raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.6% → 56.5%

Time: 1.1min

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.5% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 3.05 \cdot 10^{-54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(2 \cdot \frac{y-scale \cdot t\_0}{x-scale \cdot t\_1}\right)\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot t\_1}{x-scale \cdot t\_0}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{\sin \left(0.5 \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_1 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 3.05e-54)
     (*
      180.0
      (/ (atan (* -0.5 (* 2.0 (/ (* y-scale t_0) (* x-scale t_1))))) PI))
     (if (<= a_m 5.2e+153)
       (*
        180.0
        (/ (atan (* -0.5 (* -2.0 (/ (* y-scale t_1) (* x-scale t_0))))) PI))
       (*
        180.0
        (/
         (atan
          (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_1 (sin (* 0.5 PI)))))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 3.05e-54) {
		tmp = 180.0 * (atan((-0.5 * (2.0 * ((y_45_scale * t_0) / (x_45_scale * t_1))))) / ((double) M_PI));
	} else if (a_m <= 5.2e+153) {
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * t_1) / (x_45_scale * t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / sin((0.5 * ((double) M_PI)))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 3.05e-54)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(2.0 * Float64(Float64(y_45_scale * t_0) / Float64(x_45_scale * t_1))))) / pi));
	elseif (a_m <= 5.2e+153)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / sin(Float64(0.5 * pi))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 3.05e-54], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(2.0 * N[(N[(y$45$scale * t$95$0), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 5.2e+153], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.05000000000000006e-54

   1. Initial program 16.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if 3.05000000000000006e-54 < a < 5.1999999999999998e153

   1. Initial program 15.6%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 61.1%

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

   if 5.1999999999999998e153 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 57.6%

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

   9. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 63.8

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

   11. Applied rewrites 63.8%

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

Alternative 2: 56.4% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 6.6 \cdot 10^{-54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\mathsf{fma}\left(t\_1, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\right)}{t\_1}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a_m 6.6e-54)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           2.0
           (/
            (fma
             t_1
             (sin (fma 0.5 PI (/ PI 2.0)))
             (* (cos t_0) (sin (* 0.5 PI))))
            t_1)))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           -2.0
           (/ t_1 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a_m <= 6.6e-54) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (fma(t_1, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), (cos(t_0) * sin((0.5 * ((double) M_PI))))) / t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (a_m <= 6.6e-54)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(2.0 * Float64(fma(t_1, sin(fma(0.5, pi, Float64(pi / 2.0))), Float64(cos(t_0) * sin(Float64(0.5 * pi)))) / t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 6.6e-54], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(2.0 * N[(N[(t$95$1 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.59999999999999986e-54

   1. Initial program 16.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 46.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 49.4%

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

   if 6.59999999999999986e-54 < a

   1. Initial program 8.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 61.2%

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

Alternative 3: 55.6% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 3.05 \cdot 10^{-54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(2 \cdot \frac{y-scale \cdot t\_0}{x-scale \cdot t\_1}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{t\_0}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_1 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 3.05e-54)
     (*
      180.0
      (/ (atan (* -0.5 (* 2.0 (/ (* y-scale t_0) (* x-scale t_1))))) PI))
     (*
      180.0
      (/ (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_1 t_0))))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 3.05e-54) {
		tmp = 180.0 * (atan((-0.5 * (2.0 * ((y_45_scale * t_0) / (x_45_scale * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 3.05e-54)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(2.0 * Float64(Float64(y_45_scale * t_0) / Float64(x_45_scale * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / t_0))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 3.05e-54], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(2.0 * N[(N[(y$45$scale * t$95$0), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.05000000000000006e-54

