raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.2% → 55.9%

Time: 1.4min

Alternatives: 14

Speedup: 26.0×

• could not determine a ground truth

  1. a = -1.2188771746557745e+205
  2. b = -4.494453340828664e-56
  3. angle = 1.7775425491987684e+296
  4. x-scale = -9.12934632411864e-86
  5. y-scale = 2.6547520944105516e+205

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: 13.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 6.1 \cdot 10^{-30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot t\_0\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 14.6 \lor \neg \left(b\_m \leq 3.3 \cdot 10^{+46}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{1}{x-scale \cdot t\_0}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 6.1e-30)
     (* 180.0 (/ (atan (* (/ y-scale x-scale) t_0)) PI))
     (if (or (<= b_m 14.6) (not (<= b_m 3.3e+46)))
       (*
        180.0
        (/ (atan (* -0.5 (* y-scale (* 2.0 (/ 1.0 (* x-scale t_0)))))) PI))
       (/
        (*
         180.0
         (atan
          (*
           (/ y-scale x-scale)
           (tan (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI))))))))
        PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 6.1e-30) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * t_0)) / ((double) M_PI));
	} else if ((b_m <= 14.6) || !(b_m <= 3.3e+46)) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))))) / ((double) M_PI);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 6.1e-30) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * t_0)) / Math.PI);
	} else if ((b_m <= 14.6) || !(b_m <= 3.3e+46)) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale / x_45_scale) * Math.tan((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))))))) / Math.PI;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 6.1e-30:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * t_0)) / math.pi)
	elif (b_m <= 14.6) or not (b_m <= 3.3e+46):
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale / x_45_scale) * math.tan((0.005555555555555556 * (angle * (math.sqrt(math.pi) * math.sqrt(math.pi)))))))) / math.pi
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 6.1e-30)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * t_0)) / pi));
	elseif ((b_m <= 14.6) || !(b_m <= 3.3e+46))
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 * Float64(1.0 / Float64(x_45_scale * t_0)))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi)))))))) / pi);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 6.1e-30)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * t_0)) / pi);
	elseif ((b_m <= 14.6) || ~((b_m <= 3.3e+46)))
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * (sqrt(pi) * sqrt(pi)))))))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 6.1e-30], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b$95$m, 14.6], N[Not[LessEqual[b$95$m, 3.3e+46]], $MachinePrecision]], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 * N[(1.0 / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.09999999999999981e-30

   1. Initial program 12.0%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in x-scale around 0 25.0%

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

   7. Simplified 29.6%

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

   8. Taylor expanded in a around inf 50.6%

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

      1. times-frac 52.8%

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

   10. Simplified 52.8%

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

   11. Taylor expanded in angle around 0 53.3%

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

   if 6.09999999999999981e-30 < b < 14.5999999999999996 or 3.2999999999999998e46 < b

