raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.6% → 56.3%

Time: 52.6s

Alternatives: 11

Speedup: 26.0×

• could not determine a ground truth

  1. a = 1.1473580233227053e-35
  2. b = 1.9155932492105564e+23
  3. angle = -3.7391032485723145e+302
  4. x-scale = 3.6696410262124833e+80
  5. y-scale = -6.654977436076379e-227

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{\frac{180}{\pi}}\\ t_1 := x-scale \cdot \cos t\_0\\ \mathbf{if}\;a\_m \leq 11500000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-0.5 \cdot y-scale}{t\_1} \cdot \frac{1 + 2 \cdot \left(0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 6.8 \cdot 10^{+202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin t\_0\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{t\_1}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ angle (/ 180.0 PI))) (t_1 (* x-scale (cos t_0))))
   (if (<= a_m 11500000000.0)
     (/
      (*
       180.0
       (atan
        (*
         (/ (* -0.5 y-scale) t_1)
         (/
          (+ 1.0 (* 2.0 (* 0.5 (cos (* (* angle PI) 0.011111111111111112)))))
          (sin (* (* angle PI) 0.005555555555555556))))))
      PI)
     (if (<= a_m 6.8e+202)
       (* 180.0 (/ (atan (* (/ y-scale x-scale) (sin t_0))) PI))
       (/
        (*
         180.0
         (atan
          (/ (* y-scale (sin (* PI (* angle 0.005555555555555556)))) t_1)))
        PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle / (180.0 / ((double) M_PI));
	double t_1 = x_45_scale * cos(t_0);
	double tmp;
	if (a_m <= 11500000000.0) {
		tmp = (180.0 * atan((((-0.5 * y_45_scale) / t_1) * ((1.0 + (2.0 * (0.5 * cos(((angle * ((double) M_PI)) * 0.011111111111111112))))) / sin(((angle * ((double) M_PI)) * 0.005555555555555556)))))) / ((double) M_PI);
	} else if (a_m <= 6.8e+202) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin(t_0))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale * sin((((double) M_PI) * (angle * 0.005555555555555556)))) / t_1))) / ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle / (180.0 / Math.PI);
	double t_1 = x_45_scale * Math.cos(t_0);
	double tmp;
	if (a_m <= 11500000000.0) {
		tmp = (180.0 * Math.atan((((-0.5 * y_45_scale) / t_1) * ((1.0 + (2.0 * (0.5 * Math.cos(((angle * Math.PI) * 0.011111111111111112))))) / Math.sin(((angle * Math.PI) * 0.005555555555555556)))))) / Math.PI;
	} else if (a_m <= 6.8e+202) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.sin(t_0))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale * Math.sin((Math.PI * (angle * 0.005555555555555556)))) / t_1))) / Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = angle / (180.0 / math.pi)
	t_1 = x_45_scale * math.cos(t_0)
	tmp = 0
	if a_m <= 11500000000.0:
		tmp = (180.0 * math.atan((((-0.5 * y_45_scale) / t_1) * ((1.0 + (2.0 * (0.5 * math.cos(((angle * math.pi) * 0.011111111111111112))))) / math.sin(((angle * math.pi) * 0.005555555555555556)))))) / math.pi
	elif a_m <= 6.8e+202:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.sin(t_0))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale * math.sin((math.pi * (angle * 0.005555555555555556)))) / t_1))) / math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle / Float64(180.0 / pi))
	t_1 = Float64(x_45_scale * cos(t_0))
	tmp = 0.0
	if (a_m <= 11500000000.0)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(Float64(-0.5 * y_45_scale) / t_1) * Float64(Float64(1.0 + Float64(2.0 * Float64(0.5 * cos(Float64(Float64(angle * pi) * 0.011111111111111112))))) / sin(Float64(Float64(angle * pi) * 0.005555555555555556)))))) / pi);
	elseif (a_m <= 6.8e+202)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(t_0))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) / t_1))) / pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = angle / (180.0 / pi);
	t_1 = x_45_scale * cos(t_0);
	tmp = 0.0;
	if (a_m <= 11500000000.0)
		tmp = (180.0 * atan((((-0.5 * y_45_scale) / t_1) * ((1.0 + (2.0 * (0.5 * cos(((angle * pi) * 0.011111111111111112))))) / sin(((angle * pi) * 0.005555555555555556)))))) / pi;
	elseif (a_m <= 6.8e+202)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin(t_0))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale * sin((pi * (angle * 0.005555555555555556)))) / t_1))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 11500000000.0], N[(N[(180.0 * N[ArcTan[N[(N[(N[(-0.5 * y$45$scale), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(1.0 + N[(2.0 * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[a$95$m, 6.8e+202], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.15e10

