raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.8% → 55.7%

Time: 1.1min

Alternatives: 12

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 13.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.7% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\\ t_2 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 3.4 \cdot 10^{-10}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(2 \cdot \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}{t\_2}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(t\_2, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), t\_1\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_2}{\mathsf{fma}\left(t\_2, \cos \left(0.5 \cdot \pi\right), t\_1\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (cos t_0) (sin (* 0.5 PI))))
        (t_2 (sin t_0)))
   (if (<= a_m 3.4e-10)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           2.0
           (/ (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))) t_2)))))
       PI))
     (if (<= a_m 1.3e+148)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (*
             -2.0
             (/
              (*
               angle
               (fma
                -2.8577960676726107e-8
                (* (* angle angle) (* (* PI PI) PI))
                (* 0.005555555555555556 PI)))
              (fma t_2 (sin (fma 0.5 PI (/ PI 2.0))) t_1))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (* -2.0 (/ t_2 (fma t_2 (cos (* 0.5 PI)) t_1))))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0) * sin((0.5 * ((double) M_PI)));
	double t_2 = sin(t_0);
	double tmp;
	if (a_m <= 3.4e-10) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (2.0 * (sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)))) / t_2))))) / ((double) M_PI));
	} else if (a_m <= 1.3e+148) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * ((angle * fma(-2.8577960676726107e-8, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.005555555555555556 * ((double) M_PI)))) / fma(t_2, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), t_1)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_2 / fma(t_2, cos((0.5 * ((double) M_PI))), t_1)))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(cos(t_0) * sin(Float64(0.5 * pi)))
	t_2 = sin(t_0)
	tmp = 0.0
	if (a_m <= 3.4e-10)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(2.0 * Float64(sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))) / t_2))))) / pi));
	elseif (a_m <= 1.3e+148)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.005555555555555556 * pi))) / fma(t_2, sin(fma(0.5, pi, Float64(pi / 2.0))), t_1)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_2 / fma(t_2, cos(Float64(0.5 * pi)), t_1)))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 3.4e-10], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.3e+148], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$2 / N[(t$95$2 * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.40000000000000015e-10

   1. Initial program 16.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 48.2%

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

   if 3.40000000000000015e-10 < a < 1.3e148

   1. Initial program 23.7%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 41.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 48.9%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. unpow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       11. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       12. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       15. lift-PI.f64 60.5

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Applied rewrites 60.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if 1.3e148 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 73.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 76.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-/.f64 N/A

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

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

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

       7. lower-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 76.9

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

   12. Applied rewrites 76.9%

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

Alternative 2: 46.4% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\\ t_2 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 4.2 \cdot 10^{-194}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_2}{t\_2}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 1.3 \cdot 10^{+148}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(t\_2, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), t\_1\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_2}{\mathsf{fma}\left(t\_2, \cos \left(0.5 \cdot \pi\right), t\_1\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (cos t_0) (sin (* 0.5 PI))))
        (t_2 (sin t_0)))
   (if (<= a_m 4.2e-194)
     (*
      180.0
      (/ (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_2 t_2))))) PI))
     (if (<= a_m 1.3e+148)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (*
             -2.0
             (/
              (*
               angle
               (fma
                -2.8577960676726107e-8
                (* (* angle angle) (* (* PI PI) PI))
                (* 0.005555555555555556 PI)))
              (fma t_2 (sin (fma 0.5 PI (/ PI 2.0))) t_1))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (* -2.0 (/ t_2 (fma t_2 (cos (* 0.5 PI)) t_1))))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0) * sin((0.5 * ((double) M_PI)));
	double t_2 = sin(t_0);
	double tmp;
	if (a_m <= 4.2e-194) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_2 / t_2))))) / ((double) M_PI));
	} else if (a_m <= 1.3e+148) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * ((angle * fma(-2.8577960676726107e-8, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.005555555555555556 * ((double) M_PI)))) / fma(t_2, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), t_1)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_2 / fma(t_2, cos((0.5 * ((double) M_PI))), t_1)))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(cos(t_0) * sin(Float64(0.5 * pi)))
	t_2 = sin(t_0)
	tmp = 0.0
	if (a_m <= 4.2e-194)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_2 / t_2))))) / pi));
	elseif (a_m <= 1.3e+148)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.005555555555555556 * pi))) / fma(t_2, sin(fma(0.5, pi, Float64(pi / 2.0))), t_1)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_2 / fma(t_2, cos(Float64(0.5 * pi)), t_1)))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 4.2e-194], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$2 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.3e+148], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$2 / N[(t$95$2 * N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.2e-194

