raw-angle from scale-rotated-ellipse

Percentage Accurate: 16.7% → 60.9%

Time: 39.0s

Alternatives: 20

Speedup: 22.2×

• could not determine a ground truth

  1. a = -2.718685185018452e+185
  2. b = 8.148339656667858e-269
  3. angle = -2.842502766617874e+178
  4. x-scale = 1.8402462977609814e+237
  5. y-scale = -4.583088508613472e-273

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: 16.7% 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: 60.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ \mathbf{if}\;b\_m \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot t\_2}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 7.2 \cdot 10^{+176}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(2 \cdot \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(-0.5, t\_0, 0.5\right), \left(b\_m \cdot b\_m\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\right) \cdot \frac{y-scale \cdot -0.5}{x-scale}}{b\_m + a}}{\left(b\_m - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_2}{\sin t\_1 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* angle (* PI 0.011111111111111112))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1)))
   (if (<= b_m 3.8e-63)
     (*
      180.0
      (/
       (atan
        (/
         (*
          y-scale
          (sin (* 0.005555555555555556 (* angle (cbrt (* PI (* PI PI)))))))
         (* x-scale t_2)))
       PI))
     (if (<= b_m 7.2e+176)
       (*
        (/ 180.0 (sqrt PI))
        (/
         (atan
          (/
           (/
            (*
             (*
              2.0
              (fma
               a
               (* a (fma -0.5 t_0 0.5))
               (* (* b_m b_m) (fma 0.5 t_0 0.5))))
             (/ (* y-scale -0.5) x-scale))
            (+ b_m a))
           (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle))))))
         (sqrt PI)))
       (*
        180.0
        (/ (atan (/ (* y-scale t_2) (* (sin t_1) (- 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((angle * (((double) M_PI) * 0.011111111111111112)));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double tmp;
	if (b_m <= 3.8e-63) {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * cbrt((((double) M_PI) * (((double) M_PI) * ((double) M_PI)))))))) / (x_45_scale * t_2))) / ((double) M_PI));
	} else if (b_m <= 7.2e+176) {
		tmp = (180.0 / sqrt(((double) M_PI))) * (atan(((((2.0 * fma(a, (a * fma(-0.5, t_0, 0.5)), ((b_m * b_m) * fma(0.5, t_0, 0.5)))) * ((y_45_scale * -0.5) / x_45_scale)) / (b_m + a)) / ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle)))))) / sqrt(((double) M_PI)));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * t_2) / (sin(t_1) * -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(angle * Float64(pi * 0.011111111111111112)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	tmp = 0.0
	if (b_m <= 3.8e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * cbrt(Float64(pi * Float64(pi * pi))))))) / Float64(x_45_scale * t_2))) / pi));
	elseif (b_m <= 7.2e+176)
		tmp = Float64(Float64(180.0 / sqrt(pi)) * Float64(atan(Float64(Float64(Float64(Float64(2.0 * fma(a, Float64(a * fma(-0.5, t_0, 0.5)), Float64(Float64(b_m * b_m) * fma(0.5, t_0, 0.5)))) * Float64(Float64(y_45_scale * -0.5) / x_45_scale)) / Float64(b_m + a)) / Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))))) / sqrt(pi)));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_2) / Float64(sin(t_1) * Float64(-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[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[b$95$m, 3.8e-63], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 7.2e+176], N[(N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[ArcTan[N[(N[(N[(N[(2.0 * N[(a * N[(a * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y$45$scale * -0.5), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] / N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$2), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.80000000000000017e-63

   1. Initial program 18.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 56.5

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

   8. Simplified 56.5%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 60.2

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

   10. Applied egg-rr 60.2%

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

   if 3.80000000000000017e-63 < b < 7.19999999999999983e176

   1. Initial program 28.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 42.2%

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

   7. Applied egg-rr 41.7%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 45.9%

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

   12. Applied egg-rr 62.4%

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

   if 7.19999999999999983e176 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 80.3

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

   8. Simplified 80.3%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+176}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(2 \cdot \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(-0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\right) \cdot \frac{y-scale \cdot -0.5}{x-scale}}{b + a}}{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ \mathbf{if}\;b\_m \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot t\_2}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{\left(b\_m + a\right) \cdot \left(\left(b\_m - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \cdot \frac{2 \cdot \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(-0.5, t\_0, 0.5\right), \left(b\_m \cdot b\_m\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)}{x-scale}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_2}{\sin t\_1 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* angle (* PI 0.011111111111111112))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1)))
   (if (<= b_m 2.7e+18)
     (*
      180.0
      (/
       (atan
        (/
         (*
          y-scale
          (sin (* 0.005555555555555556 (* angle (cbrt (* PI (* PI PI)))))))
         (* x-scale t_2)))
       PI))
     (if (<= b_m 9.5e+153)
       (*
        (/ 180.0 (sqrt PI))
        (/
         (atan
          (*
           (/
            (* y-scale -0.5)
            (*
             (+ b_m a)
             (* (- b_m a) (sin (* PI (* 0.005555555555555556 angle))))))
           (/
            (*
             2.0
             (fma
              a
              (* a (fma -0.5 t_0 0.5))
              (* (* b_m b_m) (fma 0.5 t_0 0.5))))
            x-scale)))
         (sqrt PI)))
       (*
        180.0
        (/ (atan (/ (* y-scale t_2) (* (sin t_1) (- 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((angle * (((double) M_PI) * 0.011111111111111112)));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double tmp;
	if (b_m <= 2.7e+18) {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * cbrt((((double) M_PI) * (((double) M_PI) * ((double) M_PI)))))))) / (x_45_scale * t_2))) / ((double) M_PI));
	} else if (b_m <= 9.5e+153) {
		tmp = (180.0 / sqrt(((double) M_PI))) * (atan((((y_45_scale * -0.5) / ((b_m + a) * ((b_m - a) * sin((((double) M_PI) * (0.005555555555555556 * angle)))))) * ((2.0 * fma(a, (a * fma(-0.5, t_0, 0.5)), ((b_m * b_m) * fma(0.5, t_0, 0.5)))) / x_45_scale))) / sqrt(((double) M_PI)));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * t_2) / (sin(t_1) * -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(angle * Float64(pi * 0.011111111111111112)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	tmp = 0.0
	if (b_m <= 2.7e+18)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * cbrt(Float64(pi * Float64(pi * pi))))))) / Float64(x_45_scale * t_2))) / pi));
	elseif (b_m <= 9.5e+153)
		tmp = Float64(Float64(180.0 / sqrt(pi)) * Float64(atan(Float64(Float64(Float64(y_45_scale * -0.5) / Float64(Float64(b_m + a) * Float64(Float64(b_m - a) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))))) * Float64(Float64(2.0 * fma(a, Float64(a * fma(-0.5, t_0, 0.5)), Float64(Float64(b_m * b_m) * fma(0.5, t_0, 0.5)))) / x_45_scale))) / sqrt(pi)));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_2) / Float64(sin(t_1) * Float64(-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[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[b$95$m, 2.7e+18], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 9.5e+153], N[(N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[ArcTan[N[(N[(N[(y$45$scale * -0.5), $MachinePrecision] / N[(N[(b$95$m + a), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(a * N[(a * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$2), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.7e18

