4.0)))) + (3.08641975308642e-5 * (Math.pow(a, 2.0) * Math.pow(Math.PI, 2.0)))))) / (Math.pow(a, 2.0) * (x_45_scale * Math.PI))))) / Math.PI);
	} else if (Math.abs(b) <= 5.2e+72) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (Math.sqrt(Math.pow(Math.abs(b), 4.0)) + t_0)) / (x_45_scale * (Math.cos(t_1) * (Math.sin(t_1) * (t_0 - Math.pow(a, 2.0)))))))) / Math.PI);
	} else {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan((((((0.5 + (0.5 * Math.cos((2.0 * t_2)))) + Math.sqrt(Math.pow(Math.cos(t_2), 4.0))) * y_45_scale) / (x_45_scale * Math.sin(t_2))) * -0.5))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(b), 2.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if math.fabs(b) <= 1.32e-139:
		tmp = 180.0 * (math.atan((90.0 * ((angle * (y_45_scale * (math.sqrt((9.525986892242036e-10 * (math.pow(a, 4.0) * math.pow(math.pi, 4.0)))) + (3.08641975308642e-5 * (math.pow(a, 2.0) * math.pow(math.pi, 2.0)))))) / (math.pow(a, 2.0) * (x_45_scale * math.pi))))) / math.pi)
	elif math.fabs(b) <= 5.2e+72:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (math.sqrt(math.pow(math.fabs(b), 4.0)) + t_0)) / (x_45_scale * (math.cos(t_1) * (math.sin(t_1) * (t_0 - math.pow(a, 2.0)))))))) / math.pi)
	else:
		tmp = 1.0 / (math.pi / (180.0 * math.atan((((((0.5 + (0.5 * math.cos((2.0 * t_2)))) + math.sqrt(math.pow(math.cos(t_2), 4.0))) * y_45_scale) / (x_45_scale * math.sin(t_2))) * -0.5))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 1.32e-139)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((a ^ 4.0) * (pi ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((a ^ 2.0) * (pi ^ 2.0)))))) / Float64((a ^ 2.0) * Float64(x_45_scale * pi))))) / pi));
	elseif (abs(b) <= 5.2e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(x_45_scale * Float64(cos(t_1) * Float64(sin(t_1) * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_2)))) + sqrt((cos(t_2) ^ 4.0))) * y_45_scale) / Float64(x_45_scale * sin(t_2))) * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0;
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (abs(b) <= 1.32e-139)
		tmp = 180.0 * (atan((90.0 * ((angle * (y_45_scale * (sqrt((9.525986892242036e-10 * ((a ^ 4.0) * (pi ^ 4.0)))) + (3.08641975308642e-5 * ((a ^ 2.0) * (pi ^ 2.0)))))) / ((a ^ 2.0) * (x_45_scale * pi))))) / pi);
	elseif (abs(b) <= 5.2e+72)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt((abs(b) ^ 4.0)) + t_0)) / (x_45_scale * (cos(t_1) * (sin(t_1) * (t_0 - (a ^ 2.0)))))))) / pi);
	else
		tmp = 1.0 / (pi / (180.0 * atan((((((0.5 + (0.5 * cos((2.0 * t_2)))) + sqrt((cos(t_2) ^ 4.0))) * y_45_scale) / (x_45_scale * sin(t_2))) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.32e-139], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[a, 2.0], $MachinePrecision] * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 5.2e+72], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[Sin[t$95$1], $MachinePrecision] * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$2], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(x$45$scale * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.31999999999999995e-139

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 17.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]

   if 1.31999999999999995e-139 < b < 5.19999999999999963e72

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 25.0%

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

   7. Applied rewrites 25.0%

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

   if 5.19999999999999963e72 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 43.2%