   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 11.5%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 30.2%

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

   9. Applied egg-rr 33.2%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \color{blue}{\left(2 \cdot \frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \left(\frac{2 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \frac{1}{2} + 2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(1 + 2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       8. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       12. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       13. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. Simplified 52.9%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-0.5 \cdot y-scale}{x-scale \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)} \cdot \color{blue}{\frac{1 + 2 \cdot \left(0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]

   if 1.15e10 < a < 6.8e202

   1. Initial program 18.8%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 30.1%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 50.0%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 60.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot 1}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   15. Applied egg-rr 61.5%

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

   if 6.8e202 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 0.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 72.6%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       7. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       10. PI-lowering-PI.f64 89.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. Applied egg-rr 89.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 11500000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-0.5 \cdot y-scale}{x-scale \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)} \cdot \frac{1 + 2 \cdot \left(0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)}{\pi}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{x-scale \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;b \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+39}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \left(-0.5 + -0.5 \cdot \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}{\sin t\_0 \cdot \left(x-scale \cdot \cos t\_0\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \frac{angle}{180}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \cos t\_1}{\sin t\_1 \cdot \left(0 - x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.005555555555555556))
        (t_1 (* PI (* angle 0.005555555555555556))))
   (if (<= b 1.3e-33)
     (*
      180.0
      (/ (atan (* (/ y-scale x-scale) (sin (/ angle (/ 180.0 PI))))) PI))
     (if (<= b 1.32e+39)
       (/
        (*
         180.0
         (atan
          (/
           (*
            y-scale
            (+ -0.5 (* -0.5 (cos (* angle (* PI 0.011111111111111112))))))
           (* (sin t_0) (* x-scale (cos t_0))))))
        PI)
       (if (<= b 1.95e+136)
         (*
          180.0
          (/ (atan (* (/ y-scale x-scale) (tan (* PI (/ angle 180.0))))) PI))
         (/
          (*
           180.0
           (atan (/ (* y-scale (cos t_1)) (* (sin t_1) (- 0.0 x-scale)))))
          PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 1.3e-33) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / ((double) M_PI)))))) / ((double) M_PI));
	} else if (b <= 1.32e+39) {
		tmp = (180.0 * atan(((y_45_scale * (-0.5 + (-0.5 * cos((angle * (((double) M_PI) * 0.011111111111111112)))))) / (sin(t_0) * (x_45_scale * cos(t_0)))))) / ((double) M_PI);
	} else if (b <= 1.95e+136) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((((double) M_PI) * (angle / 180.0))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale * cos(t_1)) / (sin(t_1) * (0.0 - x_45_scale))))) / ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle * Math.PI) * 0.005555555555555556;
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 1.3e-33) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.sin((angle / (180.0 / Math.PI))))) / Math.PI);
	} else if (b <= 1.32e+39) {
		tmp = (180.0 * Math.atan(((y_45_scale * (-0.5 + (-0.5 * Math.cos((angle * (Math.PI * 0.011111111111111112)))))) / (Math.sin(t_0) * (x_45_scale * Math.cos(t_0)))))) / Math.PI;
	} else if (b <= 1.95e+136) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan((Math.PI * (angle / 180.0))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale * Math.cos(t_1)) / (Math.sin(t_1) * (0.0 - x_45_scale))))) / Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle * math.pi) * 0.005555555555555556
	t_1 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if b <= 1.3e-33:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.sin((angle / (180.0 / math.pi))))) / math.pi)
	elif b <= 1.32e+39:
		tmp = (180.0 * math.atan(((y_45_scale * (-0.5 + (-0.5 * math.cos((angle * (math.pi * 0.011111111111111112)))))) / (math.sin(t_0) * (x_45_scale * math.cos(t_0)))))) / math.pi
	elif b <= 1.95e+136:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan((math.pi * (angle / 180.0))))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale * math.cos(t_1)) / (math.sin(t_1) * (0.0 - x_45_scale))))) / math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (b <= 1.3e-33)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(Float64(angle / Float64(180.0 / pi))))) / pi));
	elseif (b <= 1.32e+39)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(-0.5 + Float64(-0.5 * cos(Float64(angle * Float64(pi * 0.011111111111111112)))))) / Float64(sin(t_0) * Float64(x_45_scale * cos(t_0)))))) / pi);
	elseif (b <= 1.95e+136)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(pi * Float64(angle / 180.0))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * cos(t_1)) / Float64(sin(t_1) * Float64(0.0 - x_45_scale))))) / pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle * pi) * 0.005555555555555556;
	t_1 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (b <= 1.3e-33)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / pi))))) / pi);
	elseif (b <= 1.32e+39)
		tmp = (180.0 * atan(((y_45_scale * (-0.5 + (-0.5 * cos((angle * (pi * 0.011111111111111112)))))) / (sin(t_0) * (x_45_scale * cos(t_0)))))) / pi;
	elseif (b <= 1.95e+136)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((pi * (angle / 180.0))))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale * cos(t_1)) / (sin(t_1) * (0.0 - x_45_scale))))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3e-33], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.32e+39], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(-0.5 + N[(-0.5 * N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b, 1.95e+136], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * N[(0.0 - x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.29999999999999997e-33

