raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.8% → 56.4%

Time: 1.5min

Alternatives: 13

Speedup: 26.0×

• could not determine a ground truth

  1. a = -2.130813219389047e-79
  2. b = -6.348321620857184e-280
  3. angle = -365779940662425600.0
  4. x-scale = 6.5559733358656365e-115
  5. y-scale = -3.427165970179762e-145

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: 13.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\sin t\_0}{\cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan t\_0}}{x-scale}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 4.2e+18)
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (/ (sin t_0) (cos t_0)))) PI))
     (* 180.0 (/ (atan (/ (/ y-scale (tan t_0)) x-scale)) (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 4.2e+18) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (sin(t_0) / cos(t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 4.2e+18) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * (Math.sin(t_0) / Math.cos(t_0)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / Math.tan(t_0)) / x_45_scale)) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 4.2e+18:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * (math.sin(t_0) / math.cos(t_0)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / math.tan(t_0)) / x_45_scale)) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 4.2e+18)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(sin(t_0) / cos(t_0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / tan(t_0)) / x_45_scale)) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 4.2e+18)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (sin(t_0) / cos(t_0)))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e+18], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] / N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.2e18

   1. Initial program 12.8%

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

   3. Simplified 14.0%

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

   5. Taylor expanded in x-scale around 0 29.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in a around inf 44.5%

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

      1. times-frac 45.9%

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

   10. Simplified 45.9%

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

   if 4.2e18 < b

   1. Initial program 13.0%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in x-scale around 0 27.4%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. times-frac 52.3%

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

   10. Simplified 52.3%

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

       1. associate-*r/ 52.3%

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

       2. atan-neg 52.3%

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

       3. clear-num 52.3%

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

       4. quot-tan 52.3%

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

   12. Applied egg-rr 52.3%

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

       1. associate-*r/ 52.3%

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

       2. associate-*l/ 55.7%

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

       3. associate-*r/ 55.7%

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

   14. Simplified 55.7%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan t\_0}}{x-scale}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 1.25e+18)
     (* 180.0 (/ (atan (* y-scale (/ (sin t_0) (* x-scale (cos t_0))))) PI))
     (* 180.0 (/ (atan (/ (/ y-scale (tan t_0)) x-scale)) (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 1.25e+18) {
		tmp = 180.0 * (atan((y_45_scale * (sin(t_0) / (x_45_scale * cos(t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 1.25e+18) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.sin(t_0) / (x_45_scale * Math.cos(t_0))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / Math.tan(t_0)) / x_45_scale)) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 1.25e+18:
		tmp = 180.0 * (math.atan((y_45_scale * (math.sin(t_0) / (x_45_scale * math.cos(t_0))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / math.tan(t_0)) / x_45_scale)) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 1.25e+18)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(sin(t_0) / Float64(x_45_scale * cos(t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / tan(t_0)) / x_45_scale)) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 1.25e+18)
		tmp = 180.0 * (atan((y_45_scale * (sin(t_0) / (x_45_scale * cos(t_0))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.25e+18], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Sin[t$95$0], $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25e18

   1. Initial program 12.8%

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

   3. Simplified 14.0%

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

   5. Taylor expanded in b around 0 4.1%

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

   6. Taylor expanded in y-scale around inf 23.5%

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

      1. add-sqr-sqrt 23.2%

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

   8. Applied egg-rr 23.2%

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

   9. Taylor expanded in a around inf 44.5%

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

       1. associate-/l* 44.5%

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

   11. Simplified 44.5%

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

   if 1.25e18 < b

   1. Initial program 13.0%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in x-scale around 0 27.4%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in a around 0 55.6%