   1. Initial program 19.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.7

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

   8. Simplified 55.7%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 58.9

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

   10. Applied egg-rr 58.9%

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

   if 2.7e18 < b < 9.4999999999999995e153

   1. Initial program 30.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 50.3%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 53.9%

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

   12. Applied egg-rr 63.2%

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

   if 9.4999999999999995e153 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 76.4

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

   8. Simplified 76.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \cdot \frac{2 \cdot \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(-0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\right)}{x-scale}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 1.7 \cdot 10^{+24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin t\_0 \cdot \left(-x-scale\right)}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 1.7e+24)
     (*
      180.0
      (/
       (atan
        (/
         (*
          y-scale
          (sin (* 0.005555555555555556 (* angle (cbrt (* PI (* PI PI)))))))
         (* x-scale (cos t_0))))
       PI))
     (*
      (/ 180.0 (sqrt PI))
      (/
       (atan
        (/
         (* y-scale (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5))
         (* (sin t_0) (- x-scale))))
       (sqrt PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 1.7e+24) {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * cbrt((((double) M_PI) * (((double) M_PI) * ((double) M_PI)))))))) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else {
		tmp = (180.0 / sqrt(((double) M_PI))) * (atan(((y_45_scale * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)) / (sin(t_0) * -x_45_scale))) / sqrt(((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(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 1.7e+24)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * cbrt(Float64(pi * Float64(pi * pi))))))) / Float64(x_45_scale * cos(t_0)))) / pi));
	else
		tmp = Float64(Float64(180.0 / sqrt(pi)) * Float64(atan(Float64(Float64(y_45_scale * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)) / Float64(sin(t_0) * Float64(-x_45_scale)))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.7e+24], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[ArcTan[N[(N[(y$45$scale * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7e24

   1. Initial program 19.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.7

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

   8. Simplified 55.7%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 58.9

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

   10. Applied egg-rr 58.9%

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

   if 1.7e24 < b

   1. Initial program 18.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 30.2%

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

   7. Applied egg-rr 29.8%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 32.3%

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

   11. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

   13. Simplified 64.7%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\pi \cdot \left(\pi \cdot \pi\right)}\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin t\_0 \cdot \left(-x-scale\right)}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a 1.3e-98)
     (*
      (/ 180.0 (sqrt PI))
      (/
       (atan
        (/
         (* y-scale (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5))
         (* (sin t_0) (- x-scale))))
       (sqrt PI)))
     (*
      180.0
      (/
       (atan
        (/
         (*
          y-scale
          (sin (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI))))))
         (* x-scale (cos t_0))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a <= 1.3e-98) {
		tmp = (180.0 / sqrt(((double) M_PI))) * (atan(((y_45_scale * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)) / (sin(t_0) * -x_45_scale))) / sqrt(((double) M_PI)));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))) / (x_45_scale * cos(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 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a <= 1.3e-98)
		tmp = Float64(Float64(180.0 / sqrt(pi)) * Float64(atan(Float64(Float64(y_45_scale * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)) / Float64(sin(t_0) * Float64(-x_45_scale)))) / sqrt(pi)));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi)))))) / Float64(x_45_scale * cos(t_0)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.3e-98], N[(N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[ArcTan[N[(N[(y$45$scale * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.30000000000000007e-98

   1. Initial program 19.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 30.1%

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

   7. Applied egg-rr 28.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 31.2%

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

   11. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

   13. Simplified 44.5%

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

   if 1.30000000000000007e-98 < a

   1. Initial program 18.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 37.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 58.6

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

   8. Simplified 58.6%

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

      1. lift-PI.f64 58.6

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 63.2

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

   10. Applied egg-rr 63.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;a \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{\sin t\_0 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)}{x-scale \cdot t\_1}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (cos t_0)))
   (if (<= a 3.2e-89)
     (* 180.0 (/ (atan (/ (* y-scale t_1) (* (sin t_0) (- x-scale)))) PI))
     (*
      180.0
      (/
       (atan
        (/
         (*
          y-scale
          (sin (* 0.005555555555555556 (* angle (* (sqrt PI) (sqrt PI))))))
         (* x-scale 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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double tmp;
	if (a <= 3.2e-89) {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (sin(t_0) * -x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))) / (x_45_scale * t_1))) / ((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 t_1 = Math.cos(t_0);
	double tmp;
	if (a <= 3.2e-89) {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (Math.sin(t_0) * -x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin((0.005555555555555556 * (angle * (Math.sqrt(Math.PI) * Math.sqrt(Math.PI)))))) / (x_45_scale * t_1))) / 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)
	t_1 = math.cos(t_0)
	tmp = 0
	if a <= 3.2e-89:
		tmp = 180.0 * (math.atan(((y_45_scale * t_1) / (math.sin(t_0) * -x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin((0.005555555555555556 * (angle * (math.sqrt(math.pi) * math.sqrt(math.pi)))))) / (x_45_scale * t_1))) / 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))
	t_1 = cos(t_0)
	tmp = 0.0
	if (a <= 3.2e-89)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(sin(t_0) * Float64(-x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(pi) * sqrt(pi)))))) / Float64(x_45_scale * t_1))) / 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);
	t_1 = cos(t_0);
	tmp = 0.0;
	if (a <= 3.2e-89)
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (sin(t_0) * -x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * sin((0.005555555555555556 * (angle * (sqrt(pi) * sqrt(pi)))))) / (x_45_scale * t_1))) / 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]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a, 3.2e-89], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.19999999999999998e-89