   1. Initial program 16.1%

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

   3. Simplified 12.8%

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

   5. Taylor expanded in x-scale around 0 32.4%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in a around 0 67.5%

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

   9. Taylor expanded in angle around 0 74.4%

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

   if 14.5999999999999996 < b < 3.2999999999999998e46

   1. Initial program 0.4%

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

   3. Simplified 0.6%

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

   5. Taylor expanded in x-scale around 0 3.0%

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

   7. Simplified 3.6%

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

   8. Taylor expanded in a around inf 73.7%

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

      1. times-frac 73.7%

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

   10. Simplified 73.7%

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

       1. associate-*r/ 73.7%

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

       2. quot-tan 73.7%

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

   12. Applied egg-rr 73.7%

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

       1. add-sqr-sqrt 15.9%

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

   14. Applied egg-rr 74.9%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 14.6 \lor \neg \left(b \leq 3.3 \cdot 10^{+46}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{1}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\\ \mathbf{if}\;a \leq 2.6 \cdot 10^{-35}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{t\_0}{x-scale \cdot t\_1}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{t\_1}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI)))))))
   (if (<= a 2.6e-35)
     (*
      180.0
      (/ (atan (* -0.5 (* y-scale (* 2.0 (/ t_0 (* x-scale t_1)))))) PI))
     (* 180.0 (/ (atan (* (/ y-scale x-scale) (/ t_1 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 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))));
	double tmp;
	if (a <= 2.6e-35) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (t_0 / (x_45_scale * t_1)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_1 / 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 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_1 = Math.sin((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))));
	double tmp;
	if (a <= 2.6e-35) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 * (t_0 / (x_45_scale * t_1)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * (t_1 / 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 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_1 = math.sin((0.005555555555555556 * (angle * (math.sqrt(math.pi) * math.sqrt(math.pi)))))
	tmp = 0
	if a <= 2.6e-35:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (2.0 * (t_0 / (x_45_scale * t_1)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * (t_1 / 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 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi)))))
	tmp = 0.0
	if (a <= 2.6e-35)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 * Float64(t_0 / Float64(x_45_scale * t_1)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(t_1 / 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 = cos((0.005555555555555556 * (angle * pi)));
	t_1 = sin((0.005555555555555556 * (angle * (sqrt(pi) * sqrt(pi)))));
	tmp = 0.0;
	if (a <= 2.6e-35)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (t_0 / (x_45_scale * t_1)))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_1 / 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[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 2.6e-35], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 * N[(t$95$0 / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.60000000000000005e-35

   1. Initial program 13.7%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in x-scale around 0 28.4%

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

   7. Simplified 31.0%

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

   8. Taylor expanded in a around 0 49.5%

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

      1. add-sqr-sqrt 51.8%

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

   10. Applied egg-rr 51.8%

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

   if 2.60000000000000005e-35 < a

   1. Initial program 10.2%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in x-scale around 0 20.7%

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

   7. Simplified 25.5%

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

   8. Taylor expanded in a around inf 54.6%

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

      1. times-frac 56.2%

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

   10. Simplified 56.2%

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

       1. add-sqr-sqrt 27.6%

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

   12. Applied egg-rr 58.7%

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

Alternative 3: 49.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \mathbf{if}\;a \leq 6.5 \cdot 10^{+16}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos t\_0}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle (cbrt (pow PI 3.0))))))
   (if (<= a 6.5e+16)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          y-scale
          (*
           2.0
           (/
            (cos t_0)
            (* x-scale (sin (* 0.005555555555555556 (* angle PI)))))))))
       PI))
     (/ (* 180.0 (atan (* (/ y-scale x-scale) (tan 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 * cbrt(pow(((double) M_PI), 3.0)));
	double tmp;
	if (a <= 6.5e+16) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (cos(t_0) / (x_45_scale * sin((0.005555555555555556 * (angle * ((double) M_PI)))))))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan(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.cbrt(Math.pow(Math.PI, 3.0)));
	double tmp;
	if (a <= 6.5e+16) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 * (Math.cos(t_0) / (x_45_scale * Math.sin((0.005555555555555556 * (angle * Math.PI))))))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale / x_45_scale) * Math.tan(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 * cbrt((pi ^ 3.0))))
	tmp = 0.0
	if (a <= 6.5e+16)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 * Float64(cos(t_0) / Float64(x_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0)))) / pi);
	end
	return 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 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 6.5e+16], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] / N[(x$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.5e16

   1. Initial program 15.1%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in x-scale around 0 29.0%

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

   7. Simplified 32.0%

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

   8. Taylor expanded in a around 0 49.0%

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

      1. add-cbrt-cube 51.8%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

      2. unpow2 51.8%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{2}} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   10. Applied egg-rr 51.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{2} \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   11. Step-by-step derivation

       1. unpow2 51.8%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. unpow3 51.8%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   12. Simplified 51.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if 6.5e16 < a