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

Alternative 5: 44.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\\ t_2 := 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}{t\_1}\right)}{\pi}\\ \mathbf{if}\;y-scale \leq -9 \cdot 10^{-28}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y-scale \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;y-scale \leq 1.9 \cdot 10^{+223}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (a b 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
                (* -6.17283950617284e-5 (* (pow angle 2.0) (pow PI 2.0)))))
              t_1)))
           PI))))
   (if (<= y-scale -9e-28)
     t_2
     (if (<= y-scale 3.4e-38)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (*
             x-scale
             (*
              y-scale
              (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
            (* angle PI))))
         PI))
       (if (<= y-scale 1.9e+223)
         (* 180.0 (/ (atan (* -0.5 (/ (* y-scale 2.0) t_1))) PI))
         t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	double t_2 = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * (pow(angle, 2.0) * pow(((double) M_PI), 2.0))))) / t_1))) / ((double) M_PI));
	double tmp;
	if (y_45_scale <= -9e-28) {
		tmp = t_2;
	} else if (y_45_scale <= 3.4e-38) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else if (y_45_scale <= 1.9e+223) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / ((double) M_PI));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = 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 + (-6.17283950617284e-5 * (Math.pow(angle, 2.0) * Math.pow(Math.PI, 2.0))))) / t_1))) / Math.PI);
	double tmp;
	if (y_45_scale <= -9e-28) {
		tmp = t_2;
	} else if (y_45_scale <= 3.4e-38) {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	} else if (y_45_scale <= 1.9e+223) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / Math.PI);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, 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 + (-6.17283950617284e-5 * (math.pow(angle, 2.0) * math.pow(math.pi, 2.0))))) / t_1))) / math.pi)
	tmp = 0
	if y_45_scale <= -9e-28:
		tmp = t_2
	elif y_45_scale <= 3.4e-38:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	elif y_45_scale <= 1.9e+223:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / math.pi)
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, 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 * Float64(2.0 + Float64(-6.17283950617284e-5 * Float64((angle ^ 2.0) * (pi ^ 2.0))))) / t_1))) / pi))
	tmp = 0.0
	if (y_45_scale <= -9e-28)
		tmp = t_2;
	elseif (y_45_scale <= 3.4e-38)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	elseif (y_45_scale <= 1.9e+223)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / t_1))) / pi));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	t_2 = 180.0 * (atan((-0.5 * ((y_45_scale * (2.0 + (-6.17283950617284e-5 * ((angle ^ 2.0) * (pi ^ 2.0))))) / t_1))) / pi);
	tmp = 0.0;
	if (y_45_scale <= -9e-28)
		tmp = t_2;
	elseif (y_45_scale <= 3.4e-38)
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	elseif (y_45_scale <= 1.9e+223)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / pi);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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 * N[(2.0 + N[(-6.17283950617284e-5 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -9e-28], t$95$2, If[LessEqual[y$45$scale, 3.4e-38], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 1.9e+223], 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], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -8.9999999999999996e-28 or 1.9e223 < y-scale

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

      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. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

      5. lower-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 + \frac{-1}{16200} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

      6. lower-PI.f64 37.0%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   10. Applied rewrites 37.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(2 + -6.17283950617284 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   if -8.9999999999999996e-28 < y-scale < 3.4000000000000002e-38