   1. Initial program 17.2%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in angle around inf

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 39.4

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

   11. Applied rewrites 39.4%

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

   if 4.2e-194 < a < 1.3e148

   1. Initial program 17.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 38.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 43.3%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. unpow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       11. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       12. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       15. lift-PI.f64 46.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Applied rewrites 46.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if 1.3e148 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 73.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 76.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-/.f64 N/A

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

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

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

       7. lower-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 76.9

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

   12. Applied rewrites 76.9%

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

Alternative 3: 46.0% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 4.2 \cdot 10^{-194}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{t\_1}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;a\_m \leq 7.5 \cdot 10^{+65}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(t\_1, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a_m 4.2e-194)
     (*
      180.0
      (/ (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_1 t_1))))) PI))
     (if (<= a_m 7.5e+65)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (*
             -2.0
             (/
              (*
               angle
               (fma
                -2.8577960676726107e-8
                (* (* angle angle) (* (* PI PI) PI))
                (* 0.005555555555555556 PI)))
              (fma
               t_1
               (sin (fma 0.5 PI (/ PI 2.0)))
               (* (cos t_0) (sin (* 0.5 PI)))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/ y-scale x-scale)
            (*
             -2.0
             (/
              t_1
              (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a_m <= 4.2e-194) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / t_1))))) / ((double) M_PI));
	} else if (a_m <= 7.5e+65) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * ((angle * fma(-2.8577960676726107e-8, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.005555555555555556 * ((double) M_PI)))) / fma(t_1, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), (cos(t_0) * sin((0.5 * ((double) M_PI)))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, 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_m <= 4.2e-194)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / t_1))))) / pi));
	elseif (a_m <= 7.5e+65)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.005555555555555556 * pi))) / fma(t_1, sin(fma(0.5, pi, Float64(pi / 2.0))), Float64(cos(t_0) * sin(Float64(0.5 * pi))))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 4.2e-194], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 7.5e+65], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.2e-194

   1. Initial program 17.2%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in angle around inf

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 39.4

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

   11. Applied rewrites 39.4%

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

   if 4.2e-194 < a < 7.50000000000000006e65

   1. Initial program 19.8%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 38.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 44.5%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. unpow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       11. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       12. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       15. lift-PI.f64 44.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Applied rewrites 44.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if 7.50000000000000006e65 < a

   1. Initial program 2.3%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 9.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 65.0%

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

Alternative 4: 45.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 4.2 \cdot 10^{-194}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{t\_1}{t\_1}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(t\_1, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a_m 4.2e-194)
     (*
      180.0
      (/ (atan (* -0.5 (* (/ y-scale x-scale) (* -2.0 (/ t_1 t_1))))) PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           -2.0
           (/
            (*
             angle
             (fma
              -2.8577960676726107e-8
              (* (* angle angle) (* (* PI PI) PI))
              (* 0.005555555555555556 PI)))
            (fma
             t_1
             (sin (fma 0.5 PI (/ PI 2.0)))
             (* (cos t_0) (sin (* 0.5 PI)))))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a_m <= 4.2e-194) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (t_1 / t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * ((angle * fma(-2.8577960676726107e-8, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.005555555555555556 * ((double) M_PI)))) / fma(t_1, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), (cos(t_0) * sin((0.5 * ((double) M_PI)))))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, 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_m <= 4.2e-194)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(t_1 / t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.005555555555555556 * pi))) / fma(t_1, sin(fma(0.5, pi, Float64(pi / 2.0))), Float64(cos(t_0) * sin(Float64(0.5 * pi))))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 4.2e-194], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(t$95$1 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.2e-194

