raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.4% → 55.8%

Time: 35.3s

Alternatives: 12

Speedup: 49.1×

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: 13.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_3 := y-scale \cdot t\_1\\ \mathbf{if}\;b\_m \leq 1.95 \cdot 10^{-262}:\\ \;\;\;\;\frac{\tan^{-1} \left(\tan t\_2 \cdot \frac{y-scale}{x-scale}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 3.1 \cdot 10^{-177}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{t\_3}{x-scale \cdot \sin \left(\left(-t\_2\right) + \frac{\pi}{2}\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.25 \cdot 10^{+100}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{t\_3}{x-scale \cdot \sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {t\_0}^{2}\right)\right)}{x-scale \cdot \left(t\_0 \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))))
        (t_1 (sin (* 0.005555555555555556 (* angle PI))))
        (t_2 (* (* PI angle) 0.005555555555555556))
        (t_3 (* y-scale t_1)))
   (if (<= b_m 1.95e-262)
     (* (/ (atan (* (tan t_2) (/ y-scale x-scale))) PI) 180.0)
     (if (<= b_m 3.1e-177)
       (* 180.0 (/ (atan (/ t_3 (* x-scale (sin (+ (- t_2) (/ PI 2.0)))))) PI))
       (if (<= b_m 1.25e+100)
         (*
          180.0
          (/
           (atan
            (/
             t_3
             (*
              x-scale
              (sin
               (fma (fabs (* 0.005555555555555556 angle)) PI (/ PI 2.0))))))
           PI))
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/
              (*
               y-scale
               (+
                0.5
                (fma
                 0.5
                 (cos (* 0.011111111111111112 (* angle PI)))
                 (pow t_0 2.0))))
              (* x-scale (* t_0 t_1)))))
           PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_3 = y_45_scale * t_1;
	double tmp;
	if (b_m <= 1.95e-262) {
		tmp = (atan((tan(t_2) * (y_45_scale / x_45_scale))) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 3.1e-177) {
		tmp = 180.0 * (atan((t_3 / (x_45_scale * sin((-t_2 + (((double) M_PI) / 2.0)))))) / ((double) M_PI));
	} else if (b_m <= 1.25e+100) {
		tmp = 180.0 * (atan((t_3 / (x_45_scale * sin(fma(fabs((0.005555555555555556 * angle)), ((double) M_PI), (((double) M_PI) / 2.0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (0.5 + fma(0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), pow(t_0, 2.0)))) / (x_45_scale * (t_0 * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_3 = Float64(y_45_scale * t_1)
	tmp = 0.0
	if (b_m <= 1.95e-262)
		tmp = Float64(Float64(atan(Float64(tan(t_2) * Float64(y_45_scale / x_45_scale))) / pi) * 180.0);
	elseif (b_m <= 3.1e-177)
		tmp = Float64(180.0 * Float64(atan(Float64(t_3 / Float64(x_45_scale * sin(Float64(Float64(-t_2) + Float64(pi / 2.0)))))) / pi));
	elseif (b_m <= 1.25e+100)
		tmp = Float64(180.0 * Float64(atan(Float64(t_3 / Float64(x_45_scale * sin(fma(abs(Float64(0.005555555555555556 * angle)), pi, Float64(pi / 2.0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(0.5 + fma(0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), (t_0 ^ 2.0)))) / Float64(x_45_scale * Float64(t_0 * t_1))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = N[(y$45$scale * t$95$1), $MachinePrecision]}, If[LessEqual[b$95$m, 1.95e-262], N[(N[(N[ArcTan[N[(N[Tan[t$95$2], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 3.1e-177], N[(180.0 * N[(N[ArcTan[N[(t$95$3 / N[(x$45$scale * N[Sin[N[((-t$95$2) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.25e+100], N[(180.0 * N[(N[ArcTan[N[(t$95$3 / N[(x$45$scale * N[Sin[N[(N[Abs[N[(0.005555555555555556 * angle), $MachinePrecision]], $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.94999999999999992e-262

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.3%

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

   if 1.94999999999999992e-262 < b < 3.10000000000000018e-177

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

       3. sin-+PI/2-rev N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 N/A

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

       14. lower-/.f64 45.9

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

   11. Applied rewrites 45.9%

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

   if 3.10000000000000018e-177 < b < 1.25e100

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

       3. sin-+PI/2-rev N/A

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. fabs-mul N/A

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

       9. lift-PI.f64 N/A

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

       10. add-exp-log N/A

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

       11. exp-fabs N/A

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

       12. add-exp-log N/A

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

       13. lift-PI.f64 N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-PI.f64 N/A

