raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.9% → 55.9%

Time: 25.0s

Alternatives: 10

Speedup: 28.2×

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: 13.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;b\_m \leq 8 \cdot 10^{-61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{1 \cdot t\_1}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= b_m 8e-61)
     (* 180.0 (/ (atan (/ (* y-scale t_1) (* x-scale (cos t_0)))) PI))
     (*
      180.0
      (/ (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* 1.0 t_1)))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (b_m <= 8e-61) {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * t_1)))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (b_m <= 8e-61) {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * t_1)))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	tmp = 0
	if b_m <= 8e-61:
		tmp = 180.0 * (math.atan(((y_45_scale * t_1) / (x_45_scale * math.cos(t_0)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * t_1)))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (b_m <= 8e-61)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * cos(t_0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(1.0 * t_1)))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (b_m <= 8e-61)
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * cos(t_0)))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * t_1)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 8e-61], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000003e-61

   1. Initial program 13.9%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 29.8%

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

   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 45.1%

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

   if 8.0000000000000003e-61 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 23.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 45.0%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 45.0%

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

Alternative 2: 44.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;x-scale \leq 2.35 \cdot 10^{-166}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + -3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale}}{t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 1.0 (sin (* 0.005555555555555556 (* angle PI))))))
   (if (<= x-scale 2.35e-166)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (*
           -2.0
           (/
            (*
             y-scale
             (+ 1.0 (* -3.08641975308642e-5 (* (* angle angle) (* PI PI)))))
            x-scale))
          t_0)))
       PI))
     (* 180.0 (/ (atan (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) t_0))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 * sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (x_45_scale <= 2.35e-166) {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * ((y_45_scale * (1.0 + (-3.08641975308642e-5 * ((angle * angle) * (((double) M_PI) * ((double) M_PI)))))) / x_45_scale)) / t_0))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 * Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (x_45_scale <= 2.35e-166) {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * ((y_45_scale * (1.0 + (-3.08641975308642e-5 * ((angle * angle) * (Math.PI * Math.PI))))) / x_45_scale)) / t_0))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / t_0))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 1.0 * math.sin((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if x_45_scale <= 2.35e-166:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * ((y_45_scale * (1.0 + (-3.08641975308642e-5 * ((angle * angle) * (math.pi * math.pi))))) / x_45_scale)) / t_0))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / t_0))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(1.0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))
	tmp = 0.0
	if (x_45_scale <= 2.35e-166)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * Float64(1.0 + Float64(-3.08641975308642e-5 * Float64(Float64(angle * angle) * Float64(pi * pi))))) / x_45_scale)) / t_0))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / t_0))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 1.0 * sin((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (x_45_scale <= 2.35e-166)
		tmp = 180.0 * (atan((0.5 * ((-2.0 * ((y_45_scale * (1.0 + (-3.08641975308642e-5 * ((angle * angle) * (pi * pi))))) / x_45_scale)) / t_0))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(1.0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 2.35e-166], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(N[(y$45$scale * N[(1.0 + N[(-3.08641975308642e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.35000000000000007e-166

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 23.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 45.0%

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

   14. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       7. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       8. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + \frac{-1}{32400} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale}}{1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       9. lift-PI.f64 39.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + -3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale}}{1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   16. Applied rewrites 39.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale \cdot \left(1 + -3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x-scale}}{1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   if 2.35000000000000007e-166 < x-scale

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 23.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 45.0%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 45.0%

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

Alternative 3: 41.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.7 \cdot 10^{-233}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(2 \cdot \frac{\left(a \cdot a\right) \cdot x-scale}{angle \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(y-scale \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 7.7e-233)
   (*
    180.0
    (/
     (atan
      (*
       90.0
       (*
        2.0
        (/ (* (* a a) x-scale) (* angle (* (* b_m b_m) (* y-scale PI)))))))
     PI))
   (*
    180.0
    (/
     (atan
      (*
       0.5
       (/
        (* -2.0 (/ y-scale x-scale))
        (* 1.0 (sin (* 0.005555555555555556 (* angle PI)))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 7.7e-233) {
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 7.7e-233) {
		tmp = 180.0 * (Math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * Math.sin((0.005555555555555556 * (angle * Math.PI))))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 7.7e-233:
		tmp = 180.0 * (math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * math.sin((0.005555555555555556 * (angle * math.pi))))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 7.7e-233)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * x_45_scale) / Float64(angle * Float64(Float64(b_m * b_m) * Float64(y_45_scale * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(1.0 * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 7.7e-233)
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * pi))))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (1.0 * sin((0.005555555555555556 * (angle * pi))))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 7.7e-233], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.70000000000000007e-233