   1. Initial program 5.9%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x-scale around 0 17.9%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in a around inf 57.0%

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

      1. times-frac 58.9%

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

   10. Simplified 58.9%

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

       1. associate-*r/ 59.1%

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

       2. quot-tan 59.1%

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

   12. Applied egg-rr 59.1%

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

       1. add-cbrt-cube 28.5%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. unpow2 28.5%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{2}} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   14. Applied egg-rr 59.3%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{2} \cdot \pi}}\right)\right)\right)}{\pi} \]
   15. Step-by-step derivation

       1. unpow2 28.5%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. unpow3 28.5%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   16. Simplified 59.3%

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

Alternative 4: 56.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+46}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{t\_0}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{1}{x-scale \cdot t\_0}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 1.4e+46)
     (*
      180.0
      (/
       (atan
        (*
         (/ y-scale x-scale)
         (/
          t_0
          (cos (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI))))))))
       PI))
     (*
      180.0
      (/ (atan (* -0.5 (* y-scale (* 2.0 (/ 1.0 (* x-scale t_0)))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 1.4e+46) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_0 / cos((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 1.4e+46) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * (t_0 / Math.cos((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 1.4e+46:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * (t_0 / math.cos((0.005555555555555556 * (angle * (math.sqrt(math.pi) * math.sqrt(math.pi)))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 1.4e+46)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(t_0 / cos(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 * Float64(1.0 / Float64(x_45_scale * t_0)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 1.4e+46)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * (t_0 / cos((0.005555555555555556 * (angle * (sqrt(pi) * sqrt(pi)))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * t_0)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1.4e+46], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$0 / N[Cos[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 * N[(1.0 / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000009e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

       1. add-sqr-sqrt 36.2%

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

   12. Applied egg-rr 53.8%

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

   if 1.40000000000000009e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

   9. Taylor expanded in angle around 0 72.6%

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

Alternative 5: 56.6% accurate, 13.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 3.6 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{1}{x-scale \cdot \sin t\_0}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 3.6e+46)
     (/ (* 180.0 (atan (* (/ y-scale x-scale) (tan t_0)))) PI)
     (*
      180.0
      (/
       (atan (* -0.5 (* y-scale (* 2.0 (/ 1.0 (* x-scale (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 (b_m <= 3.6e+46) {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan(t_0)))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * 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 (b_m <= 3.6e+46) {
		tmp = (180.0 * Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0)))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * 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 b_m <= 3.6e+46:
		tmp = (180.0 * math.atan(((y_45_scale / x_45_scale) * math.tan(t_0)))) / math.pi
	else:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * 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 (b_m <= 3.6e+46)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0)))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 * Float64(1.0 / Float64(x_45_scale * 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 (b_m <= 3.6e+46)
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan(t_0)))) / pi;
	else
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 * (1.0 / (x_45_scale * 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[b$95$m, 3.6e+46], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 * N[(1.0 / N[(x$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999999e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

       1. associate-*r/ 53.1%

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

       2. quot-tan 53.1%

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

   12. Applied egg-rr 53.1%

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

   if 3.5999999999999999e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

   9. Taylor expanded in angle around 0 72.6%

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

Alternative 6: 54.0% accurate, 13.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4.7 \cdot 10^{+63}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 4.7e+63)
   (/
    (*
     180.0
     (atan
      (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI))))))
    PI)
   (*
    180.0
    (/ (atan (* -0.5 (/ (* y-scale 360.0) (* angle (* PI 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 tmp;
	if (b_m <= 4.7e+63) {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI))))))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (((double) M_PI) * 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 tmp;
	if (b_m <= 4.7e+63) {
		tmp = (180.0 * Math.atan(((y_45_scale / x_45_scale) * Math.tan((0.005555555555555556 * (angle * Math.PI)))))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (Math.PI * 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):
	tmp = 0
	if b_m <= 4.7e+63:
		tmp = (180.0 * math.atan(((y_45_scale / x_45_scale) * math.tan((0.005555555555555556 * (angle * math.pi)))))) / math.pi
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 4.7e+63)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi)))))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 360.0) / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 4.7e+63)
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * pi)))))) / pi;
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (pi * 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_] := If[LessEqual[b$95$m, 4.7e+63], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 360.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.7000000000000003e63