   1. Initial program 19.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 44.2

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

   8. Simplified 44.2%

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

   if 3.19999999999999998e-89 < a

   1. Initial program 18.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 60.1

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

   8. Simplified 60.1%

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

      1. lift-PI.f64 60.1

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 63.8

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

   10. Applied egg-rr 63.8%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos t\_0}{t\_1 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{x-scale \cdot \cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a 5.5e-89)
     (* 180.0 (/ (atan (/ (* y-scale (cos t_0)) (* t_1 (- x-scale)))) PI))
     (*
      180.0
      (/
       (atan
        (/
         (* y-scale t_1)
         (*
          x-scale
          (cos (* (sqrt PI) (* (sqrt PI) (* 0.005555555555555556 angle)))))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a <= 5.5e-89) {
		tmp = 180.0 * (atan(((y_45_scale * cos(t_0)) / (t_1 * -x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * cos((sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * (0.005555555555555556 * angle))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (a <= 5.5e-89) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.cos(t_0)) / (t_1 * -x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (x_45_scale * Math.cos((Math.sqrt(Math.PI) * (Math.sqrt(Math.PI) * (0.005555555555555556 * angle))))))) / 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)
	t_1 = math.sin(t_0)
	tmp = 0
	if a <= 5.5e-89:
		tmp = 180.0 * (math.atan(((y_45_scale * math.cos(t_0)) / (t_1 * -x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * t_1) / (x_45_scale * math.cos((math.sqrt(math.pi) * (math.sqrt(math.pi) * (0.005555555555555556 * angle))))))) / 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))
	t_1 = sin(t_0)
	tmp = 0.0
	if (a <= 5.5e-89)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * cos(t_0)) / Float64(t_1 * Float64(-x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * cos(Float64(sqrt(pi) * Float64(sqrt(pi) * Float64(0.005555555555555556 * angle))))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (a <= 5.5e-89)
		tmp = 180.0 * (atan(((y_45_scale * cos(t_0)) / (t_1 * -x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * cos((sqrt(pi) * (sqrt(pi) * (0.005555555555555556 * angle))))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a, 5.5e-89], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * N[Cos[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.50000000000000012e-89

   1. Initial program 19.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 44.2

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

   8. Simplified 44.2%

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

   if 5.50000000000000012e-89 < a

   1. Initial program 18.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 60.1

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

   8. Simplified 60.1%

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

      1. lift-PI.f64 N/A

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

      2. associate-*r* N/A

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

      3. rem-square-sqrt N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 63.7

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

   10. Applied egg-rr 63.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos t\_0}{\sin t\_0 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= a 5.5e-89)
     (*
      180.0
      (/ (atan (/ (* y-scale (cos t_0)) (* (sin t_0) (- x-scale)))) PI))
     (* (atan (* (/ y-scale x-scale) (tan t_0))) (* 180.0 (/ 1.0 PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a <= 5.5e-89) {
		tmp = 180.0 * (atan(((y_45_scale * cos(t_0)) / (sin(t_0) * -x_45_scale))) / ((double) M_PI));
	} else {
		tmp = atan(((y_45_scale / x_45_scale) * tan(t_0))) * (180.0 * (1.0 / ((double) M_PI)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a <= 5.5e-89) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.cos(t_0)) / (Math.sin(t_0) * -x_45_scale))) / Math.PI);
	} else {
		tmp = Math.atan(((y_45_scale / x_45_scale) * Math.tan(t_0))) * (180.0 * (1.0 / Math.PI));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a <= 5.5e-89:
		tmp = 180.0 * (math.atan(((y_45_scale * math.cos(t_0)) / (math.sin(t_0) * -x_45_scale))) / math.pi)
	else:
		tmp = math.atan(((y_45_scale / x_45_scale) * math.tan(t_0))) * (180.0 * (1.0 / math.pi))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a <= 5.5e-89)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * cos(t_0)) / Float64(sin(t_0) * Float64(-x_45_scale)))) / pi));
	else
		tmp = Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) * Float64(180.0 * Float64(1.0 / pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a <= 5.5e-89)
		tmp = 180.0 * (atan(((y_45_scale * cos(t_0)) / (sin(t_0) * -x_45_scale))) / pi);
	else
		tmp = atan(((y_45_scale / x_45_scale) * tan(t_0))) * (180.0 * (1.0 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.5e-89], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.50000000000000012e-89

   1. Initial program 19.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 44.2

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

   8. Simplified 44.2%

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

   if 5.50000000000000012e-89 < a

   1. Initial program 18.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 60.1

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

   8. Simplified 60.1%

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

   10. Applied egg-rr 61.1%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{-89}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan t\_0\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale \cdot \left(\sin t\_0 \cdot \left(\left(b\_m + a\right) \cdot \left(a - b\_m\right)\right)\right)}\right)}{\sqrt{\pi}}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b\_m \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 2.8e+18)
     (* (atan (* (/ y-scale x-scale) (tan t_0))) (* 180.0 (/ 1.0 PI)))
     (if (<= b_m 4.1e+132)
       (*
        (/ 180.0 (sqrt PI))
        (/
         (atan
          (/
           (* y-scale (* b_m b_m))
           (* x-scale (* (sin t_0) (* (+ b_m a) (- a b_m))))))
         (sqrt PI)))
       (if (<= b_m 5.4e+178)
         (*
          180.0
          (/
           (atan
            (/
             (* y-scale (tan (* PI (* 0.005555555555555556 angle))))
             x-scale))
           PI))
         (*
          180.0
          (/
           (atan
            (fma
             -180.0
             (/ y-scale (* angle (* PI x-scale)))
             (/
              (* 180.0 (* y-scale (- a a)))
              (* angle (* b_m (* PI x-scale))))))
           PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 2.8e+18) {
		tmp = atan(((y_45_scale / x_45_scale) * tan(t_0))) * (180.0 * (1.0 / ((double) M_PI)));
	} else if (b_m <= 4.1e+132) {
		tmp = (180.0 / sqrt(((double) M_PI))) * (atan(((y_45_scale * (b_m * b_m)) / (x_45_scale * (sin(t_0) * ((b_m + a) * (a - b_m)))))) / sqrt(((double) M_PI)));
	} else if (b_m <= 5.4e+178) {
		tmp = 180.0 * (atan(((y_45_scale * tan((((double) M_PI) * (0.005555555555555556 * angle)))) / x_45_scale)) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(fma(-180.0, (y_45_scale / (angle * (((double) M_PI) * x_45_scale))), ((180.0 * (y_45_scale * (a - a))) / (angle * (b_m * (((double) M_PI) * 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(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 2.8e+18)
		tmp = Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(t_0))) * Float64(180.0 * Float64(1.0 / pi)));
	elseif (b_m <= 4.1e+132)
		tmp = Float64(Float64(180.0 / sqrt(pi)) * Float64(atan(Float64(Float64(y_45_scale * Float64(b_m * b_m)) / Float64(x_45_scale * Float64(sin(t_0) * Float64(Float64(b_m + a) * Float64(a - b_m)))))) / sqrt(pi)));
	elseif (b_m <= 5.4e+178)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * tan(Float64(pi * Float64(0.005555555555555556 * angle)))) / x_45_scale)) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(fma(-180.0, Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))), Float64(Float64(180.0 * Float64(y_45_scale * Float64(a - a))) / Float64(angle * Float64(b_m * Float64(pi * 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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 2.8e+18], N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+132], N[(N[(180.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[ArcTan[N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+178], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $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[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(180.0 * N[(y$45$scale * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(b$95$m * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.8e18