   1. Initial program 13.0%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 26.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 49.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 49.7%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 55.6%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot 1}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   15. Applied egg-rr 56.0%

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

   if 1.29999999999999997e-33 < b < 1.32e39

   1. Initial program 41.6%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 64.2%

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

   9. Applied egg-rr 58.7%

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

   10. Taylor expanded in angle around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   11. Step-by-step derivation

       1. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. PI-lowering-PI.f64 58.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, y-scale\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. Simplified 58.7%

       \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{-0.5 \cdot y-scale}{x-scale \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)} \cdot \frac{2 \cdot \left(a \cdot \left(a \cdot \left(0.5 - 0.5 \cdot \color{blue}{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right) + \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\right)}{\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}{\pi} \]

   13. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   15. Simplified 63.0%

       \[\leadsto \frac{180 \cdot \tan^{-1} \color{blue}{\left(\frac{\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot -0.5 + -0.5\right) \cdot y-scale}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}}{\pi} \]

   if 1.32e39 < b < 1.9500000000000001e136

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 39.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 73.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   12. Applied egg-rr 57.0%

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

       1. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       8. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(\frac{angle}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       10. PI-lowering-PI.f64 73.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

   14. Applied egg-rr 73.2%

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

   if 1.9500000000000001e136 < b

   1. Initial program 4.4%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 7.7%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-1 \cdot \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      15. PI-lowering-PI.f64 54.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 54.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+39}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \left(-0.5 + -0.5 \cdot \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(x-scale \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \frac{angle}{180}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(0 - x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{angle}{\frac{180}{\pi}}\\ t_1 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a\_m \leq 5800000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(0 - 0.5\right) - 0.5}{\sin t\_1 \cdot \cos t\_1}\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 6.2 \cdot 10^{+202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin t\_0\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ angle (/ 180.0 PI)))
        (t_1 (* (* angle PI) 0.005555555555555556)))
   (if (<= a_m 5800000000.0)
     (/
      (*
       180.0
       (atan