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

       18. lower-/.f64 45.6

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

   11. Applied rewrites 45.6%

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

   if 1.25e100 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{\color{blue}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       10. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       11. lower-/.f64 43.6

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

   11. Applied rewrites 43.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       10. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       11. lower-/.f64 43.6

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

   13. Applied rewrites 43.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := y-scale \cdot t\_0\\ \mathbf{if}\;b\_m \leq 1.95 \cdot 10^{-262}:\\ \;\;\;\;\frac{\tan^{-1} \left(\tan t\_1 \cdot \frac{y-scale}{x-scale}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 3.1 \cdot 10^{-177}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{t\_2}{x-scale \cdot \sin \left(\left(-t\_1\right) + \frac{\pi}{2}\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.1 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{t\_2}{x-scale \cdot \sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {1}^{2}\right)\right)}{x-scale \cdot \left(1 \cdot t\_0\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (* y-scale t_0)))
   (if (<= b_m 1.95e-262)
     (* (/ (atan (* (tan t_1) (/ y-scale x-scale))) PI) 180.0)
     (if (<= b_m 3.1e-177)
       (* 180.0 (/ (atan (/ t_2 (* x-scale (sin (+ (- t_1) (/ PI 2.0)))))) PI))
       (if (<= b_m 3.1e+99)
         (*
          180.0
          (/
           (atan
            (/
             t_2
             (*
              x-scale
              (sin
               (fma (fabs (* 0.005555555555555556 angle)) PI (/ PI 2.0))))))
           PI))
         (*
          180.0
          (/
           (atan
            (*
             -0.5
             (/
              (*
               y-scale
               (+
                0.5
                (fma
                 0.5
                 (cos (* 0.011111111111111112 (* angle PI)))
                 (pow 1.0 2.0))))
              (* x-scale (* 1.0 t_0)))))
           PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = y_45_scale * t_0;
	double tmp;
	if (b_m <= 1.95e-262) {
		tmp = (atan((tan(t_1) * (y_45_scale / x_45_scale))) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 3.1e-177) {
		tmp = 180.0 * (atan((t_2 / (x_45_scale * sin((-t_1 + (((double) M_PI) / 2.0)))))) / ((double) M_PI));
	} else if (b_m <= 3.1e+99) {
		tmp = 180.0 * (atan((t_2 / (x_45_scale * sin(fma(fabs((0.005555555555555556 * angle)), ((double) M_PI), (((double) M_PI) / 2.0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (0.5 + fma(0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), pow(1.0, 2.0)))) / (x_45_scale * (1.0 * t_0))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(y_45_scale * t_0)
	tmp = 0.0
	if (b_m <= 1.95e-262)
		tmp = Float64(Float64(atan(Float64(tan(t_1) * Float64(y_45_scale / x_45_scale))) / pi) * 180.0);
	elseif (b_m <= 3.1e-177)
		tmp = Float64(180.0 * Float64(atan(Float64(t_2 / Float64(x_45_scale * sin(Float64(Float64(-t_1) + Float64(pi / 2.0)))))) / pi));
	elseif (b_m <= 3.1e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(t_2 / Float64(x_45_scale * sin(fma(abs(Float64(0.005555555555555556 * angle)), pi, Float64(pi / 2.0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(0.5 + fma(0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), (1.0 ^ 2.0)))) / Float64(x_45_scale * Float64(1.0 * t_0))))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale * t$95$0), $MachinePrecision]}, If[LessEqual[b$95$m, 1.95e-262], N[(N[(N[ArcTan[N[(N[Tan[t$95$1], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 3.1e-177], N[(180.0 * N[(N[ArcTan[N[(t$95$2 / N[(x$45$scale * N[Sin[N[((-t$95$1) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.1e+99], N[(180.0 * N[(N[ArcTan[N[(t$95$2 / N[(x$45$scale * N[Sin[N[(N[Abs[N[(0.005555555555555556 * angle), $MachinePrecision]], $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(1.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.94999999999999992e-262

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.3%

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

   if 1.94999999999999992e-262 < b < 3.10000000000000018e-177

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

       3. sin-+PI/2-rev N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 N/A

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

       14. lower-/.f64 45.9

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

   11. Applied rewrites 45.9%

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

   if 3.10000000000000018e-177 < b < 3.1000000000000001e99

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

       1. lift-cos.f64 N/A

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

       2. cos-fabs-rev N/A

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

       3. sin-+PI/2-rev N/A

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

       4. lower-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. fabs-mul N/A

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

       9. lift-PI.f64 N/A

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

       10. add-exp-log N/A

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

       11. exp-fabs N/A

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

       12. add-exp-log N/A

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

       13. lift-PI.f64 N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lift-PI.f64 N/A