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

      1. mul-1-neg 55.6%

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

      2. times-frac 52.3%

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

   10. Simplified 52.3%

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

       1. associate-*r/ 52.3%

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

       2. atan-neg 52.3%

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

       3. clear-num 52.3%

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

       4. quot-tan 52.3%

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

   12. Applied egg-rr 52.3%

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

       1. associate-*r/ 52.3%

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

       2. associate-*l/ 55.7%

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

       3. associate-*r/ 55.7%

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

   14. Simplified 55.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 4.9 \cdot 10^{+106}:\\ \;\;\;\;180 \cdot \frac{\left(1 - \tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin t\_0}\right)\right) + -1}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan t\_0}}{x-scale}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 4.9e+106)
     (* 180.0 (/ (+ (- 1.0 (atan (/ (/ y-scale x-scale) (sin t_0)))) -1.0) PI))
     (* 180.0 (/ (atan (/ (/ y-scale (tan t_0)) x-scale)) (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 4.9e+106) {
		tmp = 180.0 * (((1.0 - atan(((y_45_scale / x_45_scale) / sin(t_0)))) + -1.0) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 4.9e+106) {
		tmp = 180.0 * (((1.0 - Math.atan(((y_45_scale / x_45_scale) / Math.sin(t_0)))) + -1.0) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / Math.tan(t_0)) / x_45_scale)) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 4.9e+106:
		tmp = 180.0 * (((1.0 - math.atan(((y_45_scale / x_45_scale) / math.sin(t_0)))) + -1.0) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / math.tan(t_0)) / x_45_scale)) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 4.9e+106)
		tmp = Float64(180.0 * Float64(Float64(Float64(1.0 - atan(Float64(Float64(y_45_scale / x_45_scale) / sin(t_0)))) + -1.0) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / tan(t_0)) / x_45_scale)) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 4.9e+106)
		tmp = 180.0 * (((1.0 - atan(((y_45_scale / x_45_scale) / sin(t_0)))) + -1.0) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.9e+106], N[(180.0 * N[(N[(N[(1.0 - N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.89999999999999998e106

   1. Initial program 13.8%

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

   3. Simplified 14.0%

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

   5. Taylor expanded in x-scale around 0 32.0%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in a around 0 38.7%

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

      1. mul-1-neg 38.7%

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

      2. times-frac 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in angle around 0 43.2%

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

       1. expm1-log1p-u 27.6%

          \[\leadsto 180 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}}{\pi} \]

       2. expm1-undefine 26.3%

          \[\leadsto 180 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)} - 1}}{\pi} \]

       3. atan-neg 26.3%

          \[\leadsto 180 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}\right)} - 1}{\pi} \]

       4. un-div-inv 26.3%

          \[\leadsto 180 \cdot \frac{e^{\mathsf{log1p}\left(-\tan^{-1} \color{blue}{\left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}\right)} - 1}{\pi} \]

   13. Applied egg-rr 26.3%

       \[\leadsto 180 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(-\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)} - 1}}{\pi} \]
   14. Step-by-step derivation

       1. sub-neg 26.3%

          \[\leadsto 180 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(-\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)} + \left(-1\right)}}{\pi} \]

       2. log1p-undefine 26.3%

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

       3. rem-exp-log 41.9%

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

       4. unsub-neg 41.9%

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

       5. metadata-eval 41.9%

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

   15. Simplified 41.9%

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

   if 4.89999999999999998e106 < b

   1. Initial program 6.4%

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

   3. Simplified 3.3%

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

   5. Taylor expanded in x-scale around 0 10.4%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in a around 0 58.6%

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

      1. mul-1-neg 58.6%

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

      2. times-frac 52.9%

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

   10. Simplified 52.9%

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

       1. associate-*r/ 52.9%

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

       2. atan-neg 52.9%

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

       3. clear-num 52.9%

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

       4. quot-tan 52.9%

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

   12. Applied egg-rr 52.9%

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

       1. associate-*r/ 52.9%

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

       2. associate-*l/ 58.7%

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

       3. associate-*r/ 58.7%

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

   14. Simplified 58.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{+106}:\\ \;\;\;\;180 \cdot \frac{\left(1 - \tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) + -1}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;180 \cdot \left(-1 + \left(1 - \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin t\_0}\right)}{\pi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan t\_0}}{x-scale}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 4.8e+106)
     (* 180.0 (+ -1.0 (- 1.0 (/ (atan (/ (/ y-scale x-scale) (sin t_0))) PI))))
     (* 180.0 (/ (atan (/ (/ y-scale (tan t_0)) x-scale)) (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 4.8e+106) {
		tmp = 180.0 * (-1.0 + (1.0 - (atan(((y_45_scale / x_45_scale) / sin(t_0))) / ((double) M_PI))));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 4.8e+106) {
		tmp = 180.0 * (-1.0 + (1.0 - (Math.atan(((y_45_scale / x_45_scale) / Math.sin(t_0))) / Math.PI)));
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / Math.tan(t_0)) / x_45_scale)) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 4.8e+106:
		tmp = 180.0 * (-1.0 + (1.0 - (math.atan(((y_45_scale / x_45_scale) / math.sin(t_0))) / math.pi)))
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / math.tan(t_0)) / x_45_scale)) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 4.8e+106)
		tmp = Float64(180.0 * Float64(-1.0 + Float64(1.0 - Float64(atan(Float64(Float64(y_45_scale / x_45_scale) / sin(t_0))) / pi))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / tan(t_0)) / x_45_scale)) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 4.8e+106)
		tmp = 180.0 * (-1.0 + (1.0 - (atan(((y_45_scale / x_45_scale) / sin(t_0))) / pi)));
	else
		tmp = 180.0 * (atan(((y_45_scale / tan(t_0)) / x_45_scale)) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.8e+106], N[(180.0 * N[(-1.0 + N[(1.0 - N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000001e106