   1. Initial program 19.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.7

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

   8. Simplified 55.7%

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

   10. Applied egg-rr 57.5%

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

   if 2.8e18 < b < 4.09999999999999992e132

   1. Initial program 36.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 56.2%

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

   7. Applied egg-rr 55.5%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 60.6%

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

   11. Taylor expanded in angle around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 68.8

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

   13. Simplified 68.8%

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

   if 4.09999999999999992e132 < b < 5.40000000000000036e178

   1. Initial program 0.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 14.7%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 65.3

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

   8. Simplified 65.3%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 62.5

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

   10. Applied egg-rr 62.5%

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

   12. Applied egg-rr 65.3%

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

   if 5.40000000000000036e178 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 83.4

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

   8. Simplified 83.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \pi\right)\right)}\right)\right)}}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{180}{\sqrt{\pi}} \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(b \cdot b\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)}\right)}{\sqrt{\pi}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.6 \cdot 10^{-185}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b\_m \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.6e-185)
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))
   (if (<= b_m 1.05e+30)
     (*
      180.0
      (/
       (atan
        (* y-scale (/ (tan (* 0.005555555555555556 (* angle PI))) x-scale)))
       PI))
     (if (<= b_m 4.1e+132)
       (*
        180.0
        (/
         (atan
          (/
           (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (if (<= b_m 5.4e+178)
         (*
          180.0
          (/
           (atan
            (/
             (* y-scale (tan (* PI (* 0.005555555555555556 angle))))
             x-scale))
           PI))
         (*
          180.0
          (/
           (atan
            (fma
             -180.0
             (/ y-scale (* angle (* PI x-scale)))
             (/
              (* 180.0 (* y-scale (- a a)))
              (* angle (* b_m (* PI x-scale))))))
           PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 5.6e-185) {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * ((double) M_PI))))) / ((double) M_PI));
	} else if (b_m <= 1.05e+30) {
		tmp = 180.0 * (atan((y_45_scale * (tan((0.005555555555555556 * (angle * ((double) M_PI)))) / x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 4.1e+132) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else if (b_m <= 5.4e+178) {
		tmp = 180.0 * (atan(((y_45_scale * tan((((double) M_PI) * (0.005555555555555556 * angle)))) / x_45_scale)) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(fma(-180.0, (y_45_scale / (angle * (((double) M_PI) * x_45_scale))), ((180.0 * (y_45_scale * (a - a))) / (angle * (b_m * (((double) M_PI) * 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)
	tmp = 0.0
	if (b_m <= 5.6e-185)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_45_scale * pi)))) / pi));
	elseif (b_m <= 1.05e+30)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(tan(Float64(0.005555555555555556 * Float64(angle * pi))) / x_45_scale))) / pi));
	elseif (b_m <= 4.1e+132)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	elseif (b_m <= 5.4e+178)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * tan(Float64(pi * Float64(0.005555555555555556 * angle)))) / x_45_scale)) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(fma(-180.0, Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))), Float64(Float64(180.0 * Float64(y_45_scale * Float64(a - a))) / Float64(angle * Float64(b_m * Float64(pi * 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_] := If[LessEqual[b$95$m, 5.6e-185], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.05e+30], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+132], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+178], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $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[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(180.0 * N[(y$45$scale * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(b$95$m * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 5.59999999999999983e-185

   1. Initial program 18.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 32.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.9

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

   8. Simplified 55.9%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Simplified 55.4%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-/.f64 N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 56.3

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 56.3

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

   13. Applied egg-rr 56.3%

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

   if 5.59999999999999983e-185 < b < 1.05e30

   1. Initial program 21.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.9%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 53.1

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

   8. Simplified 53.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   10. Applied egg-rr 54.8%

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

   if 1.05e30 < b < 4.09999999999999992e132

   1. Initial program 41.5%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in y-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.6

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

   8. Simplified 54.6%

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

   9. Taylor expanded in y-scale around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 68.7

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

   11. Simplified 68.7%

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

   if 4.09999999999999992e132 < b < 5.40000000000000036e178

   1. Initial program 0.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 14.7%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 65.3

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

   8. Simplified 65.3%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 62.5

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

   10. Applied egg-rr 62.5%

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

   12. Applied egg-rr 65.3%

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

   if 5.40000000000000036e178 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 83.4