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 7.70000000000000007e-233 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 23.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 45.0%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 45.0%

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

Alternative 4: 39.9% accurate, 12.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale \cdot {1}^{2}}{x-scale}}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}{\pi}\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(2 \cdot \frac{\left(a \cdot a\right) \cdot x-scale}{angle \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(y-scale \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.9 \cdot 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_m \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{x-scale}}{angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          180.0
          (/
           (atan
            (*
             0.5
             (/
              (* -2.0 (/ (* y-scale (pow 1.0 2.0)) x-scale))
              (* 0.005555555555555556 (* angle PI)))))
           PI))))
   (if (<= b_m 1.6e-202)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          2.0
          (/ (* (* a a) x-scale) (* angle (* (* b_m b_m) (* y-scale PI)))))))
       PI))
     (if (<= b_m 3.9e-175)
       t_0
       (if (<= b_m 1.8e+114)
         (*
          180.0
          (/
           (atan
            (*
             90.0
             (/
              (* -2.0 (/ (* (* b_m b_m) y-scale) x-scale))
              (* angle (* PI (- (* b_m b_m) (* a a)))))))
           PI))
         t_0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (atan((0.5 * ((-2.0 * ((y_45_scale * pow(1.0, 2.0)) / x_45_scale)) / (0.005555555555555556 * (angle * ((double) M_PI)))))) / ((double) M_PI));
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * ((double) M_PI)))))))) / ((double) M_PI));
	} else if (b_m <= 3.9e-175) {
		tmp = t_0;
	} else if (b_m <= 1.8e+114) {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (((double) M_PI) * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (Math.atan((0.5 * ((-2.0 * ((y_45_scale * Math.pow(1.0, 2.0)) / x_45_scale)) / (0.005555555555555556 * (angle * Math.PI))))) / Math.PI);
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (Math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * Math.PI))))))) / Math.PI);
	} else if (b_m <= 3.9e-175) {
		tmp = t_0;
	} else if (b_m <= 1.8e+114) {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (Math.PI * ((b_m * b_m) - (a * a))))))) / Math.PI);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 180.0 * (math.atan((0.5 * ((-2.0 * ((y_45_scale * math.pow(1.0, 2.0)) / x_45_scale)) / (0.005555555555555556 * (angle * math.pi))))) / math.pi)
	tmp = 0
	if b_m <= 1.6e-202:
		tmp = 180.0 * (math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * math.pi))))))) / math.pi)
	elif b_m <= 3.9e-175:
		tmp = t_0
	elif b_m <= 1.8e+114:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (math.pi * ((b_m * b_m) - (a * a))))))) / math.pi)
	else:
		tmp = t_0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * (1.0 ^ 2.0)) / x_45_scale)) / Float64(0.005555555555555556 * Float64(angle * pi))))) / pi))
	tmp = 0.0
	if (b_m <= 1.6e-202)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * x_45_scale) / Float64(angle * Float64(Float64(b_m * b_m) * Float64(y_45_scale * pi))))))) / pi));
	elseif (b_m <= 3.9e-175)
		tmp = t_0;
	elseif (b_m <= 1.8e+114)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(Float64(b_m * b_m) * y_45_scale) / x_45_scale)) / Float64(angle * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 180.0 * (atan((0.5 * ((-2.0 * ((y_45_scale * (1.0 ^ 2.0)) / x_45_scale)) / (0.005555555555555556 * (angle * pi))))) / pi);
	tmp = 0.0;
	if (b_m <= 1.6e-202)
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * pi))))))) / pi);
	elseif (b_m <= 3.9e-175)
		tmp = t_0;
	elseif (b_m <= 1.8e+114)
		tmp = 180.0 * (atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (pi * ((b_m * b_m) - (a * a))))))) / pi);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(N[(y$45$scale * N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e-202], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.9e-175], t$95$0, If[LessEqual[b$95$m, 1.8e+114], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6000000000000001e-202