   1. Initial program 16.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if 3.05000000000000006e-54 < a

   1. Initial program 8.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 61.2%

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

Alternative 4: 55.2% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 3.05 \cdot 10^{-54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(2 \cdot \frac{y-scale \cdot t\_0}{x-scale \cdot t\_1}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot t\_1}{x-scale \cdot t\_0}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_1 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 3.05e-54)
     (*
      180.0
      (/ (atan (* -0.5 (* 2.0 (/ (* y-scale t_0) (* x-scale t_1))))) PI))
     (*
      180.0
      (/ (atan (* -0.5 (* -2.0 (/ (* y-scale t_1) (* x-scale t_0))))) PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 3.05e-54) {
		tmp = 180.0 * (atan((-0.5 * (2.0 * ((y_45_scale * t_0) / (x_45_scale * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * t_1) / (x_45_scale * t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 3.05e-54)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(2.0 * Float64(Float64(y_45_scale * t_0) / Float64(x_45_scale * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * t_0))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 3.05e-54], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(2.0 * N[(N[(y$45$scale * t$95$0), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.05000000000000006e-54

   1. Initial program 16.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if 3.05000000000000006e-54 < a

   1. Initial program 8.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 58.3%

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

Alternative 5: 45.0% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)}{\pi} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/
   (atan
    (*
     -0.5
     (*
      -2.0
      (/
       (* y-scale (sin (* 0.005555555555555556 (* angle PI))))
       (* x-scale (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))))))
   PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * sin((0.005555555555555556 * (angle * ((double) M_PI))))) / (x_45_scale * sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))))))))) / ((double) M_PI));
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) / Float64(x_45_scale * sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))))))) / pi))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.2%