   1. Initial program 12.7%

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

   3. Simplified 14.5%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in a around inf 51.5%

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

      1. times-frac 53.5%

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

   10. Simplified 53.5%

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

       1. associate-*r/ 53.5%

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

       2. quot-tan 53.5%

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

   12. Applied egg-rr 53.5%

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

   if 4.7000000000000003e63 < b

   1. Initial program 12.3%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in x-scale around 0 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in a around 0 70.5%

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

   9. Taylor expanded in angle around 0 64.9%

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

       1. associate-*r/ 65.2%

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

       2. *-commutative 65.2%

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

   11. Simplified 65.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+63}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% accurate, 13.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.1 \cdot 10^{+63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.1e+63)
   (*
    180.0
    (/
     (atan (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI)))))
     PI))
   (*
    180.0
    (/ (atan (* -0.5 (/ (* y-scale 360.0) (* angle (* PI 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 tmp;
	if (b_m <= 5.1e+63) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (((double) M_PI) * 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 tmp;
	if (b_m <= 5.1e+63) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan((0.005555555555555556 * (angle * Math.PI))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (Math.PI * 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):
	tmp = 0
	if b_m <= 5.1e+63:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan((0.005555555555555556 * (angle * math.pi))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 5.1e+63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 360.0) / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 5.1e+63)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * pi))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (pi * 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_] := If[LessEqual[b$95$m, 5.1e+63], 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], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 360.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e63

