raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.6% → 56.9%

Time: 20.5s

Alternatives: 15

Speedup: 22.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{\sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)}^{2}}{\cos t\_0 \cdot \sin t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 7.2e-65)
     (*
      180.0
      (/
       (atan
        (*
         -1.0
         (*
          (/ y-scale x-scale)
          (/
           (pow
            (sin (* angle (fma 0.005555555555555556 PI (* 0.5 (/ PI angle)))))
            2.0)
           (* (cos t_0) (sin t_0))))))
       PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (pow(sin((angle * fma(0.005555555555555556, ((double) M_PI), (0.5 * (((double) M_PI) / angle))))), 2.0) / (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 7.2e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64((sin(Float64(angle * fma(0.005555555555555556, pi, Float64(0.5 * Float64(pi / angle))))) ^ 2.0) / Float64(cos(t_0) * sin(t_0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 7.2e-65], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[Power[N[Sin[N[(angle * N[(0.005555555555555556 * Pi + N[(0.5 * N[(Pi / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / 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[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999996e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 34.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-sin.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 34.7

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 57.4%

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

   10. Taylor expanded in angle around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lift-PI.f64 57.4

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

   12. Applied rewrites 57.4%

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

   if 7.1999999999999996e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 2: 40.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a\_m \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a\_m \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \leq 100:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{x-scale}{angle} \cdot \frac{y-scale \cdot \left(-2 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right)}{\pi \cdot \left(b \cdot b - a\_m \cdot a\_m\right)}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m 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_m 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a_m t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/
          (/ (+ (pow (* a_m t_2) 2.0) (pow (* b t_1) 2.0)) x-scale)
          x-scale)))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0))))
            t_3))
          PI))
        100.0)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          (/ x-scale angle)
          (/
           (* y-scale (* -2.0 (* (/ b x-scale) (/ b x-scale))))
           (* PI (- (* b b) (* a_m a_m)))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (*
          x-scale
          (* -360.0 (/ y-scale (* angle (* (* x-scale x-scale) PI)))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a_m * t_1), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a_m * t_2), 2.0) + pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI))) <= 100.0) {
		tmp = 180.0 * (atan((90.0 * ((x_45_scale / angle) * ((y_45_scale * (-2.0 * ((b / x_45_scale) * (b / x_45_scale)))) / (((double) M_PI) * ((b * b) - (a_m * a_m))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a_m * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a_m * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI)) <= 100.0) {
		tmp = 180.0 * (Math.atan((90.0 * ((x_45_scale / angle) * ((y_45_scale * (-2.0 * ((b / x_45_scale) * (b / x_45_scale)))) / (Math.PI * ((b * b) - (a_m * a_m))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * Math.PI))))))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale
	t_4 = ((math.pow((a_m * t_1), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a_m * t_2), 2.0) + math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (180.0 * (math.atan((((t_4 - t_5) - math.sqrt((math.pow((t_5 - t_4), 2.0) + math.pow(t_3, 2.0)))) / t_3)) / math.pi)) <= 100.0:
		tmp = 180.0 * (math.atan((90.0 * ((x_45_scale / angle) * ((y_45_scale * (-2.0 * ((b / x_45_scale) * (b / x_45_scale)))) / (math.pi * ((b * b) - (a_m * a_m))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * math.pi))))))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, 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_m ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a_m * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a_m * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(x_45_scale / angle) * Float64(Float64(y_45_scale * Float64(-2.0 * Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)))) / Float64(pi * Float64(Float64(b * b) - Float64(a_m * a_m))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(x_45_scale * Float64(-360.0 * Float64(y_45_scale / Float64(angle * Float64(Float64(x_45_scale * x_45_scale) * pi))))))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, 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_m ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a_m * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a_m * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = 180.0 * (atan((90.0 * ((x_45_scale / angle) * ((y_45_scale * (-2.0 * ((b / x_45_scale) * (b / x_45_scale)))) / (pi * ((b * b) - (a_m * a_m))))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * pi))))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, 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$95$m, 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$95$m * 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$95$m * 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]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(x$45$scale / angle), $MachinePrecision] * N[(N[(y$45$scale * N[(-2.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(x$45$scale * N[(-360.0 * N[(y$45$scale / N[(angle * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.6%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 40.4