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

       18. lower-/.f64 45.6

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

   11. Applied rewrites 45.6%

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

   if 3.1000000000000001e99 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {1}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   11. Step-by-step derivation

   12. Applied rewrites 43.2%

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

   13. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {1}^{2}\right)\right)}{x-scale \cdot \left(1 \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   14. Step-by-step derivation

   15. Applied rewrites 43.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {1}^{2}\right)\right)}{x-scale \cdot \left(1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 52.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\\ t_3 := -t\_1\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 8 \cdot 10^{-215}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, t\_0, {\cos t\_4}^{2}\right)\right)}{x-scale \cdot \frac{\sin \left(t\_1 - t\_3\right) + \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, t\_3\right)\right)}{2}}\right)}{\pi}\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, t\_0, {t\_2}^{2}\right)\right)}{x-scale \cdot \left(t\_2 \cdot \sin t\_4\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\tan t\_1 \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.011111111111111112 (* angle PI))))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (sin (fma (* PI angle) 0.005555555555555556 (/ PI 2.0))))
        (t_3 (- t_1))
        (t_4 (* 0.005555555555555556 (* angle PI))))
   (if (<= a 8e-215)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (* y-scale (+ 0.5 (fma 0.5 t_0 (pow (cos t_4) 2.0))))
          (*
           x-scale
           (/
            (+
             (sin (- t_1 t_3))
             (sin (fma (* PI angle) 0.005555555555555556 t_3)))
            2.0)))))
       PI))
     (if (<= a 1.12e-169)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ 0.5 (fma 0.5 t_0 (pow t_2 2.0))))
            (* x-scale (* t_2 (sin t_4))))))
         PI))
       (/ (* 180.0 (atan (* (tan t_1) (/ y-scale x-scale)))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = sin(fma((((double) M_PI) * angle), 0.005555555555555556, (((double) M_PI) / 2.0)));
	double t_3 = -t_1;
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a <= 8e-215) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (0.5 + fma(0.5, t_0, pow(cos(t_4), 2.0)))) / (x_45_scale * ((sin((t_1 - t_3)) + sin(fma((((double) M_PI) * angle), 0.005555555555555556, t_3))) / 2.0))))) / ((double) M_PI));
	} else if (a <= 1.12e-169) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (0.5 + fma(0.5, t_0, pow(t_2, 2.0)))) / (x_45_scale * (t_2 * sin(t_4)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan((tan(t_1) * (y_45_scale / x_45_scale)))) / ((double) M_PI);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = sin(fma(Float64(pi * angle), 0.005555555555555556, Float64(pi / 2.0)))
	t_3 = Float64(-t_1)
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a <= 8e-215)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(0.5 + fma(0.5, t_0, (cos(t_4) ^ 2.0)))) / Float64(x_45_scale * Float64(Float64(sin(Float64(t_1 - t_3)) + sin(fma(Float64(pi * angle), 0.005555555555555556, t_3))) / 2.0))))) / pi));
	elseif (a <= 1.12e-169)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(0.5 + fma(0.5, t_0, (t_2 ^ 2.0)))) / Float64(x_45_scale * Float64(t_2 * sin(t_4)))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(tan(t_1) * Float64(y_45_scale / x_45_scale)))) / pi);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-t$95$1)}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 8e-215], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(0.5 + N[(0.5 * t$95$0 + N[Power[N[Cos[t$95$4], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[(N[Sin[N[(t$95$1 - t$95$3), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.12e-169], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(0.5 + N[(0.5 * t$95$0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[Tan[t$95$1], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 8.00000000000000033e-215

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{\color{blue}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}{\pi} \]

       6. sin-cos-mult N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{2}}\right)}{\pi} \]

       7. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{2}}\right)}{\pi} \]

   11. Applied rewrites 43.3%

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

   if 8.00000000000000033e-215 < a < 1.11999999999999998e-169

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{\color{blue}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       10. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       11. lower-/.f64 43.6

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

   11. Applied rewrites 43.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. sin-+PI/2-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       10. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\pi}{2}\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       11. lower-/.f64 43.6