        (*
         (/ y-scale x-scale)
         (/
          (- (* (cos (* (* angle PI) 0.011111111111111112)) (- 0.0 0.5)) 0.5)
          (* (sin t_1) (cos t_1))))))
      PI)
     (if (<= a_m 6.2e+202)
       (* 180.0 (/ (atan (* (/ y-scale x-scale) (sin t_0))) PI))
       (/
        (*
         180.0
         (atan
          (/
           (* y-scale (sin (* PI (* angle 0.005555555555555556))))
           (* x-scale (cos t_0)))))
        PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle / (180.0 / ((double) M_PI));
	double t_1 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (a_m <= 5800000000.0) {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * (((cos(((angle * ((double) M_PI)) * 0.011111111111111112)) * (0.0 - 0.5)) - 0.5) / (sin(t_1) * cos(t_1)))))) / ((double) M_PI);
	} else if (a_m <= 6.2e+202) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin(t_0))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale * sin((((double) M_PI) * (angle * 0.005555555555555556)))) / (x_45_scale * cos(t_0))))) / ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = angle / (180.0 / Math.PI);
	double t_1 = (angle * Math.PI) * 0.005555555555555556;
	double tmp;
	if (a_m <= 5800000000.0) {
		tmp = (180.0 * Math.atan(((y_45_scale / x_45_scale) * (((Math.cos(((angle * Math.PI) * 0.011111111111111112)) * (0.0 - 0.5)) - 0.5) / (Math.sin(t_1) * Math.cos(t_1)))))) / Math.PI;
	} else if (a_m <= 6.2e+202) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.sin(t_0))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale * Math.sin((Math.PI * (angle * 0.005555555555555556)))) / (x_45_scale * Math.cos(t_0))))) / Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = angle / (180.0 / math.pi)
	t_1 = (angle * math.pi) * 0.005555555555555556
	tmp = 0
	if a_m <= 5800000000.0:
		tmp = (180.0 * math.atan(((y_45_scale / x_45_scale) * (((math.cos(((angle * math.pi) * 0.011111111111111112)) * (0.0 - 0.5)) - 0.5) / (math.sin(t_1) * math.cos(t_1)))))) / math.pi
	elif a_m <= 6.2e+202:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.sin(t_0))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale * math.sin((math.pi * (angle * 0.005555555555555556)))) / (x_45_scale * math.cos(t_0))))) / math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(angle / Float64(180.0 / pi))
	t_1 = Float64(Float64(angle * pi) * 0.005555555555555556)
	tmp = 0.0
	if (a_m <= 5800000000.0)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(Float64(Float64(cos(Float64(Float64(angle * pi) * 0.011111111111111112)) * Float64(0.0 - 0.5)) - 0.5) / Float64(sin(t_1) * cos(t_1)))))) / pi);
	elseif (a_m <= 6.2e+202)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(t_0))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) / Float64(x_45_scale * cos(t_0))))) / pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = angle / (180.0 / pi);
	t_1 = (angle * pi) * 0.005555555555555556;
	tmp = 0.0;
	if (a_m <= 5800000000.0)
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * (((cos(((angle * pi) * 0.011111111111111112)) * (0.0 - 0.5)) - 0.5) / (sin(t_1) * cos(t_1)))))) / pi;
	elseif (a_m <= 6.2e+202)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin(t_0))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale * sin((pi * (angle * 0.005555555555555556)))) / (x_45_scale * cos(t_0))))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a$95$m, 5800000000.0], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * N[(0.0 - 0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[a$95$m, 6.2e+202], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.8e9

   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 11.5%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 30.2%