   1. Initial program 12.7%

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

   3. Simplified 14.5%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in a around inf 51.5%

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

      1. times-frac 53.5%

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

   10. Simplified 53.5%

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

       1. associate-*r/ 53.5%

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

       2. quot-tan 53.5%

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

   12. Applied egg-rr 53.5%

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

       1. associate-/l* 53.5%

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

       2. *-commutative 53.5%

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

   14. Applied egg-rr 53.5%

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

   if 5.0999999999999998e63 < b

   1. Initial program 12.3%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in x-scale around 0 26.7%

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

   7. Simplified 26.7%

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

   8. Taylor expanded in a around 0 70.5%

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

   9. Taylor expanded in angle around 0 64.9%

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

       1. associate-*r/ 65.2%

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

       2. *-commutative 65.2%

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

   11. Simplified 65.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{+63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.7 \cdot 10^{-69}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\pi}{x-scale}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 7.7e-69)
   (*
    180.0
    (/
     (atan (* y-scale (/ (tan (* 0.005555555555555556 (* angle PI))) x-scale)))
     PI))
   (if (<= b_m 3.1e+46)
     (/
      (*
       180.0
       (atan (* (* y-scale (/ PI x-scale)) (* 0.005555555555555556 angle))))
      PI)
     (*
      180.0
      (/ (atan (* -0.5 (/ (* y-scale 360.0) (* angle (* PI 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 tmp;
	if (b_m <= 7.7e-69) {
		tmp = 180.0 * (atan((y_45_scale * (tan((0.005555555555555556 * (angle * ((double) M_PI)))) / x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 3.1e+46) {
		tmp = (180.0 * atan(((y_45_scale * (((double) M_PI) / x_45_scale)) * (0.005555555555555556 * angle)))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (((double) M_PI) * 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 tmp;
	if (b_m <= 7.7e-69) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.tan((0.005555555555555556 * (angle * Math.PI))) / x_45_scale))) / Math.PI);
	} else if (b_m <= 3.1e+46) {
		tmp = (180.0 * Math.atan(((y_45_scale * (Math.PI / x_45_scale)) * (0.005555555555555556 * angle)))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (Math.PI * 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):
	tmp = 0
	if b_m <= 7.7e-69:
		tmp = 180.0 * (math.atan((y_45_scale * (math.tan((0.005555555555555556 * (angle * math.pi))) / x_45_scale))) / math.pi)
	elif b_m <= 3.1e+46:
		tmp = (180.0 * math.atan(((y_45_scale * (math.pi / x_45_scale)) * (0.005555555555555556 * angle)))) / math.pi
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 7.7e-69)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(tan(Float64(0.005555555555555556 * Float64(angle * pi))) / x_45_scale))) / pi));
	elseif (b_m <= 3.1e+46)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(pi / x_45_scale)) * Float64(0.005555555555555556 * angle)))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 360.0) / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 7.7e-69)
		tmp = 180.0 * (atan((y_45_scale * (tan((0.005555555555555556 * (angle * pi))) / x_45_scale))) / pi);
	elseif (b_m <= 3.1e+46)
		tmp = (180.0 * atan(((y_45_scale * (pi / x_45_scale)) * (0.005555555555555556 * angle)))) / pi;
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (pi * 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_] := If[LessEqual[b$95$m, 7.7e-69], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.1e+46], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 360.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.70000000000000008e-69

   1. Initial program 11.3%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in x-scale around 0 25.3%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in a around inf 50.2%

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

      1. times-frac 52.6%

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

   10. Simplified 52.6%

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

       1. associate-*l/ 50.2%

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

       2. quot-tan 50.2%

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

   12. Applied egg-rr 50.2%

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

       1. associate-/l* 52.4%

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

   14. Simplified 52.4%

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

   if 7.70000000000000008e-69 < b < 3.09999999999999975e46

   1. Initial program 22.3%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in x-scale around 0 30.3%

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

   7. Simplified 31.2%

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

   8. Taylor expanded in a around inf 55.8%

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

      1. times-frac 55.7%

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

   10. Simplified 55.7%

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

       1. associate-*r/ 55.7%

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

       2. quot-tan 55.7%

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

   12. Applied egg-rr 55.7%

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

   13. Taylor expanded in angle around 0 54.9%

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

       1. associate-/l* 54.8%

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

       2. associate-*r* 54.8%

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

       3. *-commutative 54.8%

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

       4. associate-*r/ 54.8%

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

   15. Simplified 54.8%

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

   if 3.09999999999999975e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

   9. Taylor expanded in angle around 0 64.4%

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

       1. associate-*r/ 64.8%

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

       2. *-commutative 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.7 \cdot 10^{-69}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\pi}{x-scale}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 24.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\pi}{x-scale}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.5e+46)
   (/
    (*
     180.0
     (atan (* (* y-scale (/ PI x-scale)) (* 0.005555555555555556 angle))))
    PI)
   (*
    180.0
    (/ (atan (* -0.5 (/ (* y-scale 360.0) (* angle (* PI 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 tmp;
	if (b_m <= 1.5e+46) {
		tmp = (180.0 * atan(((y_45_scale * (((double) M_PI) / x_45_scale)) * (0.005555555555555556 * angle)))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (((double) M_PI) * 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 tmp;
	if (b_m <= 1.5e+46) {
		tmp = (180.0 * Math.atan(((y_45_scale * (Math.PI / x_45_scale)) * (0.005555555555555556 * angle)))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (Math.PI * 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):
	tmp = 0
	if b_m <= 1.5e+46:
		tmp = (180.0 * math.atan(((y_45_scale * (math.pi / x_45_scale)) * (0.005555555555555556 * angle)))) / math.pi
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 360.0) / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.5e+46)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(pi / x_45_scale)) * Float64(0.005555555555555556 * angle)))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 360.0) / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.5e+46)
		tmp = (180.0 * atan(((y_45_scale * (pi / x_45_scale)) * (0.005555555555555556 * angle)))) / pi;
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 360.0) / (angle * (pi * 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_] := If[LessEqual[b$95$m, 1.5e+46], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 360.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000012e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