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

3. Taylor expanded in x-scale around 0

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

5. Applied rewrites 32.2%

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

6. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

8. Applied rewrites 42.6%

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

Alternative 6: 29.4% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := {t\_0}^{2}\\ t_2 := \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\\ t_3 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_4 := {t\_3}^{2}\\ t_5 := \frac{t\_1 \cdot t\_4}{{x-scale}^{2}}\\ t_6 := \sin \left(0.5 \cdot \pi\right)\\ t_7 := {t\_6}^{2}\\ t_8 := t\_6 \cdot t\_6\\ t_9 := \mathsf{fma}\left(-2, t\_5, 4 \cdot t\_5\right)\\ t_10 := \frac{{t\_0}^{4}}{{x-scale}^{4}} - 0.25 \cdot \frac{{t\_9}^{2}}{{t\_3}^{4}}\\ t_11 := \frac{\left(\pi \cdot \pi\right) \cdot t\_8}{y-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq -1.4 \cdot 10^{-79} \lor \neg \left(x-scale \leq 1.42 \cdot 10^{-78}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \mathsf{fma}\left(-1, \frac{x-scale \cdot \mathsf{fma}\left(0.5, \frac{t\_9}{t\_4}, \frac{t\_1}{{x-scale}^{2}}\right)}{t\_0 \cdot t\_3}, {y-scale}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{x-scale \cdot t\_10}{t\_0 \cdot {t\_3}^{3}}, 0.25 \cdot \frac{x-scale \cdot \left({y-scale}^{2} \cdot \left(t\_9 \cdot t\_10\right)\right)}{t\_0 \cdot {t\_3}^{7}}\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot t\_7}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2}}{{x-scale}^{2}}, 0.5 \cdot \frac{\frac{\mathsf{fma}\left(2, \left(x-scale \cdot x-scale\right) \cdot \left(t\_8 \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, t\_11, 3.08641975308642 \cdot 10^{-5} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(t\_2 \cdot t\_2\right)}{y-scale \cdot y-scale}\right)\right), \left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_11, 0.0001234567901234568 \cdot t\_11\right)\right)}{x-scale \cdot x-scale}}{t\_7}\right)\right)\right)\right)}{\pi \cdot t\_6}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (pow t_0 2.0))
        (t_2 (sin (fma 0.5 PI (/ PI 2.0))))
        (t_3 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_4 (pow t_3 2.0))
        (t_5 (/ (* t_1 t_4) (pow x-scale 2.0)))
        (t_6 (sin (* 0.5 PI)))
        (t_7 (pow t_6 2.0))
        (t_8 (* t_6 t_6))
        (t_9 (fma -2.0 t_5 (* 4.0 t_5)))
        (t_10
         (-
          (/ (pow t_0 4.0) (pow x-scale 4.0))
          (* 0.25 (/ (pow t_9 2.0) (pow t_3 4.0)))))
        (t_11 (/ (* (* PI PI) t_8) (* y-scale y-scale))))
   (if (or (<= x-scale -1.4e-79) (not (<= x-scale 1.42e-78)))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          y-scale
          (fma
           -1.0
           (/
            (* x-scale (fma 0.5 (/ t_9 t_4) (/ t_1 (pow x-scale 2.0))))
            (* t_0 t_3))
           (*
            (pow y-scale 2.0)
            (fma
             -0.5
             (/ (* x-scale t_10) (* t_0 (pow t_3 3.0)))
             (*
              0.25
              (/
               (* x-scale (* (pow y-scale 2.0) (* t_9 t_10)))
               (* t_0 (pow t_3 7.0))))))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          180.0
          (/
           (*
            angle
            (*
             x-scale
             (*
              y-scale
              (-
               (fma
                -3.08641975308642e-5
                (/ (* (pow PI 2.0) t_7) (pow y-scale 2.0))
                (*
                 3.08641975308642e-5
                 (/
                  (* (pow PI 2.0) (pow (cos (* 0.5 PI)) 2.0))
                  (pow y-scale 2.0))))
               (fma
                3.08641975308642e-5
                (/ (pow PI 2.0) (pow x-scale 2.0))
                (*
                 0.5
                 (/
                  (/
                   (fma
                    2.0
                    (*
                     (* x-scale x-scale)
                     (*
                      t_8
                      (fma
                       -3.08641975308642e-5
                       t_11
                       (*
                        3.08641975308642e-5
                        (/ (* (* PI PI) (* t_2 t_2)) (* y-scale y-scale))))))
                    (*
                     (* y-scale y-scale)
                     (fma
                      -6.17283950617284e-5
                      t_11
                      (* 0.0001234567901234568 t_11))))
                   (* x-scale x-scale))
                  t_7)))))))
           (* PI t_6)))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = pow(t_0, 2.0);
	double t_2 = sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0)));
	double t_3 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_4 = pow(t_3, 2.0);
	double t_5 = (t_1 * t_4) / pow(x_45_scale, 2.0);
	double t_6 = sin((0.5 * ((double) M_PI)));