   1. Initial program 13.8%

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

   3. Simplified 14.0%

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

   5. Taylor expanded in x-scale around 0 32.0%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in a around 0 38.7%

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

      1. mul-1-neg 38.7%

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

      2. times-frac 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in angle around 0 43.2%

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

       1. expm1-log1p-u 43.2%

          \[\leadsto 180 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

       2. expm1-undefine 41.9%

          \[\leadsto 180 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)} - 1\right)} \]

       3. atan-neg 41.9%

          \[\leadsto 180 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi}\right)} - 1\right) \]

       4. un-div-inv 41.9%

          \[\leadsto 180 \cdot \left(e^{\mathsf{log1p}\left(\frac{-\tan^{-1} \color{blue}{\left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi}\right)} - 1\right) \]

   13. Applied egg-rr 41.9%

       \[\leadsto 180 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. sub-neg 41.9%

          \[\leadsto 180 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)} + \left(-1\right)\right)} \]

       2. log1p-undefine 41.9%

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

       3. rem-exp-log 41.9%

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

       4. distribute-frac-neg 41.9%

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

       5. unsub-neg 41.9%

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

       6. metadata-eval 41.9%

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

   15. Simplified 41.9%

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

   if 4.8000000000000001e106 < b

   1. Initial program 6.4%

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

   3. Simplified 3.3%

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

   5. Taylor expanded in x-scale around 0 10.4%

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

   7. Simplified 10.6%

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

   8. Taylor expanded in a around 0 58.6%

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

      1. mul-1-neg 58.6%

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

      2. times-frac 52.9%

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

   10. Simplified 52.9%

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

       1. associate-*r/ 52.9%

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

       2. atan-neg 52.9%

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

       3. clear-num 52.9%

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

       4. quot-tan 52.9%

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

   12. Applied egg-rr 52.9%

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

       1. associate-*r/ 52.9%

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

       2. associate-*l/ 58.7%

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

       3. associate-*r/ 58.7%

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

   14. Simplified 58.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;180 \cdot \left(-1 + \left(1 - \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale \leq -5 \cdot 10^{-267}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan t\_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{-1}{\sin t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale -5e-267)
     (* 180.0 (/ -1.0 (/ PI (atan (/ (/ y-scale x-scale) (tan t_0))))))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (/ -1.0 (sin t_0)))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale <= -5e-267) {
		tmp = 180.0 * (-1.0 / (((double) M_PI) / atan(((y_45_scale / x_45_scale) / tan(t_0)))));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (-1.0 / sin(t_0)))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale <= -5e-267) {
		tmp = 180.0 * (-1.0 / (Math.PI / Math.atan(((y_45_scale / x_45_scale) / Math.tan(t_0)))));
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * (-1.0 / Math.sin(t_0)))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale <= -5e-267:
		tmp = 180.0 * (-1.0 / (math.pi / math.atan(((y_45_scale / x_45_scale) / math.tan(t_0)))))
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * (-1.0 / math.sin(t_0)))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale <= -5e-267)
		tmp = Float64(180.0 * Float64(-1.0 / Float64(pi / atan(Float64(Float64(y_45_scale / x_45_scale) / tan(t_0))))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(-1.0 / sin(t_0)))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale <= -5e-267)
		tmp = 180.0 * (-1.0 / (pi / atan(((y_45_scale / x_45_scale) / tan(t_0)))));
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (-1.0 / sin(t_0)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -5e-267], N[(180.0 * N[(-1.0 / N[(Pi / N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-1.0 / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -4.9999999999999999e-267