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

   7. Applied rewrites 40.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.2%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 39.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   7. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 40.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a\_m \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a\_m \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \leq 100:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(\frac{x-scale}{angle} \cdot \left(-2 \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot \frac{y-scale}{\pi \cdot \left(b \cdot b - a\_m \cdot a\_m\right)}\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m 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_m 2.0))) t_2) t_1) x-scale)
          y-scale))
        (t_4
         (/ (/ (+ (pow (* a_m t_1) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale))
        (t_5
         (/
          (/ (+ (pow (* a_m t_2) 2.0) (pow (* b t_1) 2.0)) x-scale)
          x-scale)))
   (if (<=
        (*
         180.0
         (/
          (atan
           (/
            (- (- t_4 t_5) (sqrt (+ (pow (- t_5 t_4) 2.0) (pow t_3 2.0))))
            t_3))
          PI))
        100.0)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          (/ x-scale angle)
          (*
           -2.0
           (*
            (* (/ b x-scale) (/ b x-scale))
            (/ y-scale (* PI (- (* b b) (* a_m a_m)))))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (*
          x-scale
          (* -360.0 (/ y-scale (* angle (* (* x-scale x-scale) PI)))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((pow((a_m * t_1), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((pow((a_m * t_2), 2.0) + pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((pow((t_5 - t_4), 2.0) + pow(t_3, 2.0)))) / t_3)) / ((double) M_PI))) <= 100.0) {
		tmp = 180.0 * (atan((90.0 * ((x_45_scale / angle) * (-2.0 * (((b / x_45_scale) * (b / x_45_scale)) * (y_45_scale / (((double) M_PI) * ((b * b) - (a_m * a_m))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a_m * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a_m * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI)) <= 100.0) {
		tmp = 180.0 * (Math.atan((90.0 * ((x_45_scale / angle) * (-2.0 * (((b / x_45_scale) * (b / x_45_scale)) * (y_45_scale / (Math.PI * ((b * b) - (a_m * a_m))))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * Math.PI))))))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, 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_m, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale
	t_4 = ((math.pow((a_m * t_1), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_5 = ((math.pow((a_m * t_2), 2.0) + math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (180.0 * (math.atan((((t_4 - t_5) - math.sqrt((math.pow((t_5 - t_4), 2.0) + math.pow(t_3, 2.0)))) / t_3)) / math.pi)) <= 100.0:
		tmp = 180.0 * (math.atan((90.0 * ((x_45_scale / angle) * (-2.0 * (((b / x_45_scale) * (b / x_45_scale)) * (y_45_scale / (math.pi * ((b * b) - (a_m * a_m))))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * math.pi))))))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, 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_m ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a_m * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64((Float64(a_m * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(180.0 * Float64(atan(Float64(Float64(Float64(t_4 - t_5) - sqrt(Float64((Float64(t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(x_45_scale / angle) * Float64(-2.0 * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * Float64(y_45_scale / Float64(pi * Float64(Float64(b * b) - Float64(a_m * a_m))))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(x_45_scale * Float64(-360.0 * Float64(y_45_scale / Float64(angle * Float64(Float64(x_45_scale * x_45_scale) * pi))))))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, 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_m ^ 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	t_4 = ((((a_m * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_5 = ((((a_m * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((180.0 * (atan((((t_4 - t_5) - sqrt((((t_5 - t_4) ^ 2.0) + (t_3 ^ 2.0)))) / t_3)) / pi)) <= 100.0)
		tmp = 180.0 * (atan((90.0 * ((x_45_scale / angle) * (-2.0 * (((b / x_45_scale) * (b / x_45_scale)) * (y_45_scale / (pi * ((b * b) - (a_m * a_m))))))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * pi))))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, 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$95$m, 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$95$m * 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$95$m * 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]}, If[LessEqual[N[(180.0 * N[(N[ArcTan[N[(N[(N[(t$95$4 - t$95$5), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 100.0], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(x$45$scale / angle), $MachinePrecision] * N[(-2.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(x$45$scale * N[(-360.0 * N[(y$45$scale / N[(angle * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.6%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 32.4%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   7. Applied rewrites 39.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.2%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 39.3