       1. associate-*r/ 53.1%

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

       2. quot-tan 53.1%

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

   12. Applied egg-rr 53.1%

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

   13. Taylor expanded in angle around 0 47.2%

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

       1. associate-/l* 52.4%

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

       2. associate-*r* 52.4%

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

       3. *-commutative 52.4%

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

       4. associate-*r/ 52.4%

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

   15. Simplified 52.4%

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

   if 1.50000000000000012e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

   9. Taylor expanded in angle around 0 64.4%

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

       1. associate-*r/ 64.8%

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

       2. *-commutative 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\pi}{x-scale}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 360}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\pi}{x-scale}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\pi \cdot x-scale} \cdot \frac{y-scale}{angle}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 3.4e+46)
   (/
    (*
     180.0
     (atan (* (* y-scale (/ PI x-scale)) (* 0.005555555555555556 angle))))
    PI)
   (* 180.0 (/ (atan (* (/ -180.0 (* PI x-scale)) (/ y-scale angle))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.4e+46) {
		tmp = (180.0 * atan(((y_45_scale * (((double) M_PI) / x_45_scale)) * (0.005555555555555556 * angle)))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan(((-180.0 / (((double) M_PI) * x_45_scale)) * (y_45_scale / angle))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.4e+46) {
		tmp = (180.0 * Math.atan(((y_45_scale * (Math.PI / x_45_scale)) * (0.005555555555555556 * angle)))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan(((-180.0 / (Math.PI * x_45_scale)) * (y_45_scale / angle))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 3.4e+46:
		tmp = (180.0 * math.atan(((y_45_scale * (math.pi / x_45_scale)) * (0.005555555555555556 * angle)))) / math.pi
	else:
		tmp = 180.0 * (math.atan(((-180.0 / (math.pi * x_45_scale)) * (y_45_scale / angle))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.4e+46)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(pi / x_45_scale)) * Float64(0.005555555555555556 * angle)))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / Float64(pi * x_45_scale)) * Float64(y_45_scale / angle))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 3.4e+46)
		tmp = (180.0 * atan(((y_45_scale * (pi / x_45_scale)) * (0.005555555555555556 * angle)))) / pi;
	else
		tmp = 180.0 * (atan(((-180.0 / (pi * x_45_scale)) * (y_45_scale / angle))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 3.4e+46], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3999999999999998e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

       1. associate-*r/ 53.1%

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

       2. quot-tan 53.1%

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

   12. Applied egg-rr 53.1%

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

   13. Taylor expanded in angle around 0 47.2%

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

       1. associate-/l* 52.4%

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

       2. associate-*r* 52.4%

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

       3. *-commutative 52.4%

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

       4. associate-*r/ 52.4%

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

   15. Simplified 52.4%

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

   if 3.3999999999999998e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

      1. add-cbrt-cube 71.9%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

      2. unpow2 71.9%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{2}} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   10. Applied egg-rr 71.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{2} \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   11. Step-by-step derivation

       1. unpow2 71.9%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. unpow3 71.9%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   12. Simplified 71.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Taylor expanded in angle around 0 64.4%