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

   9. Applied egg-rr 33.2%

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

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{y-scale \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{y-scale}{x-scale} \cdot \frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\left(\mathsf{neg}\left(\frac{y-scale}{x-scale}\right)\right) \cdot \frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{y-scale}{x-scale}\right)\right), \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\frac{y-scale}{x-scale}\right)\right), \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right)\right), \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. Simplified 53.4%

       \[\leadsto \frac{180 \cdot \tan^{-1} \color{blue}{\left(\left(-\frac{y-scale}{x-scale}\right) \cdot \frac{0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   if 5.8e9 < a < 6.19999999999999983e202

   1. Initial program 18.8%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 30.1%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 50.0%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 60.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot 1}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   15. Applied egg-rr 61.5%

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

   if 6.19999999999999983e202 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 0.0%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 72.6%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       7. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       10. PI-lowering-PI.f64 89.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. Applied egg-rr 89.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5800000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(0 - 0.5\right) - 0.5}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)}{\pi}\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+202}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{x-scale \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \cos t\_0}{\sin t\_0 \cdot \left(0 - x-scale\right)}\right)}{\pi}\\ \mathbf{if}\;b \leq 4.9 \cdot 10^{-34}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.02 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \frac{angle}{180}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556)))
        (t_1
         (/
          (*
           180.0
           (atan (/ (* y-scale (cos t_0)) (* (sin t_0) (- 0.0 x-scale)))))
          PI)))
   (if (<= b 4.9e-34)
     (*
      180.0
      (/ (atan (* (/ y-scale x-scale) (sin (/ angle (/ 180.0 PI))))) PI))
     (if (<= b 1.5e+39)
       t_1
       (if (<= b 2.02e+136)
         (*
          180.0
          (/ (atan (* (/ y-scale x-scale) (tan (* PI (/ angle 180.0))))) PI))
         t_1)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = (180.0 * atan(((y_45_scale * cos(t_0)) / (sin(t_0) * (0.0 - x_45_scale))))) / ((double) M_PI);
	double tmp;
	if (b <= 4.9e-34) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / ((double) M_PI)))))) / ((double) M_PI));
	} else if (b <= 1.5e+39) {
		tmp = t_1;
	} else if (b <= 2.02e+136) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((((double) M_PI) * (angle / 180.0))))) / ((double) M_PI));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = (180.0 * Math.atan(((y_45_scale * Math.cos(t_0)) / (Math.sin(t_0) * (0.0 - x_45_scale))))) / Math.PI;
	double tmp;
	if (b <= 4.9e-34) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.sin((angle / (180.0 / Math.PI))))) / Math.PI);
	} else if (b <= 1.5e+39) {
		tmp = t_1;
	} else if (b <= 2.02e+136) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan((Math.PI * (angle / 180.0))))) / Math.PI);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (angle * 0.005555555555555556)
	t_1 = (180.0 * math.atan(((y_45_scale * math.cos(t_0)) / (math.sin(t_0) * (0.0 - x_45_scale))))) / math.pi
	tmp = 0
	if b <= 4.9e-34:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.sin((angle / (180.0 / math.pi))))) / math.pi)
	elif b <= 1.5e+39:
		tmp = t_1
	elif b <= 2.02e+136:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan((math.pi * (angle / 180.0))))) / math.pi)
	else:
		tmp = t_1
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * cos(t_0)) / Float64(sin(t_0) * Float64(0.0 - x_45_scale))))) / pi)
	tmp = 0.0
	if (b <= 4.9e-34)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(Float64(angle / Float64(180.0 / pi))))) / pi));
	elseif (b <= 1.5e+39)
		tmp = t_1;
	elseif (b <= 2.02e+136)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(pi * Float64(angle / 180.0))))) / pi));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (angle * 0.005555555555555556);
	t_1 = (180.0 * atan(((y_45_scale * cos(t_0)) / (sin(t_0) * (0.0 - x_45_scale))))) / pi;
	tmp = 0.0;
	if (b <= 4.9e-34)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / pi))))) / pi);
	elseif (b <= 1.5e+39)
		tmp = t_1;
	elseif (b <= 2.02e+136)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((pi * (angle / 180.0))))) / pi);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * N[(0.0 - x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]}, If[LessEqual[b, 4.9e-34], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e+39], t$95$1, If[LessEqual[b, 2.02e+136], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.89999999999999962e-34