	double t_7 = pow(t_6, 2.0);
	double t_8 = t_6 * t_6;
	double t_9 = fma(-2.0, t_5, (4.0 * t_5));
	double t_10 = (pow(t_0, 4.0) / pow(x_45_scale, 4.0)) - (0.25 * (pow(t_9, 2.0) / pow(t_3, 4.0)));
	double t_11 = ((((double) M_PI) * ((double) M_PI)) * t_8) / (y_45_scale * y_45_scale);
	double tmp;
	if ((x_45_scale <= -1.4e-79) || !(x_45_scale <= 1.42e-78)) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * fma(-1.0, ((x_45_scale * fma(0.5, (t_9 / t_4), (t_1 / pow(x_45_scale, 2.0)))) / (t_0 * t_3)), (pow(y_45_scale, 2.0) * fma(-0.5, ((x_45_scale * t_10) / (t_0 * pow(t_3, 3.0))), (0.25 * ((x_45_scale * (pow(y_45_scale, 2.0) * (t_9 * t_10))) / (t_0 * pow(t_3, 7.0)))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (180.0 * ((angle * (x_45_scale * (y_45_scale * (fma(-3.08641975308642e-5, ((pow(((double) M_PI), 2.0) * t_7) / pow(y_45_scale, 2.0)), (3.08641975308642e-5 * ((pow(((double) M_PI), 2.0) * pow(cos((0.5 * ((double) M_PI))), 2.0)) / pow(y_45_scale, 2.0)))) - fma(3.08641975308642e-5, (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0)), (0.5 * ((fma(2.0, ((x_45_scale * x_45_scale) * (t_8 * fma(-3.08641975308642e-5, t_11, (3.08641975308642e-5 * (((((double) M_PI) * ((double) M_PI)) * (t_2 * t_2)) / (y_45_scale * y_45_scale)))))), ((y_45_scale * y_45_scale) * fma(-6.17283950617284e-5, t_11, (0.0001234567901234568 * t_11)))) / (x_45_scale * x_45_scale)) / t_7))))))) / (((double) M_PI) * t_6))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = t_0 ^ 2.0
	t_2 = sin(fma(0.5, pi, Float64(pi / 2.0)))
	t_3 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_4 = t_3 ^ 2.0
	t_5 = Float64(Float64(t_1 * t_4) / (x_45_scale ^ 2.0))
	t_6 = sin(Float64(0.5 * pi))
	t_7 = t_6 ^ 2.0
	t_8 = Float64(t_6 * t_6)
	t_9 = fma(-2.0, t_5, Float64(4.0 * t_5))
	t_10 = Float64(Float64((t_0 ^ 4.0) / (x_45_scale ^ 4.0)) - Float64(0.25 * Float64((t_9 ^ 2.0) / (t_3 ^ 4.0))))
	t_11 = Float64(Float64(Float64(pi * pi) * t_8) / Float64(y_45_scale * y_45_scale))
	tmp = 0.0
	if ((x_45_scale <= -1.4e-79) || !(x_45_scale <= 1.42e-78))
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * fma(-1.0, Float64(Float64(x_45_scale * fma(0.5, Float64(t_9 / t_4), Float64(t_1 / (x_45_scale ^ 2.0)))) / Float64(t_0 * t_3)), Float64((y_45_scale ^ 2.0) * fma(-0.5, Float64(Float64(x_45_scale * t_10) / Float64(t_0 * (t_3 ^ 3.0))), Float64(0.25 * Float64(Float64(x_45_scale * Float64((y_45_scale ^ 2.0) * Float64(t_9 * t_10))) / Float64(t_0 * (t_3 ^ 7.0)))))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(180.0 * Float64(Float64(angle * Float64(x_45_scale * Float64(y_45_scale * Float64(fma(-3.08641975308642e-5, Float64(Float64((pi ^ 2.0) * t_7) / (y_45_scale ^ 2.0)), Float64(3.08641975308642e-5 * Float64(Float64((pi ^ 2.0) * (cos(Float64(0.5 * pi)) ^ 2.0)) / (y_45_scale ^ 2.0)))) - fma(3.08641975308642e-5, Float64((pi ^ 2.0) / (x_45_scale ^ 2.0)), Float64(0.5 * Float64(Float64(fma(2.0, Float64(Float64(x_45_scale * x_45_scale) * Float64(t_8 * fma(-3.08641975308642e-5, t_11, Float64(3.08641975308642e-5 * Float64(Float64(Float64(pi * pi) * Float64(t_2 * t_2)) / Float64(y_45_scale * y_45_scale)))))), Float64(Float64(y_45_scale * y_45_scale) * fma(-6.17283950617284e-5, t_11, Float64(0.0001234567901234568 * t_11)))) / Float64(x_45_scale * x_45_scale)) / t_7))))))) / Float64(pi * t_6))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 * t$95$4), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$6, 2.0], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 * t$95$6), $MachinePrecision]}, Block[{t$95$9 = N[(-2.0 * t$95$5 + N[(4.0 * t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[Power[t$95$9, 2.0], $MachinePrecision] / N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[(N[(Pi * Pi), $MachinePrecision] * t$95$8), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x$45$scale, -1.4e-79], N[Not[LessEqual[x$45$scale, 1.42e-78]], $MachinePrecision]], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(-1.0 * N[(N[(x$45$scale * N[(0.5 * N[(t$95$9 / t$95$4), $MachinePrecision] + N[(t$95$1 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[(-0.5 * N[(N[(x$45$scale * t$95$10), $MachinePrecision] / N[(t$95$0 * N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(N[(x$45$scale * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[(t$95$9 * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Power[t$95$3, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(180.0 * N[(N[(angle * N[(x$45$scale * N[(y$45$scale * N[(N[(-3.08641975308642e-5 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * t$95$7), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[(2.0 * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(t$95$8 * N[(-3.08641975308642e-5 * t$95$11 + N[(3.08641975308642e-5 * N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(-6.17283950617284e-5 * t$95$11 + N[(0.0001234567901234568 * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -1.40000000000000006e-79 or 1.41999999999999999e-78 < x-scale