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.1%

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

   if 3.4000000000000002e-38 < y-scale < 1.9e223

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 43.1%

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

Alternative 6: 44.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|b\right| \leq 5.6 \cdot 10^{-95}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\left(\mathsf{fma}\left(\cos \left(t\_0 \cdot 2\right), 0.5, 0.5\right) + \sqrt{{\cos t\_0}^{4}}\right) \cdot y-scale}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\left|\pi \cdot angle\right|, 0.005555555555555556, \pi \cdot 0.5\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs b) 5.6e-95)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           angle
           (*
            y-scale
            (+
             (sqrt (* 9.525986892242036e-10 (* (pow a 4.0) (pow PI 4.0))))
             (* 3.08641975308642e-5 (* (pow a 2.0) (pow PI 2.0))))))
          (* (pow a 2.0) (* x-scale PI)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           (+ (fma (cos (* t_0 2.0)) 0.5 0.5) (sqrt (pow (cos t_0) 4.0)))
           y-scale)
          (*
           x-scale
           (*
            (sin (fma (fabs (* PI angle)) 0.005555555555555556 (* PI 0.5)))
            (sin (* 0.005555555555555556 (* angle PI))))))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 5.6e-95) {
		tmp = 180.0 * (atan((90.0 * ((angle * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(a, 4.0) * pow(((double) M_PI), 4.0)))) + (3.08641975308642e-5 * (pow(a, 2.0) * pow(((double) M_PI), 2.0)))))) / (pow(a, 2.0) * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (((fma(cos((t_0 * 2.0)), 0.5, 0.5) + sqrt(pow(cos(t_0), 4.0))) * y_45_scale) / (x_45_scale * (sin(fma(fabs((((double) M_PI) * angle)), 0.005555555555555556, (((double) M_PI) * 0.5))) * sin((0.005555555555555556 * (angle * ((double) M_PI))))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 5.6e-95)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((a ^ 4.0) * (pi ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((a ^ 2.0) * (pi ^ 2.0)))))) / Float64((a ^ 2.0) * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(fma(cos(Float64(t_0 * 2.0)), 0.5, 0.5) + sqrt((cos(t_0) ^ 4.0))) * y_45_scale) / Float64(x_45_scale * Float64(sin(fma(abs(Float64(pi * angle)), 0.005555555555555556, Float64(pi * 0.5))) * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.6e-95], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[a, 2.0], $MachinePrecision] * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(x$45$scale * N[(N[Sin[N[(N[Abs[N[(Pi * angle), $MachinePrecision]], $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999998e-95

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 17.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]

   if 5.5999999999999998e-95 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fabs-mul N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 43.5%

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

   9. Applied rewrites 43.5%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.4%

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

   11. Applied rewrites 43.4%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-fabs.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. mult-flip N/A

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

       16. metadata-eval N/A

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

       17. lower-*.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   15. Applied rewrites 43.6%

