raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.6% → 54.5%

Time: 1.5min

Alternatives: 5

Speedup: 22.2×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 7.6 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale}{x-scale \cdot t\_1}\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 8.5 \cdot 10^{+123}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(0.005555555555555556 \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{\cos t\_0}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a_m 7.6e-119)
     (* 180.0 (/ (atan (* -1.0 (/ y-scale (* x-scale t_1)))) PI))
     (if (<= a_m 8.5e+123)
       (*
        180.0
        (/
         (atan (* angle (* 0.005555555555555556 (/ (* y-scale PI) x-scale))))
         PI))
       (*
        180.0
        (/
         (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_1 (cos t_0))))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a_m <= 7.6e-119) {
		tmp = 180.0 * (atan((-1.0 * (y_45_scale / (x_45_scale * t_1)))) / ((double) M_PI));
	} else if (a_m <= 8.5e+123) {
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / cos(t_0)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (a_m <= 7.6e-119) {
		tmp = 180.0 * (Math.atan((-1.0 * (y_45_scale / (x_45_scale * t_1)))) / Math.PI);
	} else if (a_m <= 8.5e+123) {
		tmp = 180.0 * (Math.atan((angle * (0.005555555555555556 * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / Math.cos(t_0)))))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	tmp = 0
	if a_m <= 7.6e-119:
		tmp = 180.0 * (math.atan((-1.0 * (y_45_scale / (x_45_scale * t_1)))) / math.pi)
	elif a_m <= 8.5e+123:
		tmp = 180.0 * (math.atan((angle * (0.005555555555555556 * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / math.cos(t_0)))))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (a_m <= 7.6e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(y_45_scale / Float64(x_45_scale * t_1)))) / pi));
	elseif (a_m <= 8.5e+123)
		tmp = Float64(180.0 * Float64(atan(Float64(angle * Float64(0.005555555555555556 * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / 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 / cos(t_0)))))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (a_m <= 7.6e-119)
		tmp = 180.0 * (atan((-1.0 * (y_45_scale / (x_45_scale * t_1)))) / pi);
	elseif (a_m <= 8.5e+123)
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * pi) / x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / cos(t_0)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 7.6e-119], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(y$45$scale / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 8.5e+123], N[(180.0 * N[(N[ArcTan[N[(angle * N[(0.005555555555555556 * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $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[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.59999999999999949e-119

   1. Initial program 18.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 58.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 58.7%

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

   if 7.59999999999999949e-119 < a < 8.5e123

   1. Initial program 19.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 42.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      7. lift-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      8. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   10. Applied rewrites 34.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \color{blue}{\frac{y-scale \cdot \pi}{x-scale}}, {angle}^{2} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale} - -8.573388203017833 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-/.f64 44.1

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

   13. Applied rewrites 44.1%

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

   if 8.5e123 < a

   1. Initial program 1.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 16.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 63.6

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

   7. Applied rewrites 63.6%

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

Alternative 2: 54.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.6 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{y-scale}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(0.005555555555555556 \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 7.6e-119)
   (*
    180.0
    (/
     (atan
      (*
       -1.0
       (/ y-scale (* x-scale (sin (* 0.005555555555555556 (* angle PI)))))))
     PI))
   (*
    180.0
    (/
     (atan (* angle (* 0.005555555555555556 (/ (* y-scale PI) x-scale))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 7.6e-119) {
		tmp = 180.0 * (atan((-1.0 * (y_45_scale / (x_45_scale * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 7.6e-119) {
		tmp = 180.0 * (Math.atan((-1.0 * (y_45_scale / (x_45_scale * Math.sin((0.005555555555555556 * (angle * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((angle * (0.005555555555555556 * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 7.6e-119:
		tmp = 180.0 * (math.atan((-1.0 * (y_45_scale / (x_45_scale * math.sin((0.005555555555555556 * (angle * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((angle * (0.005555555555555556 * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 7.6e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(-1.0 * Float64(y_45_scale / Float64(x_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(angle * Float64(0.005555555555555556 * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 7.6e-119)
		tmp = 180.0 * (atan((-1.0 * (y_45_scale / (x_45_scale * sin((0.005555555555555556 * (angle * pi))))))) / pi);
	else
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * pi) / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 7.6e-119], N[(180.0 * N[(N[ArcTan[N[(-1.0 * N[(y$45$scale / N[(x$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(angle * N[(0.005555555555555556 * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.59999999999999949e-119