   1. Initial program 13.0%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 26.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 49.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 49.7%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 55.6%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot 1}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   15. Applied egg-rr 56.0%

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

   if 4.89999999999999962e-34 < b < 1.5e39 or 2.02000000000000002e136 < b

   1. Initial program 19.1%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 30.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-1 \cdot \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      15. PI-lowering-PI.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(y-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 58.3%

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

   if 1.5e39 < b < 2.02000000000000002e136

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 39.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 73.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   12. Applied egg-rr 57.0%

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

       1. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       8. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\left(\frac{angle}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

       10. PI-lowering-PI.f64 73.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

   14. Applied egg-rr 73.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-34}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(0 - x-scale\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.02 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \frac{angle}{180}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(0 - x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= b 9.8e+72)
   (* 180.0 (/ (atan (* (/ y-scale x-scale) (sin (/ angle (/ 180.0 PI))))) PI))
   (/ (* 180.0 (atan (/ (* y-scale -180.0) (* angle (* x-scale PI))))) PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 9.8e+72) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 9.8e+72) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.sin((angle / (180.0 / Math.PI))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * Math.PI))))) / Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 9.8e+72:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.sin((angle / (180.0 / math.pi))))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * math.pi))))) / math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 9.8e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * sin(Float64(angle / Float64(180.0 / pi))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(x_45_scale * pi))))) / pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 9.8e+72)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * sin((angle / (180.0 / pi))))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * pi))))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 9.8e+72], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Sin[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.80000000000000012e72

   1. Initial program 14.9%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 29.5%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 48.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 48.7%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 55.1%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot 1}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   15. Applied egg-rr 55.8%

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

   if 9.80000000000000012e72 < b

   1. Initial program 5.5%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)\right), \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 5.1%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-180 \cdot y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. PI-lowering-PI.f64 49.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 49.1%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(\frac{\tan \left(\frac{angle}{\frac{180}{\pi}}\right)}{\frac{x-scale}{y-scale}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 2e+17)
   (/ (* 180.0 (atan (/ (* y-scale -180.0) (* angle (* x-scale PI))))) PI)
   (*
    (/ 180.0 PI)
    (atan (/ (tan (/ angle (/ 180.0 PI))) (/ x-scale y-scale))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2e+17) {
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI);
	} else {
		tmp = (180.0 / ((double) M_PI)) * atan((tan((angle / (180.0 / ((double) M_PI)))) / (x_45_scale / y_45_scale)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2e+17) {
		tmp = (180.0 * Math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * Math.PI))))) / Math.PI;
	} else {
		tmp = (180.0 / Math.PI) * Math.atan((Math.tan((angle / (180.0 / Math.PI))) / (x_45_scale / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 2e+17:
		tmp = (180.0 * math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * math.pi))))) / math.pi
	else:
		tmp = (180.0 / math.pi) * math.atan((math.tan((angle / (180.0 / math.pi))) / (x_45_scale / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 2e+17)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(x_45_scale * pi))))) / pi);
	else
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(tan(Float64(angle / Float64(180.0 / pi))) / Float64(x_45_scale / y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 2e+17)
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * pi))))) / pi;
	else
		tmp = (180.0 / pi) * atan((tan((angle / (180.0 / pi))) / (x_45_scale / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 2e+17], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(N[Tan[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2e17

   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 11.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)\right), \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 11.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-180 \cdot y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. PI-lowering-PI.f64 44.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 44.1%

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

   if 2e17 < a

   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 14.4%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 20.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 57.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 57.1%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

   12. Applied egg-rr 67.7%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)\right), \color{blue}{\left(\frac{180}{\mathsf{PI}\left(\right)}\right)}\right) \]