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

   8. Simplified 83.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \pi\right)\right)}\right)\right)}}{\pi} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-185}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b\_m \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.05e+30)
   (*
    (atan (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI)))))
    (* 180.0 (/ 1.0 PI)))
   (if (<= b_m 4.1e+132)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (if (<= b_m 5.4e+178)
       (*
        180.0
        (/
         (atan
          (/ (* y-scale (tan (* PI (* 0.005555555555555556 angle)))) x-scale))
         PI))
       (*
        180.0
        (/
         (atan
          (fma
           -180.0
           (/ y-scale (* angle (* PI x-scale)))
           (/ (* 180.0 (* y-scale (- a a))) (* angle (* b_m (* PI x-scale))))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.05e+30) {
		tmp = atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI)))))) * (180.0 * (1.0 / ((double) M_PI)));
	} else if (b_m <= 4.1e+132) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else if (b_m <= 5.4e+178) {
		tmp = 180.0 * (atan(((y_45_scale * tan((((double) M_PI) * (0.005555555555555556 * angle)))) / x_45_scale)) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(fma(-180.0, (y_45_scale / (angle * (((double) M_PI) * x_45_scale))), ((180.0 * (y_45_scale * (a - a))) / (angle * (b_m * (((double) M_PI) * 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)
	tmp = 0.0
	if (b_m <= 1.05e+30)
		tmp = Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi))))) * Float64(180.0 * Float64(1.0 / pi)));
	elseif (b_m <= 4.1e+132)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	elseif (b_m <= 5.4e+178)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * tan(Float64(pi * Float64(0.005555555555555556 * angle)))) / x_45_scale)) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(fma(-180.0, Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))), Float64(Float64(180.0 * Float64(y_45_scale * Float64(a - a))) / Float64(angle * Float64(b_m * Float64(pi * 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_] := If[LessEqual[b$95$m, 1.05e+30], N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+132], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+178], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $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[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(180.0 * N[(y$45$scale * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(b$95$m * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.05e30

   1. Initial program 19.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.1%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.2

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

   8. Simplified 55.2%

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

   10. Applied egg-rr 57.3%

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

   if 1.05e30 < b < 4.09999999999999992e132

   1. Initial program 41.5%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in y-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.6

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

   8. Simplified 54.6%

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

   9. Taylor expanded in y-scale around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 68.7

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

   11. Simplified 68.7%

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

   if 4.09999999999999992e132 < b < 5.40000000000000036e178

   1. Initial program 0.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 14.7%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 65.3

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

   8. Simplified 65.3%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 62.5

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

   10. Applied egg-rr 62.5%

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

   12. Applied egg-rr 65.3%

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

   if 5.40000000000000036e178 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 83.4

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

   8. Simplified 83.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \pi\right)\right)}\right)\right)}}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b\_m \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.05e+30)
   (/
    (*
     180.0
     (atan
      (* (/ y-scale x-scale) (tan (* 0.005555555555555556 (* angle PI))))))
    PI)
   (if (<= b_m 4.1e+132)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (if (<= b_m 5.4e+178)
       (*
        180.0
        (/
         (atan
          (/ (* y-scale (tan (* PI (* 0.005555555555555556 angle)))) x-scale))
         PI))
       (*
        180.0
        (/
         (atan
          (fma
           -180.0
           (/ y-scale (* angle (* PI x-scale)))
           (/ (* 180.0 (* y-scale (- a a))) (* angle (* b_m (* PI x-scale))))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.05e+30) {
		tmp = (180.0 * atan(((y_45_scale / x_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI))))))) / ((double) M_PI);
	} else if (b_m <= 4.1e+132) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else if (b_m <= 5.4e+178) {
		tmp = 180.0 * (atan(((y_45_scale * tan((((double) M_PI) * (0.005555555555555556 * angle)))) / x_45_scale)) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(fma(-180.0, (y_45_scale / (angle * (((double) M_PI) * x_45_scale))), ((180.0 * (y_45_scale * (a - a))) / (angle * (b_m * (((double) M_PI) * 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)
	tmp = 0.0
	if (b_m <= 1.05e+30)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi)))))) / pi);
	elseif (b_m <= 4.1e+132)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	elseif (b_m <= 5.4e+178)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * tan(Float64(pi * Float64(0.005555555555555556 * angle)))) / x_45_scale)) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(fma(-180.0, Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))), Float64(Float64(180.0 * Float64(y_45_scale * Float64(a - a))) / Float64(angle * Float64(b_m * Float64(pi * 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_] := If[LessEqual[b$95$m, 1.05e+30], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+132], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+178], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $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[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(180.0 * N[(y$45$scale * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(b$95$m * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.05e30

   1. Initial program 19.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 33.1%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.2

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

   8. Simplified 55.2%

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

   10. Applied egg-rr 57.2%

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

   if 1.05e30 < b < 4.09999999999999992e132

   1. Initial program 41.5%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 36.3%

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

   6. Taylor expanded in y-scale around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 54.6

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

   8. Simplified 54.6%

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

   9. Taylor expanded in y-scale around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 68.7

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

   11. Simplified 68.7%

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

   if 4.09999999999999992e132 < b < 5.40000000000000036e178

   1. Initial program 0.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 14.7%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 65.3

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

   8. Simplified 65.3%

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

      1. add-cbrt-cube N/A

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

      2. lower-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-*.f64 62.5

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

   10. Applied egg-rr 62.5%

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

   12. Applied egg-rr 65.3%

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

   if 5.40000000000000036e178 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 83.4

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

   8. Simplified 83.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \pi\right)\right)}\right)\right)}}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+132}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+178}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.6 \cdot 10^{-185}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.6e-185)
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))
   (if (<= b_m 4.7e+30)
     (*
      180.0
      (/
       (atan
        (* y-scale (/ (tan (* 0.005555555555555556 (* angle PI))) x-scale)))
       PI))
     (* 180.0 (/ (atan (* y-scale (/ -180.0 (* angle (* PI x-scale))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 5.6e-185) {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * ((double) M_PI))))) / ((double) M_PI));
	} else if (b_m <= 4.7e+30) {
		tmp = 180.0 * (atan((y_45_scale * (tan((0.005555555555555556 * (angle * ((double) M_PI)))) / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 5.6e-185) {
		tmp = 180.0 * (Math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * Math.PI)))) / Math.PI);
	} else if (b_m <= 4.7e+30) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.tan((0.005555555555555556 * (angle * Math.PI))) / x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((y_45_scale * (-180.0 / (angle * (Math.PI * x_45_scale))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 5.6e-185:
		tmp = 180.0 * (math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * math.pi)))) / math.pi)
	elif b_m <= 4.7e+30:
		tmp = 180.0 * (math.atan((y_45_scale * (math.tan((0.005555555555555556 * (angle * math.pi))) / x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((y_45_scale * (-180.0 / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 5.6e-185)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_45_scale * pi)))) / pi));
	elseif (b_m <= 4.7e+30)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(tan(Float64(0.005555555555555556 * Float64(angle * pi))) / x_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-180.0 / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 5.6e-185)
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * pi)))) / pi);
	elseif (b_m <= 4.7e+30)
		tmp = 180.0 * (atan((y_45_scale * (tan((0.005555555555555556 * (angle * pi))) / x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (pi * x_45_scale))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 5.6e-185], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.7e+30], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-180.0 / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.59999999999999983e-185