   1. Initial program 17.2%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 44.8%

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

   9. Taylor expanded in angle around inf

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 39.4

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

   11. Applied rewrites 39.4%

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

   if 4.2e-194 < a

   1. Initial program 12.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 21.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 50.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. unpow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       11. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       12. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       15. lift-PI.f64 52.7

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Applied rewrites 52.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 45.7% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;x-scale \leq -3.3 \cdot 10^{+54}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot t\_1}{x-scale \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(t\_1, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos t\_0 \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= x-scale -3.3e+54)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -2.0
          (/
           (* y-scale t_1)
           (*
            x-scale
            (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           -2.0
           (/
            (*
             angle
             (fma
              -2.8577960676726107e-8
              (* (* angle angle) (* (* PI PI) PI))
              (* 0.005555555555555556 PI)))
            (fma
             t_1
             (sin (fma 0.5 PI (/ PI 2.0)))
             (* (cos t_0) (sin (* 0.5 PI)))))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (x_45_scale <= -3.3e+54) {
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * t_1) / (x_45_scale * sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * ((angle * fma(-2.8577960676726107e-8, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.005555555555555556 * ((double) M_PI)))) / fma(t_1, sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), (cos(t_0) * sin((0.5 * ((double) M_PI)))))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (x_45_scale <= -3.3e+54)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(Float64(angle * fma(-2.8577960676726107e-8, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.005555555555555556 * pi))) / fma(t_1, sin(fma(0.5, pi, Float64(pi / 2.0))), Float64(cos(t_0) * sin(Float64(0.5 * pi))))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale, -3.3e+54], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(N[(angle * N[(-2.8577960676726107e-8 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -3.3e54

   1. Initial program 8.8%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if -3.3e54 < x-scale

   1. Initial program 16.6%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 47.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sin-sum N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 49.2%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. unpow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       11. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       12. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       13. lift-PI.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, \frac{\pi}{2}\right)\right), \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       15. lift-PI.f64 51.6