       4. atan-lowering-atan.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)\right)\right), \left(\frac{\color{blue}{180}}{\mathsf{PI}\left(\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\left(\tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{y-scale}{x-scale}\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\left(\tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{\frac{x-scale}{y-scale}}\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\left(\frac{\tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)}{\frac{x-scale}{y-scale}}\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\tan \left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right), \left(\frac{x-scale}{y-scale}\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       9. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\frac{angle}{\frac{180}{\mathsf{PI}\left(\right)}}\right)\right), \left(\frac{x-scale}{y-scale}\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right)\right), \left(\frac{x-scale}{y-scale}\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{x-scale}{y-scale}\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       12. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{x-scale}{y-scale}\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(x-scale, y-scale\right)\right)\right), \left(\frac{180}{\mathsf{PI}\left(\right)}\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(x-scale, y-scale\right)\right)\right), \mathsf{/.f64}\left(180, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       15. PI-lowering-PI.f64 67.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{/.f64}\left(x-scale, y-scale\right)\right)\right), \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right) \]

   14. Applied egg-rr 67.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(\frac{\tan \left(\frac{angle}{\frac{180}{\pi}}\right)}{\frac{x-scale}{y-scale}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{angle}{\frac{180}{\pi}}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 1.32e+16)
   (/ (* 180.0 (atan (/ (* y-scale -180.0) (* angle (* x-scale PI))))) PI)
   (*
    (/ 180.0 PI)
    (atan (* (/ y-scale x-scale) (tan (/ angle (/ 180.0 PI))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.32e+16) {
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI);
	} else {
		tmp = (180.0 / ((double) M_PI)) * atan(((y_45_scale / x_45_scale) * tan((angle / (180.0 / ((double) M_PI))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.32e+16) {
		tmp = (180.0 * Math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * Math.PI))))) / Math.PI;
	} else {
		tmp = (180.0 / Math.PI) * Math.atan(((y_45_scale / x_45_scale) * Math.tan((angle / (180.0 / Math.PI)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 1.32e+16:
		tmp = (180.0 * math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * math.pi))))) / math.pi
	else:
		tmp = (180.0 / math.pi) * math.atan(((y_45_scale / x_45_scale) * math.tan((angle / (180.0 / math.pi)))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 1.32e+16)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(x_45_scale * pi))))) / pi);
	else
		tmp = Float64(Float64(180.0 / pi) * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(angle / Float64(180.0 / pi))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 1.32e+16)
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * pi))))) / pi;
	else
		tmp = (180.0 / pi) * atan(((y_45_scale / x_45_scale) * tan((angle / (180.0 / pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 1.32e+16], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(180.0 / Pi), $MachinePrecision] * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.32e16

   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 11.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)\right), \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 11.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-180 \cdot y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. PI-lowering-PI.f64 44.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 44.1%

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

   if 1.32e16 < a

   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 14.4%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 20.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 57.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 57.1%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \color{blue}{\left(\frac{180}{\mathsf{PI}\left(\right)}\right)}\right) \]

   12. Applied egg-rr 67.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\pi} \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\frac{angle}{\frac{180}{\pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 24.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2800000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(angle \cdot \frac{0.005555555555555556 \cdot \left(y-scale \cdot \pi\right)}{x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 2800000000.0)
   (/ (* 180.0 (atan (/ (* y-scale -180.0) (* angle (* x-scale PI))))) PI)
   (/
    (*
     180.0
     (atan (* angle (/ (* 0.005555555555555556 (* y-scale PI)) x-scale))))
    PI)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2800000000.0) {
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI);
	} else {
		tmp = (180.0 * atan((angle * ((0.005555555555555556 * (y_45_scale * ((double) M_PI))) / x_45_scale)))) / ((double) M_PI);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2800000000.0) {
		tmp = (180.0 * Math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * Math.PI))))) / Math.PI;
	} else {
		tmp = (180.0 * Math.atan((angle * ((0.005555555555555556 * (y_45_scale * Math.PI)) / x_45_scale)))) / Math.PI;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 2800000000.0:
		tmp = (180.0 * math.atan(((y_45_scale * -180.0) / (angle * (x_45_scale * math.pi))))) / math.pi
	else:
		tmp = (180.0 * math.atan((angle * ((0.005555555555555556 * (y_45_scale * math.pi)) / x_45_scale)))) / math.pi
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 2800000000.0)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(x_45_scale * pi))))) / pi);
	else
		tmp = Float64(Float64(180.0 * atan(Float64(angle * Float64(Float64(0.005555555555555556 * Float64(y_45_scale * pi)) / x_45_scale)))) / pi);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 2800000000.0)
		tmp = (180.0 * atan(((y_45_scale * -180.0) / (angle * (x_45_scale * pi))))) / pi;
	else
		tmp = (180.0 * atan((angle * ((0.005555555555555556 * (y_45_scale * pi)) / x_45_scale)))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 2800000000.0], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(angle * N[(N[(0.005555555555555556 * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.8e9