   1. Initial program 18.5%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Simplified 32.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 55.9

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

   8. Simplified 55.9%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Simplified 55.4%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)}{\mathsf{PI}\left(\right)} \]

       6. clear-num N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       7. un-div-inv N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{180} \cdot angle}}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-/.f64 56.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\color{blue}{\frac{x-scale}{y-scale \cdot \pi}}}\right)}{\pi} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       12. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 56.3

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\color{blue}{\pi \cdot y-scale}}}\right)}{\pi} \]

   13. Applied egg-rr 56.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\pi \cdot y-scale}}\right)}}{\pi} \]

   if 5.59999999999999983e-185 < b < 4.6999999999999999e30

   1. Initial program 21.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 33.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 53.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 53.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(y-scale \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      11. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot y-scale\right)}}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot y-scale\right)}}{\mathsf{PI}\left(\right)} \]

   10. Applied egg-rr 54.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale} \cdot y-scale\right)}}{\pi} \]

   if 4.6999999999999999e30 < b

   1. Initial program 19.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 19.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 27.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 27.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]

   11. Simplified 27.9%

       \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(y-scale \cdot \frac{180 \cdot \left(\frac{x-scale \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}} \]

   12. Taylor expanded in a around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-PI.f64 58.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   14. Simplified 58.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-185}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 16.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b\_m \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 4e-63)
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))
   (if (<= b_m 2.45e+114)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (*
      180.0
      (/
       (atan
        (fma
         -180.0
         (/ y-scale (* angle (* PI x-scale)))
         (/ (* 180.0 (* y-scale (- a a))) (* angle (* b_m (* PI x-scale))))))
       PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 4e-63) {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * ((double) M_PI))))) / ((double) M_PI));
	} else if (b_m <= 2.45e+114) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(fma(-180.0, (y_45_scale / (angle * (((double) M_PI) * x_45_scale))), ((180.0 * (y_45_scale * (a - a))) / (angle * (b_m * (((double) M_PI) * 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)
	tmp = 0.0
	if (b_m <= 4e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_45_scale * pi)))) / pi));
	elseif (b_m <= 2.45e+114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(fma(-180.0, Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))), Float64(Float64(180.0 * Float64(y_45_scale * Float64(a - a))) / Float64(angle * Float64(b_m * Float64(pi * 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_] := If[LessEqual[b$95$m, 4e-63], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.45e+114], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(180.0 * N[(y$45$scale * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(b$95$m * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.00000000000000027e-63

   1. Initial program 18.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 33.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 56.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 56.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 55.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 55.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)}{\mathsf{PI}\left(\right)} \]

       6. clear-num N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       7. un-div-inv N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{180} \cdot angle}}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-/.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\color{blue}{\frac{x-scale}{y-scale \cdot \pi}}}\right)}{\pi} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       12. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\color{blue}{\pi \cdot y-scale}}}\right)}{\pi} \]

   13. Applied egg-rr 56.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\pi \cdot y-scale}}\right)}}{\pi} \]

   if 4.00000000000000027e-63 < b < 2.45e114

   1. Initial program 28.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 21.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 28.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 28.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\color{blue}{-180 \cdot \left({b}^{2} \cdot y-scale\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-*.f64 46.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   11. Simplified 46.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{x-scale}}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   if 2.45e114 < b

   1. Initial program 10.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 13.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)} + 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}, 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}, 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}, 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}, 180 \cdot \frac{y-scale \cdot \left(a + -1 \cdot a\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{180 \cdot \left(y-scale \cdot \left(a + -1 \cdot a\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{180 \cdot \left(y-scale \cdot \left(a + -1 \cdot a\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{\color{blue}{180 \cdot \left(y-scale \cdot \left(a + -1 \cdot a\right)\right)}}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \color{blue}{\left(y-scale \cdot \left(a + -1 \cdot a\right)\right)}}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \color{blue}{\left(a + -1 \cdot a\right)}\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. mul-1-neg N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-neg.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{\color{blue}{angle \cdot \left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{angle \cdot \color{blue}{\left(b \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{angle \cdot \left(b \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-PI.f64 66.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \color{blue}{\pi}\right)\right)}\right)\right)}{\pi} \]

   8. Simplified 66.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a + \left(-a\right)\right)\right)}{angle \cdot \left(b \cdot \left(x-scale \cdot \pi\right)\right)}\right)\right)}}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(-180, \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}, \frac{180 \cdot \left(y-scale \cdot \left(a - a\right)\right)}{angle \cdot \left(b \cdot \left(\pi \cdot x-scale\right)\right)}\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.7% accurate, 17.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 4e-63)
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))
   (if (<= b_m 2.45e+114)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (* 180.0 (/ (atan (* y-scale (/ -180.0 (* angle (* PI x-scale))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 4e-63) {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * ((double) M_PI))))) / ((double) M_PI));
	} else if (b_m <= 2.45e+114) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 4e-63) {
		tmp = 180.0 * (Math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * Math.PI)))) / Math.PI);
	} else if (b_m <= 2.45e+114) {
		tmp = 180.0 * (Math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((y_45_scale * (-180.0 / (angle * (Math.PI * x_45_scale))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 4e-63:
		tmp = 180.0 * (math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * math.pi)))) / math.pi)
	elif b_m <= 2.45e+114:
		tmp = 180.0 * (math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((y_45_scale * (-180.0 / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 4e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_45_scale * pi)))) / pi));
	elseif (b_m <= 2.45e+114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-180.0 / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 4e-63)
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * pi)))) / pi);
	elseif (b_m <= 2.45e+114)
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (pi * x_45_scale))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 4e-63], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.45e+114], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-180.0 / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.00000000000000027e-63

   1. Initial program 18.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 33.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 56.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 56.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 55.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 55.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)}{\mathsf{PI}\left(\right)} \]