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   7. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 40.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{\cos t\_0 \cdot \sin t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 7.2e-65)
     (*
      180.0
      (/
       (atan
        (*
         -1.0
         (*
          (/ y-scale x-scale)
          (/
           (pow (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))) 2.0)
           (* (cos t_0) (sin t_0))))))
       PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (pow(sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)))), 2.0) / (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 7.2e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64((sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))) ^ 2.0) / Float64(cos(t_0) * sin(t_0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 7.2e-65], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / 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[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999996e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 34.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-sin.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 34.7

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 57.4%

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

   if 7.1999999999999996e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 5: 56.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{\sin \left(0.5 \cdot \pi\right)}^{2}}{\cos t\_0 \cdot \sin t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 7.2e-65)
     (*
      180.0
      (/
       (atan
        (*
         -1.0
         (*
          (/ y-scale x-scale)
          (/ (pow (sin (* 0.5 PI)) 2.0) (* (cos t_0) (sin t_0))))))
       PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (pow(sin((0.5 * ((double) M_PI))), 2.0) / (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (Math.atan((-1.0 * ((y_45_scale / x_45_scale) * (Math.pow(Math.sin((0.5 * Math.PI)), 2.0) / (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a_m <= 7.2e-65:
		tmp = 180.0 * (math.atan((-1.0 * ((y_45_scale / x_45_scale) * (math.pow(math.sin((0.5 * math.pi)), 2.0) / (math.cos(t_0) * math.sin(t_0)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan(t_0))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 7.2e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64((sin(Float64(0.5 * pi)) ^ 2.0) / Float64(cos(t_0) * sin(t_0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a_m <= 7.2e-65)
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * ((sin((0.5 * pi)) ^ 2.0) / (cos(t_0) * sin(t_0)))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 7.2e-65], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[Power[N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / 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[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999996e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 34.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-sin.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 34.7

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 57.4%

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

   10. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 56.9

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

   12. Applied rewrites 56.9%

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

   if 7.1999999999999996e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 6: 56.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\cos t\_0}{\sin t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 7.2e-65)
     (*
      180.0
      (/ (atan (* -1.0 (* (/ y-scale x-scale) (/ (cos t_0) (sin t_0))))) PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (cos(t_0) / sin(t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (Math.atan((-1.0 * ((y_45_scale / x_45_scale) * (Math.cos(t_0) / Math.sin(t_0))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a_m <= 7.2e-65:
		tmp = 180.0 * (math.atan((-1.0 * ((y_45_scale / x_45_scale) * (math.cos(t_0) / math.sin(t_0))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan(t_0))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 7.2e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(cos(t_0) / sin(t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a_m <= 7.2e-65)
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (cos(t_0) / sin(t_0))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 7.2e-65], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999996e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

   if 7.1999999999999996e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 7: 57.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 7.2 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale \cdot \cos t\_0}{x-scale \cdot \sin t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 7.2e-65)
     (*
      180.0
      (/ (atan (* -1.0 (/ (* y-scale (cos t_0)) (* x-scale (sin t_0))))) PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale * cos(t_0)) / (x_45_scale * sin(t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a_m <= 7.2e-65) {
		tmp = 180.0 * (Math.atan((-1.0 * ((y_45_scale * Math.cos(t_0)) / (x_45_scale * Math.sin(t_0))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a_m <= 7.2e-65:
		tmp = 180.0 * (math.atan((-1.0 * ((y_45_scale * math.cos(t_0)) / (x_45_scale * math.sin(t_0))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan(t_0))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 7.2e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale * cos(t_0)) / Float64(x_45_scale * sin(t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a_m <= 7.2e-65)
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale * cos(t_0)) / (x_45_scale * sin(t_0))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 7.2e-65], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999996e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 57.8%