   1. Initial program 18.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 58.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 58.7%

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

   if 7.59999999999999949e-119 < a

   1. Initial program 11.6%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 51.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      7. lift-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      8. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   10. Applied rewrites 41.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \color{blue}{\frac{y-scale \cdot \pi}{x-scale}}, {angle}^{2} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale} - -8.573388203017833 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-/.f64 52.3

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

   13. Applied rewrites 52.3%

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

Alternative 3: 52.4% accurate, 21.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 3.2 \cdot 10^{-122}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(0.005555555555555556 \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 3.2e-122)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (*
    180.0
    (/
     (atan (* angle (* 0.005555555555555556 (/ (* y-scale PI) x-scale))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 3.2e-122) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 3.2e-122) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((angle * (0.005555555555555556 * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 3.2e-122:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((angle * (0.005555555555555556 * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 3.2e-122)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(angle * Float64(0.005555555555555556 * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 3.2e-122)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	else
		tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * pi) / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 3.2e-122], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(angle * N[(0.005555555555555556 * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2000000000000002e-122

   1. Initial program 18.0%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if 3.2000000000000002e-122 < a

   1. Initial program 11.6%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 31.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      7. lift-pow.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

      8. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   10. Applied rewrites 41.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \color{blue}{\frac{y-scale \cdot \pi}{x-scale}}, {angle}^{2} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale} - -8.573388203017833 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-/.f64 52.2

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

   13. Applied rewrites 52.2%

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

Alternative 4: 45.5% accurate, 22.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* angle (* 0.005555555555555556 (/ (* y-scale PI) x-scale)))) 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((angle * (0.005555555555555556 * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((angle * (0.005555555555555556 * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((angle * (0.005555555555555556 * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)

Julia

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

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((angle * (0.005555555555555556 * ((y_45_scale * pi) / x_45_scale)))) / 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[(angle * N[(0.005555555555555556 * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 31.4%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

7. Applied rewrites 45.3%

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

8. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)}\right)\right)}{\pi} \]

   2. lower-fma.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   3. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   4. lift-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   5. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   6. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   7. lift-pow.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

   8. lower--.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \frac{y-scale \cdot \pi}{x-scale}, {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale} - \frac{-1}{11664000} \cdot \frac{y-scale \cdot {\mathsf{PI}\left(\right)}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

10. Applied rewrites 35.9%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \color{blue}{\frac{y-scale \cdot \pi}{x-scale}}, {angle}^{2} \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale} - -8.573388203017833 \cdot 10^{-8} \cdot \frac{y-scale \cdot {\pi}^{3}}{x-scale}\right)\right)\right)}{\pi} \]

11. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-/.f64 45.5

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

13. Applied rewrites 45.5%

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

Alternative 5: 40.5% accurate, 22.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* 0.005555555555555556 (/ (* angle (* y-scale PI)) x-scale))) 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((0.005555555555555556 * ((angle * (y_45_scale * ((double) M_PI))) / x_45_scale))) / ((double) M_PI));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((0.005555555555555556 * ((angle * (y_45_scale * Math.PI)) / x_45_scale))) / Math.PI);
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((0.005555555555555556 * ((angle * (y_45_scale * math.pi)) / x_45_scale))) / math.pi)

Julia

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

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((0.005555555555555556 * ((angle * (y_45_scale * pi)) / x_45_scale))) / 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[(0.005555555555555556 * N[(N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 31.4%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

7. Applied rewrites 45.3%

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

8. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 40.5

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

10. Applied rewrites 40.5%

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

Reproduce

herbie shell --seed 2025092 
(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)))