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

   13. Applied rewrites 43.6%

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

   if 1.11999999999999998e-169 < a

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.4%

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

Alternative 4: 49.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), \frac{\cos \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, t\_1\right)\right) + \cos \left(\left|t\_0\right| - t\_1\right)}{2}\right)\right)}{x-scale \cdot \left(\cos t\_2 \cdot \sin t\_2\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\tan t\_0 \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2 (* 0.005555555555555556 (* angle PI))))
   (if (<= a 1.12e-169)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (*
           y-scale
           (+
            0.5
            (fma
             0.5
             (cos (* 0.011111111111111112 (* angle PI)))
             (/
              (+
               (cos (fma (fabs (* 0.005555555555555556 angle)) PI t_1))
               (cos (- (fabs t_0) t_1)))
              2.0))))
          (* x-scale (* (cos t_2) (sin t_2))))))
       PI))
     (/ (* 180.0 (atan (* (tan t_0) (/ y-scale x-scale)))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a <= 1.12e-169) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (0.5 + fma(0.5, cos((0.011111111111111112 * (angle * ((double) M_PI)))), ((cos(fma(fabs((0.005555555555555556 * angle)), ((double) M_PI), t_1)) + cos((fabs(t_0) - t_1))) / 2.0)))) / (x_45_scale * (cos(t_2) * sin(t_2)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan((tan(t_0) * (y_45_scale / x_45_scale)))) / ((double) M_PI);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a <= 1.12e-169)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(0.5 + fma(0.5, cos(Float64(0.011111111111111112 * Float64(angle * pi))), Float64(Float64(cos(fma(abs(Float64(0.005555555555555556 * angle)), pi, t_1)) + cos(Float64(abs(t_0) - t_1))) / 2.0)))) / Float64(x_45_scale * Float64(cos(t_2) * sin(t_2)))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(tan(t_0) * Float64(y_45_scale / x_45_scale)))) / pi);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.12e-169], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Cos[N[(N[Abs[N[(0.005555555555555556 * angle), $MachinePrecision]], $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[Abs[t$95$0], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$2], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[Tan[t$95$0], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.11999999999999998e-169

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{\color{blue}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       2. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       3. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       4. cos-fabs-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right|\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       5. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right|\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       6. cos-neg-rev N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right|\right) \cdot \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       7. cos-mult N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{\cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right| + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right) + \cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right| - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right), \frac{\cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right| + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right) + \cos \left(\left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right| - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]

   11. Applied rewrites 43.6%

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

   if 1.11999999999999998e-169 < a

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.4%

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

Alternative 5: 49.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 1.12 \cdot 10^{-169}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\tan \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a 1.12e-169)
     (*
      180.0
      (/
       (atan (* -0.5 (/ (* y-scale 2.0) (* x-scale (* (cos t_0) (sin t_0))))))
       PI))
     (/
      (*
       180.0
       (atan
        (* (tan (* (* PI angle) 0.005555555555555556)) (/ y-scale x-scale))))
      PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a <= 1.12e-169) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan((tan(((((double) M_PI) * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / ((double) M_PI);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a <= 1.12e-169) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan((Math.tan(((Math.PI * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / Math.PI;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a <= 1.12e-169:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * math.sin(t_0)))))) / math.pi)
	else:
		tmp = (180.0 * math.atan((math.tan(((math.pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / math.pi
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a <= 1.12e-169)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(tan(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(y_45_scale / x_45_scale)))) / pi);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a <= 1.12e-169)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / pi);
	else
		tmp = (180.0 * atan((tan(((pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.12e-169], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[Tan[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.11999999999999998e-169

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\pi} \]

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 43.1%

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

   if 1.11999999999999998e-169 < a

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.4%

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

Alternative 6: 48.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-224}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\tan \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 2.6e-224)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (/
    (*
     180.0
     (atan
      (* (tan (* (* PI angle) 0.005555555555555556)) (/ y-scale x-scale))))
    PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.6e-224) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan((tan(((((double) M_PI) * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / ((double) M_PI);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.6e-224) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan((Math.tan(((Math.PI * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / Math.PI;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 2.6e-224:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = (180.0 * math.atan((math.tan(((math.pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / math.pi
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 2.6e-224)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(tan(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(y_45_scale / x_45_scale)))) / pi);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 2.6e-224)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	else
		tmp = (180.0 * atan((tan(((pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale)))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 2.6e-224], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[Tan[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6000000000000002e-224

   1. Initial program 13.4%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.2%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 12.2

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 12.7%

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

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if 2.6000000000000002e-224 < a