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

   if 7.1999999999999996e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 8: 54.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.3 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \pi - -0.000925925925925926 \cdot \pi, 180 \cdot {\pi}^{-1}\right)}{angle}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 1.3e-63)
   (*
    180.0
    (/
     (atan
      (*
       -1.0
       (*
        (/ y-scale x-scale)
        (/
         (fma
          (* angle angle)
          (- (* -0.002777777777777778 PI) (* -0.000925925925925926 PI))
          (* 180.0 (pow PI -1.0)))
         angle))))
     PI))
   (*
    180.0
    (/
     (atan (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI)))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.3e-63) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (fma((angle * angle), ((-0.002777777777777778 * ((double) M_PI)) - (-0.000925925925925926 * ((double) M_PI))), (180.0 * pow(((double) M_PI), -1.0))) / angle)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 1.3e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(Float64(angle * angle), Float64(Float64(-0.002777777777777778 * pi) - Float64(-0.000925925925925926 * pi)), Float64(180.0 * (pi ^ -1.0))) / angle)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 1.3e-63], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(N[(angle * angle), $MachinePrecision] * N[(N[(-0.002777777777777778 * Pi), $MachinePrecision] - N[(-0.000925925925925926 * Pi), $MachinePrecision]), $MachinePrecision] + N[(180.0 * N[Power[Pi, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3000000000000001e-63

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right)\right) + 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{angle}\right)\right)}{\pi} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left({angle}^{2}, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      3. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      5. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \mathsf{PI}\left(\right) - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      7. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \mathsf{PI}\left(\right), 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \pi, 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \pi, 180 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{angle}\right)\right)}{\pi} \]

      11. inv-pow N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \pi, 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\pi} \]

      12. lower-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, \frac{-1}{360} \cdot \pi - \frac{-1}{1080} \cdot \pi, 180 \cdot {\mathsf{PI}\left(\right)}^{-1}\right)}{angle}\right)\right)}{\pi} \]

      13. lift-PI.f64 51.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \pi - -0.000925925925925926 \cdot \pi, 180 \cdot {\pi}^{-1}\right)}{angle}\right)\right)}{\pi} \]

   10. Applied rewrites 51.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(angle \cdot angle, -0.002777777777777778 \cdot \pi - -0.000925925925925926 \cdot \pi, 180 \cdot {\pi}^{-1}\right)}{angle}\right)\right)}{\pi} \]

   if 1.3000000000000001e-63 < a

   1. Initial program 10.1%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 9: 56.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a_m 1.4e-63)
     (* 180.0 (/ (atan (* -1.0 (* (/ y-scale x-scale) (/ 1.0 (sin t_0))))) PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (tan t_0))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 1.4e-63) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (1.0 / sin(t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a_m <= 1.4e-63) {
		tmp = 180.0 * (Math.atan((-1.0 * ((y_45_scale / x_45_scale) * (1.0 / Math.sin(t_0))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a_m <= 1.4e-63:
		tmp = 180.0 * (math.atan((-1.0 * ((y_45_scale / x_45_scale) * (1.0 / math.sin(t_0))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan(t_0))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 1.4e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(1.0 / sin(t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a_m <= 1.4e-63)
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (1.0 / sin(t_0))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan(t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 1.4e-63], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(1.0 / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4000000000000001e-63