       6. clear-num N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       7. un-div-inv N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{180} \cdot angle}}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-/.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\color{blue}{\frac{x-scale}{y-scale \cdot \pi}}}\right)}{\pi} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       12. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\color{blue}{\pi \cdot y-scale}}}\right)}{\pi} \]

   13. Applied egg-rr 56.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\pi \cdot y-scale}}\right)}}{\pi} \]

   if 4.00000000000000027e-63 < b < 2.45e114

   1. Initial program 28.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 21.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 28.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 28.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\color{blue}{-180 \cdot \left({b}^{2} \cdot y-scale\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-*.f64 46.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   11. Simplified 46.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{x-scale}}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   if 2.45e114 < b

   1. Initial program 10.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 13.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 17.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 17.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]

   11. Simplified 17.5%

       \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(y-scale \cdot \frac{180 \cdot \left(\frac{x-scale \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}} \]

   12. Taylor expanded in a around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-PI.f64 66.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   14. Simplified 66.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.7% accurate, 17.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 4e-63)
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))
   (if (<= b_m 2.45e+114)
     (*
      180.0
      (/
       (atan
        (/
         (* -180.0 (/ (* y-scale (* b_m b_m)) x-scale))
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (* 180.0 (/ (atan (* y-scale (/ -180.0 (* angle (* PI x-scale))))) PI)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 4e-63) {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * ((double) M_PI))))) / ((double) M_PI));
	} else if (b_m <= 2.45e+114) {
		tmp = 180.0 * (atan(((-180.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale)) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 4e-63) {
		tmp = 180.0 * (Math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * Math.PI)))) / Math.PI);
	} else if (b_m <= 2.45e+114) {
		tmp = 180.0 * (Math.atan(((-180.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale)) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((y_45_scale * (-180.0 / (angle * (Math.PI * x_45_scale))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 4e-63:
		tmp = 180.0 * (math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * math.pi)))) / math.pi)
	elif b_m <= 2.45e+114:
		tmp = 180.0 * (math.atan(((-180.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale)) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((y_45_scale * (-180.0 / (angle * (math.pi * x_45_scale))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 4e-63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_45_scale * pi)))) / pi));
	elseif (b_m <= 2.45e+114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale)) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-180.0 / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 4e-63)
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_45_scale * pi)))) / pi);
	elseif (b_m <= 2.45e+114)
		tmp = 180.0 * (atan(((-180.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale)) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (pi * x_45_scale))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 4e-63], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.45e+114], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-180.0 / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.00000000000000027e-63

   1. Initial program 18.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 33.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 56.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 56.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 55.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 55.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)}{\mathsf{PI}\left(\right)} \]

       6. clear-num N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       7. un-div-inv N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{180} \cdot angle}}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-/.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\color{blue}{\frac{x-scale}{y-scale \cdot \pi}}}\right)}{\pi} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       12. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 56.2

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\color{blue}{\pi \cdot y-scale}}}\right)}{\pi} \]

   13. Applied egg-rr 56.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\pi \cdot y-scale}}\right)}}{\pi} \]

   if 4.00000000000000027e-63 < b < 2.45e114

   1. Initial program 28.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 21.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 46.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 46.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{x-scale}}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   if 2.45e114 < b

   1. Initial program 10.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 13.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 17.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 17.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]

   11. Simplified 17.5%

       \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(y-scale \cdot \frac{180 \cdot \left(\frac{x-scale \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}} \]

   12. Taylor expanded in a around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-PI.f64 66.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   14. Simplified 66.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 1.15e-100)
   (* 180.0 (/ (atan (* y-scale (/ -180.0 (* angle (* PI x-scale))))) PI))
   (*
    180.0
    (/
     (atan (/ (* 0.005555555555555556 angle) (/ x-scale (* y-scale PI))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1.15e-100) {
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_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 (a <= 1.15e-100) {
		tmp = 180.0 * (Math.atan((y_45_scale * (-180.0 / (angle * (Math.PI * x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_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 a <= 1.15e-100:
		tmp = 180.0 * (math.atan((y_45_scale * (-180.0 / (angle * (math.pi * x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((0.005555555555555556 * angle) / (x_45_scale / (y_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 (a <= 1.15e-100)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-180.0 / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(0.005555555555555556 * angle) / Float64(x_45_scale / Float64(y_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 (a <= 1.15e-100)
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (pi * x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan(((0.005555555555555556 * angle) / (x_45_scale / (y_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[a, 1.15e-100], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-180.0 / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(x$45$scale / N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999997e-100

   1. Initial program 19.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 15.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 16.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 16.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]

   11. Simplified 16.6%

       \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(y-scale \cdot \frac{180 \cdot \left(\frac{x-scale \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}} \]

   12. Taylor expanded in a around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-PI.f64 40.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   14. Simplified 40.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}}\right)}{\pi} \]

   if 1.14999999999999997e-100 < a

   1. Initial program 18.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 37.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 58.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 58.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 55.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 55.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)}{\mathsf{PI}\left(\right)} \]

       6. clear-num N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       7. un-div-inv N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\frac{1}{180} \cdot angle}}{\frac{x-scale}{y-scale \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-/.f64 57.4

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\color{blue}{\frac{x-scale}{y-scale \cdot \pi}}}\right)}{\pi} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}}\right)}{\mathsf{PI}\left(\right)} \]

       12. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{180} \cdot angle}{\frac{x-scale}{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 57.4

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\color{blue}{\pi \cdot y-scale}}}\right)}{\pi} \]

   13. Applied egg-rr 57.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{\pi \cdot y-scale}}\right)}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0.005555555555555556 \cdot angle}{\frac{x-scale}{y-scale \cdot \pi}}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.0% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \frac{y-scale}{x-scale}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 1.15e-100)
   (* 180.0 (/ (atan (* y-scale (/ -180.0 (* angle (* PI x-scale))))) PI))
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (* PI (/ 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 <= 1.15e-100) {
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (((double) M_PI) * (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 <= 1.15e-100) {
		tmp = 180.0 * (Math.atan((y_45_scale * (-180.0 / (angle * (Math.PI * x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * (Math.PI * (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 <= 1.15e-100:
		tmp = 180.0 * (math.atan((y_45_scale * (-180.0 / (angle * (math.pi * x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * (math.pi * (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 <= 1.15e-100)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-180.0 / Float64(angle * Float64(pi * x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(pi * 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 <= 1.15e-100)
		tmp = 180.0 * (atan((y_45_scale * (-180.0 / (angle * (pi * x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (pi * (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, 1.15e-100], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-180.0 / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(Pi * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999997e-100