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 1.6000000000000001e-202 < b < 3.89999999999999998e-175 or 1.8e114 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 23.2%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.7%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 45.0%

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

   14. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 39.4

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

   16. Applied rewrites 39.4%

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

   if 3.89999999999999998e-175 < b < 1.8e114

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 24.5

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

   7. Applied rewrites 24.5%

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

Alternative 5: 39.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(x-scale \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(2 \cdot \frac{\left(a \cdot a\right) \cdot x-scale}{angle \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(y-scale \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.9 \cdot 10^{-175}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{y-scale \cdot y-scale}{t\_0}}{y-scale}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.5 \cdot 10^{+113}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{x-scale}}{angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* angle (* x-scale PI))))
   (if (<= b_m 1.6e-202)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          2.0
          (/ (* (* a a) x-scale) (* angle (* (* b_m b_m) (* y-scale PI)))))))
       PI))
     (if (<= b_m 3.9e-175)
       (*
        180.0
        (/
         (atan (* 90.0 (/ (* -2.0 (/ (* y-scale y-scale) t_0)) y-scale)))
         PI))
       (if (<= b_m 2.5e+113)
         (*
          180.0
          (/
           (atan
            (*
             90.0
             (/
              (* -2.0 (/ (* (* b_m b_m) y-scale) x-scale))
              (* angle (* PI (- (* b_m b_m) (* a a)))))))
           PI))
         (* 180.0 (/ (atan (* -180.0 (/ y-scale t_0))) PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * ((double) M_PI));
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * ((double) M_PI)))))))) / ((double) M_PI));
	} else if (b_m <= 3.9e-175) {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / ((double) M_PI));
	} else if (b_m <= 2.5e+113) {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (((double) M_PI) * ((b_m * b_m) - (a * a))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * Math.PI);
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (Math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * Math.PI))))))) / Math.PI);
	} else if (b_m <= 3.9e-175) {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / Math.PI);
	} else if (b_m <= 2.5e+113) {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (Math.PI * ((b_m * b_m) - (a * a))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / t_0))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = angle * (x_45_scale * math.pi)
	tmp = 0
	if b_m <= 1.6e-202:
		tmp = 180.0 * (math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * math.pi))))))) / math.pi)
	elif b_m <= 3.9e-175:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / math.pi)
	elif b_m <= 2.5e+113:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (math.pi * ((b_m * b_m) - (a * a))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / t_0))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle * Float64(x_45_scale * pi))
	tmp = 0.0
	if (b_m <= 1.6e-202)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * x_45_scale) / Float64(angle * Float64(Float64(b_m * b_m) * Float64(y_45_scale * pi))))))) / pi));
	elseif (b_m <= 3.9e-175)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi));
	elseif (b_m <= 2.5e+113)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(Float64(b_m * b_m) * y_45_scale) / x_45_scale)) / Float64(angle * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / t_0))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = angle * (x_45_scale * pi);
	tmp = 0.0;
	if (b_m <= 1.6e-202)
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * pi))))))) / pi);
	elseif (b_m <= 3.9e-175)
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi);
	elseif (b_m <= 2.5e+113)
		tmp = 180.0 * (atan((90.0 * ((-2.0 * (((b_m * b_m) * y_45_scale) / x_45_scale)) / (angle * (pi * ((b_m * b_m) - (a * a))))))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e-202], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.9e-175], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.5e+113], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.6000000000000001e-202

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 1.6000000000000001e-202 < b < 3.89999999999999998e-175