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

       1. associate-*r/ 64.8%

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

       2. *-commutative 64.8%

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

       3. times-frac 64.6%

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

       4. *-commutative 64.6%

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

   15. Simplified 64.6%

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

Alternative 11: 53.0% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{+46}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\pi \cdot x-scale} \cdot \frac{y-scale}{angle}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2e+46)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (* y-scale (/ PI x-scale)))))
     PI))
   (* 180.0 (/ (atan (* (/ -180.0 (* PI x-scale)) (/ y-scale angle))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2e+46) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (((double) M_PI) / x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((-180.0 / (((double) M_PI) * x_45_scale)) * (y_45_scale / angle))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2e+46) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * (y_45_scale * (Math.PI / x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((-180.0 / (Math.PI * x_45_scale)) * (y_45_scale / angle))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 2e+46:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * (y_45_scale * (math.pi / x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((-180.0 / (math.pi * x_45_scale)) * (y_45_scale / angle))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 2e+46)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(y_45_scale * Float64(pi / x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 / Float64(pi * x_45_scale)) * Float64(y_45_scale / angle))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 2e+46)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (pi / x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan(((-180.0 / (pi * x_45_scale)) * (y_45_scale / angle))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 2e+46], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

   11. Taylor expanded in angle around 0 47.1%

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

       1. associate-/l* 52.3%

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

       2. associate-/l* 52.4%

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

   13. Simplified 52.4%

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

   if 2e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around 0 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in a around 0 67.8%

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

      1. add-cbrt-cube 71.9%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

      2. unpow2 71.9%

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{2}} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   10. Applied egg-rr 71.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{2} \cdot \pi}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   11. Step-by-step derivation

       1. unpow2 71.9%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \pi}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. unpow3 71.9%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   12. Simplified 71.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Taylor expanded in angle around 0 64.4%

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

       1. associate-*r/ 64.8%

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

       2. *-commutative 64.8%

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

       3. times-frac 64.6%

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

       4. *-commutative 64.6%

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

   15. Simplified 64.6%

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

Alternative 12: 53.0% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.5 \cdot 10^{+46}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 3.5e+46)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (* y-scale (/ PI x-scale)))))
     PI))
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* PI 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 tmp;
	if (b_m <= 3.5e+46) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (((double) M_PI) / x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (((double) M_PI) * 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 tmp;
	if (b_m <= 3.5e+46) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * (y_45_scale * (Math.PI / x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (Math.PI * 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):
	tmp = 0
	if b_m <= 3.5e+46:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * (y_45_scale * (math.pi / x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.5e+46)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(y_45_scale * Float64(pi / x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 3.5e+46)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (pi / x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (pi * 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_] := If[LessEqual[b$95$m, 3.5e+46], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999985e46

   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 14.7%

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

   5. Taylor expanded in x-scale around 0 26.0%

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

   7. Simplified 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 inf 51.0%

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

      1. times-frac 53.0%

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

   10. Simplified 53.0%

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

   11. Taylor expanded in angle around 0 47.1%

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

       1. associate-/l* 52.3%

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

       2. associate-/l* 52.4%

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

   13. Simplified 52.4%

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

   if 3.49999999999999985e46 < b

   1. Initial program 11.8%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in angle around 0 17.6%

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

      1. associate-/l* 17.6%

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

      2. distribute-lft-out-- 17.6%

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

   7. Simplified 17.6%

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

   8. Taylor expanded in a around 0 64.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+46}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.6% accurate, 26.0× speedup

Math

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

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 (* PI 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) {
	return 180.0 * (atan((-180.0 * (y_45_scale / (angle * (((double) M_PI) * x_45_scale))))) / ((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 * (Math.PI * x_45_scale))))) / 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 * (math.pi * x_45_scale))))) / 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(pi * x_45_scale))))) / 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 * (pi * x_45_scale))))) / 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[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.6%

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

3. Simplified 12.9%

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

5. Taylor expanded in angle around 0 11.0%

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

   1. associate-/l* 11.5%

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

   2. distribute-lft-out-- 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in a around 0 37.5%

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

9. Final simplification 37.5%

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

Alternative 14: 12.4% 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 12.6%

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

3. Simplified 12.9%

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

5. Taylor expanded in angle around 0 11.0%

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

   1. associate-/l* 11.5%

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

   2. distribute-lft-out-- 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in a around inf 12.1%

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

Reproduce

herbie shell --seed 2024114 
(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)))