   1. Initial program 19.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 15.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in y-scale around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\mathsf{fma}\left(-180, \frac{{b}^{2}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \color{blue}{\frac{{b}^{2}}{x-scale}}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{\color{blue}{b \cdot b}}{x-scale}, 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \color{blue}{\frac{180 \cdot \left({a}^{2} \cdot x-scale\right)}{{y-scale}^{2}}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{\color{blue}{180 \cdot \left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \color{blue}{\left({a}^{2} \cdot x-scale\right)}}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot x-scale\right)}{{y-scale}^{2}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 16.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{\color{blue}{y-scale \cdot y-scale}}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   8. Simplified 16.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \mathsf{fma}\left(-180, \frac{b \cdot b}{x-scale}, \frac{180 \cdot \left(\left(a \cdot a\right) \cdot x-scale\right)}{y-scale \cdot y-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

      \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{y-scale \cdot \left(-180 \cdot \frac{{b}^{2}}{x-scale} + 180 \cdot \frac{{a}^{2} \cdot x-scale}{{y-scale}^{2}}\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)}} \]

   11. Simplified 16.6%

       \[\leadsto 180 \cdot \color{blue}{\frac{\tan^{-1} \left(y-scale \cdot \frac{180 \cdot \left(\frac{x-scale \cdot \left(a \cdot a\right)}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}} \]

   12. Taylor expanded in a around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-PI.f64 40.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-180}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   14. Simplified 40.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(\pi \cdot x-scale\right)}}\right)}{\pi} \]

   if 1.14999999999999997e-100 < a

   1. Initial program 18.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 37.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 58.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 58.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 55.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 55.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
   12. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-/.f64 55.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\frac{y-scale}{x-scale}}\right)\right)\right)}{\pi} \]

   13. Applied egg-rr 55.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.7% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 6.5 \cdot 10^{+147}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{y-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 6.5e+147)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale))))
     PI))
   (* 180.0 (/ (atan (* x-scale (/ -180.0 (* y-scale (* angle PI))))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 6.5e+147) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((x_45_scale * (-180.0 / (y_45_scale * (angle * ((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 <= 6.5e+147) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((x_45_scale * (-180.0 / (y_45_scale * (angle * 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 <= 6.5e+147:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((x_45_scale * (-180.0 / (y_45_scale * (angle * 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 <= 6.5e+147)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(-180.0 / Float64(y_45_scale * Float64(angle * 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 <= 6.5e+147)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan((x_45_scale * (-180.0 / (y_45_scale * (angle * 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, 6.5e+147], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(-180.0 / N[(y$45$scale * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5e147

   1. Initial program 21.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 35.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 54.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Simplified 54.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

       2. associate-/l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-PI.f64 52.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

   11. Simplified 52.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]

   if 6.5e147 < b

   1. Initial program 0.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Simplified 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(2 \cdot \left(\frac{a \cdot a}{y-scale \cdot y-scale} - \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot x-scale}}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot x-scale}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot x-scale}{angle \cdot \color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 25.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\pi}\right)}\right)}{\pi} \]

   8. Simplified 25.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot x-scale}{angle \cdot \color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      3. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{x-scale \cdot -180}}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{x-scale \cdot -180}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-/l* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(x-scale \cdot \frac{-180}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(x-scale \cdot \frac{-180}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      7. lower-/.f64 25.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \color{blue}{\frac{-180}{angle \cdot \left(y-scale \cdot \pi\right)}}\right)}{\pi} \]

      8. lift-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{\color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right) \cdot angle}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{\color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)} \cdot angle}\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{\color{blue}{y-scale \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      12. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{y-scale \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{y-scale \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-*.f64 25.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{\color{blue}{y-scale \cdot \left(angle \cdot \pi\right)}}\right)}{\pi} \]

   10. Applied egg-rr 25.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(x-scale \cdot \frac{-180}{y-scale \cdot \left(angle \cdot \pi\right)}\right)}}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 45.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* 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(angle * Float64(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[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.2%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

5. Simplified 32.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

6. Taylor expanded in b around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   6. lower-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   11. lower-PI.f64 51.1

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

8. Simplified 51.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

9. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

    2. associate-/l* N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

    4. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

    6. lower-PI.f64 49.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

11. Simplified 49.8%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
12. Add Preprocessing

Alternative 20: 45.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \frac{y-scale}{x-scale}\right)\right)\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* 0.005555555555555556 (* angle (* PI (/ 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) {
	return 180.0 * (atan((0.005555555555555556 * (angle * (((double) M_PI) * (y_45_scale / 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 * (Math.PI * (y_45_scale / 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 * (math.pi * (y_45_scale / 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(angle * Float64(pi * Float64(y_45_scale / 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 * (pi * (y_45_scale / 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[(angle * N[(Pi * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.2%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in x-scale around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{-1}{2} \cdot \left(y-scale \cdot \left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)}{x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

5. Simplified 32.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

6. Taylor expanded in b around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   6. lower-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   11. lower-PI.f64 51.1

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

8. Simplified 51.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

9. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{180} \cdot \frac{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}{x-scale}\right)}}{\mathsf{PI}\left(\right)} \]

    2. associate-/l* N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right)}\right)}{\mathsf{PI}\left(\right)} \]

    4. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}}\right)\right)}{\mathsf{PI}\left(\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{y-scale \cdot \mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

    6. lower-PI.f64 49.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\pi}}{x-scale}\right)\right)}{\pi} \]

11. Simplified 49.8%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}}{\pi} \]
12. Step-by-step derivation

    1. lift-PI.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

    2. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot y-scale}}{x-scale}\right)\right)}{\mathsf{PI}\left(\right)} \]

    3. associate-/l* N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

    5. lower-/.f64 49.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\frac{y-scale}{x-scale}}\right)\right)\right)}{\pi} \]

13. Applied egg-rr 49.8%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{y-scale}{x-scale}\right)}\right)\right)}{\pi} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))