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 57.5%

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

   if 1.4000000000000001e-63 < a

   1. Initial program 10.1%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 10: 53.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7 \cdot 10^{-65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 7e-65)
   (*
    180.0
    (/
     (atan
      (*
       0.5
       (*
        x-scale
        (* -360.0 (/ y-scale (* angle (* (* x-scale x-scale) PI)))))))
     PI))
   (*
    180.0
    (/
     (atan (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI)))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 7e-65) {
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 7e-65) {
		tmp = 180.0 * (Math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan((0.005555555555555556 * (angle * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 7e-65:
		tmp = 180.0 * (math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan((0.005555555555555556 * (angle * math.pi))))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 7e-65)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(x_45_scale * Float64(-360.0 * Float64(y_45_scale / Float64(angle * Float64(Float64(x_45_scale * x_45_scale) * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi))))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 7e-65)
		tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * pi))))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 7e-65], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(x$45$scale * N[(-360.0 * N[(y$45$scale / N[(angle * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.00000000000000009e-65

   1. Initial program 18.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 47.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   7. Applied rewrites 47.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.9%

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

   if 7.00000000000000009e-65 < a

   1. Initial program 10.2%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 30.5%

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

   5. Taylor expanded in a around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 11: 41.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{y-scale \cdot \pi}{x-scale}\\ \mathbf{if}\;x-scale \leq -7.1 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \pi}\right)\right)}{\pi}\\ \mathbf{elif}\;x-scale \leq 3.9 \cdot 10^{-148}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\left(x-scale + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\mathsf{fma}\left(180, \frac{y-scale}{x-scale \cdot \pi}, \left(angle \cdot angle\right) \cdot \left(-0.002777777777777778 \cdot t\_0 - -0.000925925925925926 \cdot t\_0\right)\right)}{angle}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* y-scale PI) x-scale)))
   (if (<= x-scale -7.1e+103)
     (*
      180.0
      (/ (atan (* -1.0 (* (/ y-scale x-scale) (/ 180.0 (* angle PI))))) PI))
     (if (<= x-scale 3.9e-148)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (*
            (+
             x-scale
             (* 1.54320987654321e-5 (* (* angle angle) (* x-scale (* PI PI)))))
            (* -360.0 (/ y-scale (* angle (* (* x-scale x-scale) PI)))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -1.0
           (/
            (fma
             180.0
             (/ y-scale (* x-scale PI))
             (*
              (* angle angle)
              (- (* -0.002777777777777778 t_0) (* -0.000925925925925926 t_0))))
            angle)))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * ((double) M_PI)) / x_45_scale;
	double tmp;
	if (x_45_scale <= -7.1e+103) {
		tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (180.0 / (angle * ((double) M_PI)))))) / ((double) M_PI));
	} else if (x_45_scale <= 3.9e-148) {
		tmp = 180.0 * (atan((0.5 * ((x_45_scale + (1.54320987654321e-5 * ((angle * angle) * (x_45_scale * (((double) M_PI) * ((double) M_PI)))))) * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-1.0 * (fma(180.0, (y_45_scale / (x_45_scale * ((double) M_PI))), ((angle * angle) * ((-0.002777777777777778 * t_0) - (-0.000925925925925926 * t_0)))) / angle))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale * pi) / x_45_scale)
	tmp = 0.0
	if (x_45_scale <= -7.1e+103)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 / Float64(angle * pi))))) / pi));
	elseif (x_45_scale <= 3.9e-148)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(x_45_scale + Float64(1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(x_45_scale * Float64(pi * pi))))) * Float64(-360.0 * Float64(y_45_scale / Float64(angle * Float64(Float64(x_45_scale * x_45_scale) * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(fma(180.0, Float64(y_45_scale / Float64(x_45_scale * pi)), Float64(Float64(angle * angle) * Float64(Float64(-0.002777777777777778 * t_0) - Float64(-0.000925925925925926 * t_0)))) / angle))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[x$45$scale, -7.1e+103], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 3.9e-148], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(x$45$scale + N[(1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(x$45$scale * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-360.0 * N[(y$45$scale / N[(angle * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(180.0 * N[(y$45$scale / N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(angle * angle), $MachinePrecision] * N[(N[(-0.002777777777777778 * t$95$0), $MachinePrecision] - N[(-0.000925925925925926 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -7.1000000000000002e103