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

Alternative 7: 43.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|b\right| \leq 1.32 \cdot 10^{-139}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{\left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \sqrt{{\cos t\_0}^{4}}\right) \cdot y-scale}{x-scale \cdot \sin t\_0} \cdot -0.5\right)}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs b) 1.32e-139)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           angle
           (*
            y-scale
            (+
             (sqrt (* 9.525986892242036e-10 (* (pow a 4.0) (pow PI 4.0))))
             (* 3.08641975308642e-5 (* (pow a 2.0) (pow PI 2.0))))))
          (* (pow a 2.0) (* x-scale PI)))))
       PI))
     (/
      1.0
      (/
       PI
       (*
        180.0
        (atan
         (*
          (/
           (*
            (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (sqrt (pow (cos t_0) 4.0)))
            y-scale)
           (* x-scale (sin t_0)))
          -0.5))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 1.32e-139) {
		tmp = 180.0 * (atan((90.0 * ((angle * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(a, 4.0) * pow(((double) M_PI), 4.0)))) + (3.08641975308642e-5 * (pow(a, 2.0) * pow(((double) M_PI), 2.0)))))) / (pow(a, 2.0) * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan((((((0.5 + (0.5 * cos((2.0 * t_0)))) + sqrt(pow(cos(t_0), 4.0))) * y_45_scale) / (x_45_scale * sin(t_0))) * -0.5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (Math.abs(b) <= 1.32e-139) {
		tmp = 180.0 * (Math.atan((90.0 * ((angle * (y_45_scale * (Math.sqrt((9.525986892242036e-10 * (Math.pow(a, 4.0) * Math.pow(Math.PI, 4.0)))) + (3.08641975308642e-5 * (Math.pow(a, 2.0) * Math.pow(Math.PI, 2.0)))))) / (Math.pow(a, 2.0) * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan((((((0.5 + (0.5 * Math.cos((2.0 * t_0)))) + Math.sqrt(Math.pow(Math.cos(t_0), 4.0))) * y_45_scale) / (x_45_scale * Math.sin(t_0))) * -0.5))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if math.fabs(b) <= 1.32e-139:
		tmp = 180.0 * (math.atan((90.0 * ((angle * (y_45_scale * (math.sqrt((9.525986892242036e-10 * (math.pow(a, 4.0) * math.pow(math.pi, 4.0)))) + (3.08641975308642e-5 * (math.pow(a, 2.0) * math.pow(math.pi, 2.0)))))) / (math.pow(a, 2.0) * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = 1.0 / (math.pi / (180.0 * math.atan((((((0.5 + (0.5 * math.cos((2.0 * t_0)))) + math.sqrt(math.pow(math.cos(t_0), 4.0))) * y_45_scale) / (x_45_scale * math.sin(t_0))) * -0.5))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 1.32e-139)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((a ^ 4.0) * (pi ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((a ^ 2.0) * (pi ^ 2.0)))))) / Float64((a ^ 2.0) * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) + sqrt((cos(t_0) ^ 4.0))) * y_45_scale) / Float64(x_45_scale * sin(t_0))) * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (abs(b) <= 1.32e-139)
		tmp = 180.0 * (atan((90.0 * ((angle * (y_45_scale * (sqrt((9.525986892242036e-10 * ((a ^ 4.0) * (pi ^ 4.0)))) + (3.08641975308642e-5 * ((a ^ 2.0) * (pi ^ 2.0)))))) / ((a ^ 2.0) * (x_45_scale * pi))))) / pi);
	else
		tmp = 1.0 / (pi / (180.0 * atan((((((0.5 + (0.5 * cos((2.0 * t_0)))) + sqrt((cos(t_0) ^ 4.0))) * y_45_scale) / (x_45_scale * sin(t_0))) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.32e-139], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[a, 2.0], $MachinePrecision] * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(x$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.31999999999999995e-139

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \left({a}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)} + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{{a}^{2} \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\pi} \]

   10. Applied rewrites 17.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \left({a}^{4} \cdot {\pi}^{4}\right)} + 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)\right)}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]

   if 1.31999999999999995e-139 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 43.2%

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

Alternative 8: 42.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|a\right| \leq 2.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{2 \cdot y-scale}{\left(x-scale \cdot \cos t\_0\right) \cdot \sin t\_0} \cdot -0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs a) 2.5e-60)
     (/
      1.0
      (/
       PI
       (*
        180.0
        (atan
         (* (/ (* 2.0 y-scale) (* (* x-scale (cos t_0)) (sin t_0))) -0.5)))))
     (*
      180.0
      (/
       (atan
        (*
         -90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
          (* angle PI))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(a) <= 2.5e-60) {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan((((2.0 * y_45_scale) / ((x_45_scale * cos(t_0)) * sin(t_0))) * -0.5))));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (Math.abs(a) <= 2.5e-60) {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan((((2.0 * y_45_scale) / ((x_45_scale * Math.cos(t_0)) * Math.sin(t_0))) * -0.5))));
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if math.fabs(a) <= 2.5e-60:
		tmp = 1.0 / (math.pi / (180.0 * math.atan((((2.0 * y_45_scale) / ((x_45_scale * math.cos(t_0)) * math.sin(t_0))) * -0.5))))
	else:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(a) <= 2.5e-60)
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(Float64(2.0 * y_45_scale) / Float64(Float64(x_45_scale * cos(t_0)) * sin(t_0))) * -0.5)))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (abs(a) <= 2.5e-60)
		tmp = 1.0 / (pi / (180.0 * atan((((2.0 * y_45_scale) / ((x_45_scale * cos(t_0)) * sin(t_0))) * -0.5))));
	else
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.5e-60], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(N[(2.0 * y$45$scale), $MachinePrecision] / N[(N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.5000000000000001e-60