   1. Initial program 12.4%

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

   3. Simplified 14.9%

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

   5. Taylor expanded in x-scale around 0 31.6%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in a around 0 42.5%

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

      1. mul-1-neg 42.5%

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

      2. times-frac 46.1%

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

   10. Simplified 46.1%

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

       1. clear-num 46.2%

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

       2. inv-pow 46.2%

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

       3. atan-neg 46.2%

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

       4. clear-num 46.2%

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

       5. quot-tan 46.2%

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

   12. Applied egg-rr 46.2%

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

       1. unpow-1 46.2%

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

       2. associate-*r/ 46.2%

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

       3. *-rgt-identity 46.2%

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

   14. Simplified 46.2%

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

   if -4.9999999999999999e-267 < x-scale

   1. Initial program 13.1%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in x-scale around 0 27.7%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in a around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. times-frac 40.9%

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

   10. Simplified 40.9%

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

   11. Taylor expanded in angle around 0 46.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -5 \cdot 10^{-267}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{-1}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale \leq -6.4 \cdot 10^{-267}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan t\_0}\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin t\_0}\right)}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale -6.4e-267)
     (* 180.0 (/ -1.0 (/ PI (atan (/ (/ y-scale x-scale) (tan t_0))))))
     (* 180.0 (/ -1.0 (/ PI (atan (/ (/ y-scale x-scale) (sin t_0)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale <= -6.4e-267) {
		tmp = 180.0 * (-1.0 / (((double) M_PI) / atan(((y_45_scale / x_45_scale) / tan(t_0)))));
	} else {
		tmp = 180.0 * (-1.0 / (((double) M_PI) / atan(((y_45_scale / x_45_scale) / sin(t_0)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale <= -6.4e-267) {
		tmp = 180.0 * (-1.0 / (Math.PI / Math.atan(((y_45_scale / x_45_scale) / Math.tan(t_0)))));
	} else {
		tmp = 180.0 * (-1.0 / (Math.PI / Math.atan(((y_45_scale / x_45_scale) / Math.sin(t_0)))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale <= -6.4e-267:
		tmp = 180.0 * (-1.0 / (math.pi / math.atan(((y_45_scale / x_45_scale) / math.tan(t_0)))))
	else:
		tmp = 180.0 * (-1.0 / (math.pi / math.atan(((y_45_scale / x_45_scale) / math.sin(t_0)))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale <= -6.4e-267)
		tmp = Float64(180.0 * Float64(-1.0 / Float64(pi / atan(Float64(Float64(y_45_scale / x_45_scale) / tan(t_0))))));
	else
		tmp = Float64(180.0 * Float64(-1.0 / Float64(pi / atan(Float64(Float64(y_45_scale / x_45_scale) / sin(t_0))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale <= -6.4e-267)
		tmp = 180.0 * (-1.0 / (pi / atan(((y_45_scale / x_45_scale) / tan(t_0)))));
	else
		tmp = 180.0 * (-1.0 / (pi / atan(((y_45_scale / x_45_scale) / sin(t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -6.4e-267], N[(180.0 * N[(-1.0 / N[(Pi / N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(-1.0 / N[(Pi / N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -6.39999999999999973e-267

   1. Initial program 12.4%

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

   3. Simplified 14.9%

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

   5. Taylor expanded in x-scale around 0 31.6%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in a around 0 42.5%

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

      1. mul-1-neg 42.5%

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

      2. times-frac 46.1%

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

   10. Simplified 46.1%

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

       1. clear-num 46.2%

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

       2. inv-pow 46.2%

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

       3. atan-neg 46.2%

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

       4. clear-num 46.2%

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

       5. quot-tan 46.2%

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

   12. Applied egg-rr 46.2%

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

       1. unpow-1 46.2%

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

       2. associate-*r/ 46.2%

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

       3. *-rgt-identity 46.2%

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

   14. Simplified 46.2%

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

   if -6.39999999999999973e-267 < x-scale

   1. Initial program 13.1%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in x-scale around 0 27.7%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in a around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. times-frac 40.9%