   1. Initial program 5.3%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 16.8%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in angle around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if -7.1000000000000002e103 < x-scale < 3.89999999999999994e-148

   1. Initial program 17.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 39.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   7. Applied rewrites 39.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left({angle}^{2} \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left({angle}^{2} \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left({angle}^{2} \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left({angle}^{2} \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      7. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(\left(x-scale + \frac{1}{64800} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

      10. lift-PI.f64 42.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\left(x-scale + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   10. Applied rewrites 42.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\left(x-scale + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(x-scale \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

   if 3.89999999999999994e-148 < x-scale

   1. Initial program 12.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 23.8%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)} + {angle}^{2} \cdot \left(\frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}{angle}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)} + {angle}^{2} \cdot \left(\frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}{angle}\right)}{\pi} \]

   10. Applied rewrites 36.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\mathsf{fma}\left(180, \frac{y-scale}{x-scale \cdot \pi}, \left(angle \cdot angle\right) \cdot \left(-0.002777777777777778 \cdot \frac{y-scale \cdot \pi}{x-scale} - -0.000925925925925926 \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{angle}\right)}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.0% accurate, 20.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(x-scale \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/
   (atan
    (*
     0.5
     (* x-scale (* -360.0 (/ y-scale (* angle (* (* x-scale x-scale) PI)))))))
   PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * ((double) M_PI)))))))) / ((double) M_PI));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * Math.PI))))))) / Math.PI);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * math.pi))))))) / math.pi)

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(0.5 * Float64(x_45_scale * Float64(-360.0 * Float64(y_45_scale / Float64(angle * Float64(Float64(x_45_scale * x_45_scale) * pi))))))) / pi))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((0.5 * (x_45_scale * (-360.0 * (y_45_scale / (angle * ((x_45_scale * x_45_scale) * pi))))))) / pi);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(x$45$scale * N[(-360.0 * N[(y$45$scale / N[(angle * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 26.9%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 39.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]

7. Applied rewrites 39.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \left(\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(-360 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}}\right)\right)\right)}{\pi} \]

8. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \left(x-scale \cdot \left(\color{blue}{-360} \cdot \frac{y-scale}{angle \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \pi\right)}\right)\right)\right)}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 40.0%

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

Alternative 13: 40.2% accurate, 20.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{180}{angle \cdot \pi}\right)\right)}{\pi} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* -1.0 (* (/ y-scale x-scale) (/ 180.0 (* angle PI))))) PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (180.0 / (angle * ((double) M_PI)))))) / ((double) M_PI));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-1.0 * ((y_45_scale / x_45_scale) * (180.0 / (angle * Math.PI))))) / Math.PI);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-1.0 * ((y_45_scale / x_45_scale) * (180.0 / (angle * math.pi))))) / math.pi)

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 / Float64(angle * pi))))) / pi))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-1.0 * ((y_45_scale / x_45_scale) * (180.0 / (angle * pi))))) / pi);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 26.9%

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

5. Taylor expanded in x-scale around 0

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

   1. lower-*.f64 N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

7. Applied rewrites 45.2%

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

8. Taylor expanded in angle around 0

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

   1. lower-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 40.2

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

10. Applied rewrites 40.2%

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

Alternative 14: 38.5% accurate, 22.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 11.2%

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

5. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 38.5

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

7. Applied rewrites 38.5%

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

Alternative 15: 11.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}{\pi} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ x-scale (* angle (* y-scale PI))))) PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(x$45$scale / N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 11.2%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 11.8

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

7. Applied rewrites 11.8%

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

Reproduce

herbie shell --seed 2025107 
(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)))