   1. Initial program 13.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \color{blue}{\left(-1 \cdot \frac{x-scale \cdot \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{x-scale}^{2}}\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + {y-scale}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{x-scale \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}} + \frac{1}{4} \cdot \frac{x-scale \cdot \left({y-scale}^{2} \cdot \left(\left(-2 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}} + 4 \cdot \frac{{\sin \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) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{7}}\right)\right)}\right)\right)}{\pi} \]

   8. Applied rewrites 32.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x-scale \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, {y-scale}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{x-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}} - 0.25 \cdot \frac{{\left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{2}}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{3}}, 0.25 \cdot \frac{x-scale \cdot \left({y-scale}^{2} \cdot \left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}} - 0.25 \cdot \frac{{\left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{2}}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{7}}\right)\right)}\right)\right)}{\pi} \]

   if -1.40000000000000006e-79 < x-scale < 1.41999999999999999e-78

   1. Initial program 15.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(180 \cdot \color{blue}{\frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\left(\frac{-1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right) - \left(\frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{{y-scale}^{2} \cdot \left(-2 \cdot \frac{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(\frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}} - \left(\frac{-1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}{{y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]

   8. Applied rewrites 3.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \color{blue}{\frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2}}{{x-scale}^{2}}, 0.5 \cdot \frac{{y-scale}^{2} \cdot \mathsf{fma}\left(-2, \frac{{\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}} - \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right)\right)}{{y-scale}^{2}}, 0.0001234567901234568 \cdot \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}{{\sin \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(180 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(\frac{-1}{32400}, \frac{{\pi}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{y-scale}^{2}}, \frac{1}{32400} \cdot \frac{{\pi}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(\frac{1}{32400}, \frac{{\pi}^{2}}{{x-scale}^{2}}, \frac{1}{2} \cdot \frac{\frac{2 \cdot \left({x-scale}^{2} \cdot \left({\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)\right) + {y-scale}^{2} \cdot \left(\frac{-1}{16200} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{x-scale}^{2}}}{{\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{\pi \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}\right)\right)}{\pi} \]