   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 11.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - a \cdot a\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)\right), \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 11.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(-180 \cdot y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. PI-lowering-PI.f64 44.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-180, y-scale\right), \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(x-scale, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 44.1%

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

   if 2.8e9 < a

   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 14.4%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. Simplified 20.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

      13. PI-lowering-PI.f64 57.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. Simplified 57.1%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 62.1%

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

   14. Taylor expanded in angle around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
   15. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale} \cdot \frac{1}{180}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) \cdot \frac{1}{180}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} \cdot \frac{1}{180}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\left(\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

       10. PI-lowering-PI.f64 62.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI.f64}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   16. Simplified 62.7%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2800000000:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(angle \cdot \frac{0.005555555555555556 \cdot \left(y-scale \cdot \pi\right)}{x-scale}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.7% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.5%

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

3. Simplified 12.1%

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

5. Taylor expanded in x-scale around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

7. Simplified 28.1%

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

8. Taylor expanded in a around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   13. PI-lowering-PI.f64 47.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

10. Simplified 47.8%

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

11. Taylor expanded in angle around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \color{blue}{1}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
12. Step-by-step derivation

13. Simplified 51.6%

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

14. Taylor expanded in angle around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
15. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale} \cdot \frac{1}{180}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) \cdot \frac{1}{180}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} \cdot \frac{1}{180}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\left(\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

    10. PI-lowering-PI.f64 46.2%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI.f64}\left(\right)\right)\right), x-scale\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

16. Simplified 46.2%

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

Alternative 10: 45.7% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.5%

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

3. Simplified 12.1%

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

5. Taylor expanded in x-scale around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

7. Simplified 28.1%

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

8. Taylor expanded in a around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   13. PI-lowering-PI.f64 47.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

10. Simplified 47.8%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

12. Applied egg-rr 49.5%

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

13. Taylor expanded in angle around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]
14. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    3. PI-lowering-PI.f64 46.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y-scale, x-scale\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

15. Simplified 46.2%

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

16. Final simplification 46.2%

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

Alternative 11: 45.7% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.5%

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

3. Simplified 12.1%

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

5. Taylor expanded in x-scale around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

7. Simplified 28.1%

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

8. Taylor expanded in a around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{180} \cdot angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   13. PI-lowering-PI.f64 47.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(180, \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

10. Simplified 47.8%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}{x-scale \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{180}\right) \]

12. Applied egg-rr 49.5%

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

13. Taylor expanded in angle around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\left(\frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale} \cdot \frac{1}{180}\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\left(\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) \cdot \frac{1}{180}\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} \cdot \frac{1}{180}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\left(angle \cdot \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{180} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\left(\frac{1}{180} \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(y-scale \cdot \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI}\left(\right)\right)\right), x-scale\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

    10. PI-lowering-PI.f64 46.1%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(y-scale, \mathsf{PI.f64}\left(\right)\right)\right), x-scale\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), 180\right) \]

15. Simplified 46.1%

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

16. Final simplification 46.1%

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

Reproduce

herbie shell --seed 2024155 
(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)))