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.4%

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

Alternative 7: 48.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-224}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\tan \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \frac{y-scale}{x-scale}\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 2.6e-224)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (*
    (/
     (atan (* (tan (* (* PI angle) 0.005555555555555556)) (/ y-scale x-scale)))
     PI)
    180.0)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.6e-224) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = (atan((tan(((((double) M_PI) * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale))) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.6e-224) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = (Math.atan((Math.tan(((Math.PI * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale))) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 2.6e-224:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = (math.atan((math.tan(((math.pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale))) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 2.6e-224)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(Float64(atan(Float64(tan(Float64(Float64(pi * angle) * 0.005555555555555556)) * Float64(y_45_scale / x_45_scale))) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 2.6e-224)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	else
		tmp = (atan((tan(((pi * angle) * 0.005555555555555556)) * (y_45_scale / x_45_scale))) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 2.6e-224], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[Tan[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6000000000000002e-224

   1. Initial program 13.4%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.2%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 12.2

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 12.7%

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

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if 2.6000000000000002e-224 < a

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   11. Applied rewrites 47.3%

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

Alternative 8: 47.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.8 \cdot 10^{+100}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.8e+100)
   (*
    180.0
    (/
     (atan (/ (* y-scale (sin (* 0.005555555555555556 (* angle PI)))) x-scale))
     PI))
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.8e+100) {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * ((double) M_PI))))) / x_45_scale)) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.8e+100) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin((0.005555555555555556 * (angle * Math.PI)))) / x_45_scale)) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.8e+100:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin((0.005555555555555556 * (angle * math.pi)))) / x_45_scale)) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.8e+100)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) / x_45_scale)) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.8e+100)
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * pi)))) / x_45_scale)) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.8e+100], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e100

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 45.2%

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

   if 1.8e100 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.2%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 12.2

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 12.7%

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

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 37.1

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

   11. Applied rewrites 37.1%

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

Alternative 9: 44.1% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 480000000:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \frac{angle \cdot \left(y-scale \cdot \pi\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 480000000.0)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (/ (* angle (* y-scale PI)) x-scale)))
     PI))
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 480000000.0) {
		tmp = 180.0 * (atan((0.005555555555555556 * ((angle * (y_45_scale * ((double) M_PI))) / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 480000000.0) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * ((angle * (y_45_scale * Math.PI)) / x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 480000000.0:
		tmp = 180.0 * (math.atan((0.005555555555555556 * ((angle * (y_45_scale * math.pi)) / x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 480000000.0)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(Float64(angle * Float64(y_45_scale * pi)) / x_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 480000000.0)
		tmp = 180.0 * (atan((0.005555555555555556 * ((angle * (y_45_scale * pi)) / x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 480000000.0], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8e8

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 40.7

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

   12. Applied rewrites 40.7%

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

   if 4.8e8 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.2%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 12.2

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 12.7%

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

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 37.1

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

   11. Applied rewrites 37.1%

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

Alternative 10: 44.1% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.2 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \frac{angle \cdot \left(y-scale \cdot \pi\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 3.2e+99)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (/ (* angle (* y-scale PI)) x-scale)))
     PI))
   (* 180.0 (/ (atan (* -90.0 (/ y-scale (* angle (* x-scale PI))))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.2e+99) {
		tmp = 180.0 * (atan((0.005555555555555556 * ((angle * (y_45_scale * ((double) M_PI))) / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.2e+99) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * ((angle * (y_45_scale * Math.PI)) / x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 3.2e+99:
		tmp = 180.0 * (math.atan((0.005555555555555556 * ((angle * (y_45_scale * math.pi)) / x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.2e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(Float64(angle * Float64(y_45_scale * pi)) / x_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 3.2e+99)
		tmp = 180.0 * (atan((0.005555555555555556 * ((angle * (y_45_scale * pi)) / x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 3.2e+99], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.19999999999999999e99

   1. Initial program 13.4%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 23.4%

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

   6. Applied rewrites 26.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 45.7%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 40.7

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

   12. Applied rewrites 40.7%

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

   if 3.19999999999999999e99 < b

   1. Initial program 13.4%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.2%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-fabs.f64 12.2

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow-prod-down N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 12.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\pi} \]

      5. lower-PI.f64 33.9

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

   9. Applied rewrites 33.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.7% accurate, 28.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.4%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 23.4%

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

6. Applied rewrites 26.0%

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

7. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 45.7%

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

10. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 40.7

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

12. Applied rewrites 40.7%

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

Alternative 12: 19.1% accurate, 49.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} 0}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan 0.0) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(0.0) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(0.0) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(0.0) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(0.0) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(0.0) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.4%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.2%

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

5. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]

7. Applied rewrites 6.9%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \pi}}\right)}{\pi} \]

8. Taylor expanded in y-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 19.1%

    \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))