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

   if 3.89999999999999998e-175 < b < 2.5e113

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 24.5

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

   7. Applied rewrites 24.5%

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

   if 2.5e113 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 11.5%

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

   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 37.7

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

   7. Applied rewrites 37.7%

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

Alternative 6: 39.8% accurate, 14.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(x-scale \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(2 \cdot \frac{\left(a \cdot a\right) \cdot x-scale}{angle \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(y-scale \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.15 \cdot 10^{-170}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{y-scale \cdot y-scale}{t\_0}}{y-scale}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 0.0026:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(b\_m \cdot b\_m - a \cdot a\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* angle (* x-scale PI))))
   (if (<= b_m 1.6e-202)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          2.0
          (/ (* (* a a) x-scale) (* angle (* (* b_m b_m) (* y-scale PI)))))))
       PI))
     (if (<= b_m 1.15e-170)
       (*
        180.0
        (/
         (atan (* 90.0 (/ (* -2.0 (/ (* y-scale y-scale) t_0)) y-scale)))
         PI))
       (if (<= b_m 0.0026)
         (*
          180.0
          (/
           (atan
            (*
             -180.0
             (/
              (* (* b_m b_m) y-scale)
              (* angle (* x-scale (* PI (- (* b_m b_m) (* a a))))))))
           PI))
         (* 180.0 (/ (atan (* -180.0 (/ y-scale t_0))) PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * ((double) M_PI));
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * ((double) M_PI)))))))) / ((double) M_PI));
	} else if (b_m <= 1.15e-170) {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / ((double) M_PI));
	} else if (b_m <= 0.0026) {
		tmp = 180.0 * (atan((-180.0 * (((b_m * b_m) * y_45_scale) / (angle * (x_45_scale * (((double) M_PI) * ((b_m * b_m) - (a * a)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * Math.PI);
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (Math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * Math.PI))))))) / Math.PI);
	} else if (b_m <= 1.15e-170) {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / Math.PI);
	} else if (b_m <= 0.0026) {
		tmp = 180.0 * (Math.atan((-180.0 * (((b_m * b_m) * y_45_scale) / (angle * (x_45_scale * (Math.PI * ((b_m * b_m) - (a * a)))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / t_0))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = angle * (x_45_scale * math.pi)
	tmp = 0
	if b_m <= 1.6e-202:
		tmp = 180.0 * (math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * math.pi))))))) / math.pi)
	elif b_m <= 1.15e-170:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / math.pi)
	elif b_m <= 0.0026:
		tmp = 180.0 * (math.atan((-180.0 * (((b_m * b_m) * y_45_scale) / (angle * (x_45_scale * (math.pi * ((b_m * b_m) - (a * a)))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / t_0))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle * Float64(x_45_scale * pi))
	tmp = 0.0
	if (b_m <= 1.6e-202)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * x_45_scale) / Float64(angle * Float64(Float64(b_m * b_m) * Float64(y_45_scale * pi))))))) / pi));
	elseif (b_m <= 1.15e-170)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi));
	elseif (b_m <= 0.0026)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(Float64(Float64(b_m * b_m) * y_45_scale) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(Float64(b_m * b_m) - Float64(a * a)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / t_0))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = angle * (x_45_scale * pi);
	tmp = 0.0;
	if (b_m <= 1.6e-202)
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * pi))))))) / pi);
	elseif (b_m <= 1.15e-170)
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi);
	elseif (b_m <= 0.0026)
		tmp = 180.0 * (atan((-180.0 * (((b_m * b_m) * y_45_scale) / (angle * (x_45_scale * (pi * ((b_m * b_m) - (a * a)))))))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e-202], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.15e-170], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 0.0026], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(N[(b$95$m * b$95$m), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.6000000000000001e-202

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 1.6000000000000001e-202 < b < 1.14999999999999993e-170

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

   if 1.14999999999999993e-170 < b < 0.0025999999999999999

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 24.3

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

   7. Applied rewrites 24.3%

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

   if 0.0025999999999999999 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 11.5%