   11. Applied rewrites 21.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2}}{{x-scale}^{2}}, 0.5 \cdot \frac{\frac{\mathsf{fma}\left(2, \left(x-scale \cdot x-scale\right) \cdot \left(\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)}{y-scale \cdot y-scale}\right)\right), \left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 0.0001234567901234568 \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)}{x-scale \cdot x-scale}}{{\sin \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.4 \cdot 10^{-79} \lor \neg \left(x-scale \leq 1.42 \cdot 10^{-78}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \mathsf{fma}\left(-1, \frac{x-scale \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, {y-scale}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{x-scale \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}} - 0.25 \cdot \frac{{\left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{2}}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{3}}, 0.25 \cdot \frac{x-scale \cdot \left({y-scale}^{2} \cdot \left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{x-scale}^{4}} - 0.25 \cdot \frac{{\left(\mathsf{fma}\left(-2, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}, 4 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{2}}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{7}}\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2}}{{x-scale}^{2}}, 0.5 \cdot \frac{\frac{\mathsf{fma}\left(2, \left(x-scale \cdot x-scale\right) \cdot \left(\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)}{y-scale \cdot y-scale}\right)\right), \left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 0.0001234567901234568 \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)}{x-scale \cdot x-scale}}{{\sin \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 19.1% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \frac{\left(\pi \cdot \pi\right) \cdot t\_1}{y-scale \cdot y-scale}\\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \left(\left(-1 \cdot \frac{angle}{x-scale}\right) \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_2, 0.0001234567901234568 \cdot t\_2\right)}{t\_1}\right)}{\pi \cdot t\_0}\right)\right)\right)}{\pi} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI)))
        (t_1 (* t_0 t_0))
        (t_2 (/ (* (* PI PI) t_1) (* y-scale y-scale))))
   (*
    180.0
    (/
     (atan
      (*
       -0.5
       (*
        180.0
        (*
         (* -1.0 (/ angle x-scale))
         (/
          (*
           y-scale
           (fma
            3.08641975308642e-5
            (* PI PI)
            (*
             0.5
             (/
              (*
               (* y-scale y-scale)
               (fma -6.17283950617284e-5 t_2 (* 0.0001234567901234568 t_2)))
              t_1))))
          (* PI t_0))))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double t_1 = t_0 * t_0;
	double t_2 = ((((double) M_PI) * ((double) M_PI)) * t_1) / (y_45_scale * y_45_scale);
	return 180.0 * (atan((-0.5 * (180.0 * ((-1.0 * (angle / x_45_scale)) * ((y_45_scale * fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (((y_45_scale * y_45_scale) * fma(-6.17283950617284e-5, t_2, (0.0001234567901234568 * t_2))) / t_1)))) / (((double) M_PI) * t_0)))))) / ((double) M_PI));
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(Float64(pi * pi) * t_1) / Float64(y_45_scale * y_45_scale))
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(180.0 * Float64(Float64(-1.0 * Float64(angle / x_45_scale)) * Float64(Float64(y_45_scale * fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(Float64(Float64(y_45_scale * y_45_scale) * fma(-6.17283950617284e-5, t_2, Float64(0.0001234567901234568 * t_2))) / t_1)))) / Float64(pi * t_0)))))) / pi))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(Pi * Pi), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(180.0 * N[(N[(-1.0 * N[(angle / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(y$45$scale * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(-6.17283950617284e-5 * t$95$2 + N[(0.0001234567901234568 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 14.2%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 16.0%

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

6. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(180 \cdot \color{blue}{\frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\left(\frac{-1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right) - \left(\frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{{y-scale}^{2} \cdot \left(-2 \cdot \frac{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(\frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}} - \left(\frac{-1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}{{y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]

8. Applied rewrites 6.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \color{blue}{\frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right) - \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2}}{{x-scale}^{2}}, 0.5 \cdot \frac{{y-scale}^{2} \cdot \mathsf{fma}\left(-2, \frac{{\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}} - \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}, 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2} \cdot {\cos \left(0.5 \cdot \pi\right)}^{2}}{{y-scale}^{2}}\right)\right)}{{y-scale}^{2}}, 0.0001234567901234568 \cdot \frac{{\pi}^{2} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}{{\sin \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}\right)\right)}{\pi} \]

9. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(180 \cdot \left(-1 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{{y-scale}^{2} \cdot \left(\frac{-1}{16200} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{\color{blue}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)\right)}{\pi} \]

11. Applied rewrites 19.5%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \left(-1 \cdot \left(\frac{angle}{x-scale} \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 0.0001234567901234568 \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)}{\pi} \]

12. Final simplification 19.5%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(180 \cdot \left(\left(-1 \cdot \frac{angle}{x-scale}\right) \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}, 0.0001234567901234568 \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)}{\pi} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))