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

   10. Simplified 40.9%

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

   11. Taylor expanded in angle around 0 46.4%

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

       1. clear-num 46.4%

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

       2. inv-pow 46.4%

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

       3. atan-neg 46.4%

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

       4. un-div-inv 46.4%

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

   13. Applied egg-rr 46.4%

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

       1. unpow-1 46.4%

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

   15. Simplified 46.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -6.4 \cdot 10^{-267}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale \leq -1.95 \cdot 10^{-267}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin t\_0}\right)}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale -1.95e-267)
     (* -180.0 (/ (atan (/ (/ y-scale x-scale) (tan t_0))) PI))
     (* 180.0 (/ -1.0 (/ PI (atan (/ (/ y-scale x-scale) (sin t_0)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale <= -1.95e-267) {
		tmp = -180.0 * (atan(((y_45_scale / x_45_scale) / tan(t_0))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (-1.0 / (((double) M_PI) / atan(((y_45_scale / x_45_scale) / sin(t_0)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale <= -1.95e-267) {
		tmp = -180.0 * (Math.atan(((y_45_scale / x_45_scale) / Math.tan(t_0))) / Math.PI);
	} else {
		tmp = 180.0 * (-1.0 / (Math.PI / Math.atan(((y_45_scale / x_45_scale) / Math.sin(t_0)))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale <= -1.95e-267:
		tmp = -180.0 * (math.atan(((y_45_scale / x_45_scale) / math.tan(t_0))) / math.pi)
	else:
		tmp = 180.0 * (-1.0 / (math.pi / math.atan(((y_45_scale / x_45_scale) / math.sin(t_0)))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale <= -1.95e-267)
		tmp = Float64(-180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) / tan(t_0))) / pi));
	else
		tmp = Float64(180.0 * Float64(-1.0 / Float64(pi / atan(Float64(Float64(y_45_scale / x_45_scale) / sin(t_0))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale <= -1.95e-267)
		tmp = -180.0 * (atan(((y_45_scale / x_45_scale) / tan(t_0))) / pi);
	else
		tmp = 180.0 * (-1.0 / (pi / atan(((y_45_scale / x_45_scale) / sin(t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.95e-267], N[(-180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(-1.0 / N[(Pi / N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -1.94999999999999988e-267

   1. Initial program 12.4%

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

   3. Simplified 14.9%

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

   5. Taylor expanded in x-scale around 0 31.6%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in a around 0 42.5%

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

      1. mul-1-neg 42.5%

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

      2. times-frac 46.1%

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

   10. Simplified 46.1%

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

       1. log1p-expm1-u 32.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

       2. log1p-undefine 34.1%

          \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

   12. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

   13. Taylor expanded in y-scale around 0 42.5%

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

       1. associate-/r* 46.1%

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

   15. Simplified 46.1%

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

   if -1.94999999999999988e-267 < x-scale

   1. Initial program 13.1%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in x-scale around 0 27.7%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in a around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. times-frac 40.9%

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

   10. Simplified 40.9%

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

   11. Taylor expanded in angle around 0 46.4%

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

       1. clear-num 46.4%

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

       2. inv-pow 46.4%

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

       3. atan-neg 46.4%

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

       4. un-div-inv 46.4%

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

   13. Applied egg-rr 46.4%

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

       1. unpow-1 46.4%

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

   15. Simplified 46.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.95 \cdot 10^{-267}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{-1}{\frac{\pi}{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.3% accurate, 13.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale \leq -5 \cdot 10^{-267}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin t\_0}\right) \cdot -180}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale -5e-267)
     (* -180.0 (/ (atan (/ (/ y-scale x-scale) (tan t_0))) PI))
     (/ (* (atan (/ (/ y-scale x-scale) (sin t_0))) -180.0) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale <= -5e-267) {
		tmp = -180.0 * (atan(((y_45_scale / x_45_scale) / tan(t_0))) / ((double) M_PI));
	} else {
		tmp = (atan(((y_45_scale / x_45_scale) / sin(t_0))) * -180.0) / ((double) M_PI);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale <= -5e-267) {
		tmp = -180.0 * (Math.atan(((y_45_scale / x_45_scale) / Math.tan(t_0))) / Math.PI);
	} else {
		tmp = (Math.atan(((y_45_scale / x_45_scale) / Math.sin(t_0))) * -180.0) / Math.PI;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale <= -5e-267:
		tmp = -180.0 * (math.atan(((y_45_scale / x_45_scale) / math.tan(t_0))) / math.pi)
	else:
		tmp = (math.atan(((y_45_scale / x_45_scale) / math.sin(t_0))) * -180.0) / math.pi
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale <= -5e-267)
		tmp = Float64(-180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) / tan(t_0))) / pi));
	else
		tmp = Float64(Float64(atan(Float64(Float64(y_45_scale / x_45_scale) / sin(t_0))) * -180.0) / pi);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale <= -5e-267)
		tmp = -180.0 * (atan(((y_45_scale / x_45_scale) / tan(t_0))) / pi);
	else
		tmp = (atan(((y_45_scale / x_45_scale) / sin(t_0))) * -180.0) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -5e-267], N[(-180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -180.0), $MachinePrecision] / Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -4.9999999999999999e-267