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

   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 37.7

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

   7. Applied rewrites 37.7%

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

Alternative 7: 38.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(x-scale \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(2 \cdot \frac{\left(a \cdot a\right) \cdot x-scale}{angle \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(y-scale \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{y-scale \cdot y-scale}{t\_0}}{y-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* angle (* x-scale PI))))
   (if (<= b_m 1.6e-202)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (*
          2.0
          (/ (* (* a a) x-scale) (* angle (* (* b_m b_m) (* y-scale PI)))))))
       PI))
     (if (<= b_m 1.6e-7)
       (*
        180.0
        (/
         (atan (* 90.0 (/ (* -2.0 (/ (* y-scale y-scale) t_0)) y-scale)))
         PI))
       (* 180.0 (/ (atan (* -180.0 (/ y-scale t_0))) PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * ((double) M_PI));
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * ((double) M_PI)))))))) / ((double) M_PI));
	} else if (b_m <= 1.6e-7) {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * Math.PI);
	double tmp;
	if (b_m <= 1.6e-202) {
		tmp = 180.0 * (Math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * Math.PI))))))) / Math.PI);
	} else if (b_m <= 1.6e-7) {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / t_0))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = angle * (x_45_scale * math.pi)
	tmp = 0
	if b_m <= 1.6e-202:
		tmp = 180.0 * (math.atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * math.pi))))))) / math.pi)
	elif b_m <= 1.6e-7:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / t_0))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle * Float64(x_45_scale * pi))
	tmp = 0.0
	if (b_m <= 1.6e-202)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * x_45_scale) / Float64(angle * Float64(Float64(b_m * b_m) * Float64(y_45_scale * pi))))))) / pi));
	elseif (b_m <= 1.6e-7)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / t_0))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = angle * (x_45_scale * pi);
	tmp = 0.0;
	if (b_m <= 1.6e-202)
		tmp = 180.0 * (atan((90.0 * (2.0 * (((a * a) * x_45_scale) / (angle * ((b_m * b_m) * (y_45_scale * pi))))))) / pi);
	elseif (b_m <= 1.6e-7)
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.6e-202], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-7], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6000000000000001e-202

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 1.6000000000000001e-202 < b < 1.6e-7

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

   if 1.6e-7 < b

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 11.5%

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

   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 37.7

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

   7. Applied rewrites 37.7%

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

Alternative 8: 38.8% accurate, 19.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := angle \cdot \left(x-scale \cdot \pi\right)\\ \mathbf{if}\;x-scale \leq 1.6 \cdot 10^{-59}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{-2 \cdot \frac{y-scale \cdot y-scale}{t\_0}}{y-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* angle (* x-scale PI))))
   (if (<= x-scale 1.6e-59)
     (* 180.0 (/ (atan (* -180.0 (/ y-scale t_0))) PI))
     (*
      180.0
      (/
       (atan (* 90.0 (/ (* -2.0 (/ (* y-scale y-scale) t_0)) y-scale)))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * ((double) M_PI));
	double tmp;
	if (x_45_scale <= 1.6e-59) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle * (x_45_scale * Math.PI);
	double tmp;
	if (x_45_scale <= 1.6e-59) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / t_0))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = angle * (x_45_scale * math.pi)
	tmp = 0
	if x_45_scale <= 1.6e-59:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / t_0))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle * Float64(x_45_scale * pi))
	tmp = 0.0
	if (x_45_scale <= 1.6e-59)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / t_0))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(-2.0 * Float64(Float64(y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = angle * (x_45_scale * pi);
	tmp = 0.0;
	if (x_45_scale <= 1.6e-59)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / t_0))) / pi);
	else
		tmp = 180.0 * (atan((90.0 * ((-2.0 * ((y_45_scale * y_45_scale) / t_0)) / y_45_scale))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 1.6e-59], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(-2.0 * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.6e-59

   1. Initial program 13.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 11.5%

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

   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 37.7

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

   7. Applied rewrites 37.7%

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

   if 1.6e-59 < x-scale

   1. Initial program 13.9%

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

   2. Taylor expanded in 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 11.5%

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

   5. Taylor expanded in y-scale around 0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

Alternative 9: 37.7% accurate, 28.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-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 13.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 11.5%

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

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 37.7

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

7. Applied rewrites 37.7%

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

Alternative 10: 12.2% accurate, 28.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-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 13.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 11.5%

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

5. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\pi} \]
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 12.2

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

7. Applied rewrites 12.2%

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

Reproduce

herbie shell --seed 2025124 
(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)))