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 43.1%

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

   if 2.5000000000000001e-60 < a

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|a\right| \leq 2.5 \cdot 10^{-60}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs a) 2.5e-60)
     (*
      180.0
      (/
       (atan (* -0.5 (/ (* y-scale 2.0) (* x-scale (* (cos t_0) (sin t_0))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -90.0
         (/
          (*
           x-scale
           (*
            y-scale
            (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
          (* angle PI))))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(a) <= 2.5e-60) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(a) <= 2.5e-60) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(a) <= 2.5e-60:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * math.sin(t_0)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(a) <= 2.5e-60)
		tmp = 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));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(a) <= 2.5e-60)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / pi);
	else
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.5e-60], 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], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.5000000000000001e-60

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 43.1%

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

   if 2.5000000000000001e-60 < a

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 1250:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 1250.0)
   (/
    1.0
    (/
     PI
     (* 180.0 (atan (* (* 360.0 (/ y-scale (* angle (* x-scale PI)))) -0.5)))))
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/
        (*
         x-scale
         (*
          y-scale
          (+ (sqrt (/ 1.0 (pow x-scale 4.0))) (/ 1.0 (pow x-scale 2.0)))))
        (* angle PI))))
     PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 1250.0) {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan(((360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))) * -0.5))));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 1250.0) {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan(((360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))) * -0.5))));
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 1250.0:
		tmp = 1.0 / (math.pi / (180.0 * math.atan(((360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))) * -0.5))))
	else:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1250.0)
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))) * -0.5)))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 1250.0)
		tmp = 1.0 / (pi / (180.0 * atan(((360.0 * (y_45_scale / (angle * (x_45_scale * pi)))) * -0.5))));
	else
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 1250.0], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1250

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       5. lower-PI.f64 37.8%

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]

   12. Applied rewrites 37.8%

       \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]

   if 1250 < a

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.1% accurate, 20.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 1.3e+148)
   (/
    1.0
    (/
     PI
     (* 180.0 (atan (* (* 360.0 (/ y-scale (* angle (* x-scale PI)))) -0.5)))))
   (* 180.0 (/ (atan 0.0) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 1.3e+148) {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan(((360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))) * -0.5))));
	} else {
		tmp = 180.0 * (atan(0.0) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 1.3e+148) {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan(((360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))) * -0.5))));
	} else {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 1.3e+148:
		tmp = 1.0 / (math.pi / (180.0 * math.atan(((360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))) * -0.5))))
	else:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.3e+148)
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))) * -0.5)))));
	else
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 1.3e+148)
		tmp = 1.0 / (pi / (180.0 * atan(((360.0 * (y_45_scale / (angle * (x_45_scale * pi)))) * -0.5))));
	else
		tmp = 180.0 * (atan(0.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 1.3e+148], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e148

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{-1}{2}\right)}} \]

       5. lower-PI.f64 37.8%

          \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]

   12. Applied rewrites 37.8%

       \[\leadsto \frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot -0.5\right)}} \]

   if 1.3e148 < a

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      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 7.2%

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

   8. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.1% accurate, 22.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 1.3e+148)
   (*
    180.0
    (/ (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI)))))) PI))
   (* 180.0 (/ (atan 0.0) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 1.3e+148) {
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(0.0) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 1.3e+148) {
		tmp = 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 1.3e+148:
		tmp = 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.3e+148)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 1.3e+148)
		tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
	else
		tmp = 180.0 * (atan(0.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 1.3e+148], 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], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e148

   1. Initial program 14.3%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

      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. lower-PI.f64 37.8%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   10. Applied rewrites 37.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]

   if 1.3e148 < a

   1. Initial program 14.3%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      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 7.2%

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

   8. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 18.6% accurate, 49.1× speedup

Math

\[180 \cdot \frac{\tan^{-1} 0}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan 0.0) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(0.0) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(0.0) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(0.0) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(0.0) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(0.0) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.3%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.5%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   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 7.2%

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

8. Taylor expanded in y-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 18.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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)))