   1. Initial program 12.4%

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

   3. Simplified 14.9%

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

   5. Taylor expanded in x-scale around 0 31.6%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in a around 0 42.5%

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

      1. mul-1-neg 42.5%

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

      2. times-frac 46.1%

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

   10. Simplified 46.1%

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

       1. log1p-expm1-u 32.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

       2. log1p-undefine 34.1%

          \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

   12. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

   13. Taylor expanded in y-scale around 0 42.5%

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

       1. associate-/r* 46.1%

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

   15. Simplified 46.1%

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

   if -4.9999999999999999e-267 < x-scale

   1. Initial program 13.1%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in x-scale around 0 27.7%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in a around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. times-frac 40.9%

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

   10. Simplified 40.9%

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

   11. Taylor expanded in angle around 0 46.4%

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

       1. associate-*r/ 46.4%

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

       2. atan-neg 46.4%

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

       3. un-div-inv 46.4%

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

   13. Applied egg-rr 46.4%

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

       1. distribute-rgt-neg-out 46.4%

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

       2. distribute-lft-neg-in 46.4%

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

       3. metadata-eval 46.4%

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

   15. Simplified 46.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -5 \cdot 10^{-267}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot -180}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.7% accurate, 13.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  -180.0
  (/
   (atan (/ (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI)))))
   PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -180.0 * (atan(((y_45_scale / x_45_scale) / tan((0.005555555555555556 * (angle * ((double) M_PI)))))) / ((double) M_PI));
}

Java

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

Python

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

Julia

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

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = -180.0 * (atan(((y_45_scale / x_45_scale) / tan((0.005555555555555556 * (angle * pi))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(-180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] / N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.9%

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

3. Simplified 12.7%

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

5. Taylor expanded in x-scale around 0 29.3%

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

7. Simplified 33.8%

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

8. Taylor expanded in a around 0 41.2%

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

   1. mul-1-neg 41.2%

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

   2. times-frac 43.1%

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

10. Simplified 43.1%

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

    1. log1p-expm1-u 29.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

    2. log1p-undefine 27.9%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

12. Applied egg-rr 27.9%

    \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

13. Taylor expanded in y-scale around 0 41.2%

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

    1. associate-/r* 43.1%

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

15. Simplified 43.1%

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

Alternative 10: 43.9% accurate, 13.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  -180.0
  (/
   (atan (/ y-scale (* x-scale (tan (* 0.005555555555555556 (* angle PI))))))
   PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return -180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * ((double) M_PI))))))) / ((double) M_PI));
}

Java

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

Python

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

Julia

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

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = -180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * pi)))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(-180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.9%

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

3. Simplified 12.7%

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

5. Taylor expanded in x-scale around 0 29.3%

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

7. Simplified 33.8%

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

8. Taylor expanded in a around 0 41.2%

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

   1. mul-1-neg 41.2%

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

   2. times-frac 43.1%

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

10. Simplified 43.1%

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

    1. log1p-expm1-u 29.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

    2. log1p-undefine 27.9%

       \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

12. Applied egg-rr 27.9%

    \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\right)\right)} \]

13. Taylor expanded in y-scale around 0 41.2%

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

Alternative 11: 39.5% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 12.9%

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

3. Simplified 12.7%

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

5. Taylor expanded in x-scale around 0 29.3%

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

7. Simplified 33.8%

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

8. Taylor expanded in a around 0 41.2%

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

   1. mul-1-neg 41.2%

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

   2. times-frac 43.1%

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

10. Simplified 43.1%

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

11. Taylor expanded in angle around 0 44.2%

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

12. Taylor expanded in angle around 0 35.9%

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

    1. associate-*r/ 35.9%

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

    2. times-frac 37.8%

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

14. Simplified 37.8%

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

Alternative 12: 37.8% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 12.9%

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

3. Simplified 12.7%

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

5. Taylor expanded in angle around 0 11.7%

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

   1. associate-/l* 11.7%

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

   2. distribute-lft-out-- 11.7%

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

7. Simplified 11.7%

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

8. Taylor expanded in a around 0 35.9%

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

Alternative 13: 12.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 12.9%

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

3. Simplified 12.7%

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

5. Taylor expanded in angle around 0 11.7%

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

   1. associate-/l* 11.7%

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

   2. distribute-lft-out-- 11.7%

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

7. Simplified 11.7%

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

8. Taylor expanded in a around inf 10.5%

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

Reproduce

herbie shell --seed 2024113 
(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)))