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   13. Applied rewrites 51.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \frac{angle \cdot \mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.005555555555555556 \cdot \pi\right)}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 45.2% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(0.5 \cdot \pi\right)\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \left(\pi \cdot \pi\right) \cdot t\_2\\ t_4 := t\_0 \cdot t\_0\\ t_5 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_6 := t\_5 \cdot t\_0\\ t_7 := t\_5 \cdot t\_5\\ t_8 := t\_7 \cdot t\_4\\ \mathbf{if}\;angle \leq -280000000:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_7 \cdot \left(t\_0 \cdot t\_1\right), 4 \cdot t\_8\right)}{t\_4}, t\_7\right)}{t\_6}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;angle \leq 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_3, 0.0001234567901234568 \cdot t\_3\right)}{t\_2}\right)}{t\_1}\right)\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_8, 4 \cdot \left(t\_7 \cdot \left(t\_0 \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\right)}{t\_4}, t\_7\right)}{t\_6}\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_1 (sin (* 0.5 PI)))
        (t_2 (* t_1 t_1))
        (t_3 (* (* PI PI) t_2))
        (t_4 (* t_0 t_0))
        (t_5 (sin (* 0.005555555555555556 (* angle PI))))
        (t_6 (* t_5 t_0))
        (t_7 (* t_5 t_5))
        (t_8 (* t_7 t_4)))
   (if (<= angle -280000000.0)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           (/ y-scale x-scale)
           (/
            (fma 0.5 (/ (fma -2.0 (* t_7 (* t_0 t_1)) (* 4.0 t_8)) t_4) t_7)
            t_6)))))
       PI))
     (if (<= angle 1e-130)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            -1.0
            (*
             (/ y-scale x-scale)
             (*
              180.0
              (*
               (/ angle PI)
               (/
                (fma
                 3.08641975308642e-5
                 (* PI PI)
                 (*
                  0.5
                  (/
                   (fma -6.17283950617284e-5 t_3 (* 0.0001234567901234568 t_3))
                   t_2)))
                t_1)))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            -1.0
            (*
             (/ y-scale x-scale)
             (/
              (fma
               0.5
               (/
                (fma
                 -2.0
                 t_8
                 (*
                  4.0
                  (*
                   t_7
                   (*
                    t_0
                    (sin
                     (*
                      angle
                      (fma 0.005555555555555556 PI (* 0.5 (/ PI angle)))))))))
                t_4)
               t_7)
              t_6)))))
         PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_1 = sin((0.5 * ((double) M_PI)));
	double t_2 = t_1 * t_1;
	double t_3 = (((double) M_PI) * ((double) M_PI)) * t_2;
	double t_4 = t_0 * t_0;
	double t_5 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_6 = t_5 * t_0;
	double t_7 = t_5 * t_5;
	double t_8 = t_7 * t_4;
	double tmp;
	if (angle <= -280000000.0) {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (fma(0.5, (fma(-2.0, (t_7 * (t_0 * t_1)), (4.0 * t_8)) / t_4), t_7) / t_6))))) / ((double) M_PI));
	} else if (angle <= 1e-130) {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (180.0 * ((angle / ((double) M_PI)) * (fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (fma(-6.17283950617284e-5, t_3, (0.0001234567901234568 * t_3)) / t_2))) / t_1))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (fma(0.5, (fma(-2.0, t_8, (4.0 * (t_7 * (t_0 * sin((angle * fma(0.005555555555555556, ((double) M_PI), (0.5 * (((double) M_PI) / angle))))))))) / t_4), t_7) / t_6))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_1 = sin(Float64(0.5 * pi))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(Float64(pi * pi) * t_2)
	t_4 = Float64(t_0 * t_0)
	t_5 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_6 = Float64(t_5 * t_0)
	t_7 = Float64(t_5 * t_5)
	t_8 = Float64(t_7 * t_4)
	tmp = 0.0
	if (angle <= -280000000.0)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(0.5, Float64(fma(-2.0, Float64(t_7 * Float64(t_0 * t_1)), Float64(4.0 * t_8)) / t_4), t_7) / t_6))))) / pi));
	elseif (angle <= 1e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 * Float64(Float64(angle / pi) * Float64(fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(fma(-6.17283950617284e-5, t_3, Float64(0.0001234567901234568 * t_3)) / t_2))) / t_1))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(0.5, Float64(fma(-2.0, t_8, Float64(4.0 * Float64(t_7 * Float64(t_0 * sin(Float64(angle * fma(0.005555555555555556, pi, Float64(0.5 * Float64(pi / angle))))))))) / t_4), t_7) / t_6))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * t$95$0), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 * t$95$5), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$7 * t$95$4), $MachinePrecision]}, If[LessEqual[angle, -280000000.0], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(0.5 * N[(N[(-2.0 * N[(t$95$7 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(4.0 * t$95$8), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + t$95$7), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1e-130], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 * N[(N[(angle / Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(-6.17283950617284e-5 * t$95$3 + N[(0.0001234567901234568 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(0.5 * N[(N[(-2.0 * t$95$8 + N[(4.0 * N[(t$95$7 * N[(t$95$0 * N[Sin[N[(angle * N[(0.005555555555555556 * Pi + N[(0.5 * N[(Pi / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + t$95$7), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -2.8e8

   1. Initial program 4.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 23.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 15.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 39.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lift-PI.f64 44.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   14. Applied rewrites 44.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if -2.8e8 < angle < 1.0000000000000001e-130

   1. Initial program 22.6%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 8.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 11.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 33.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \frac{angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

   14. Applied rewrites 52.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)\right)}{\pi} \]

   if 1.0000000000000001e-130 < angle

   1. Initial program 13.8%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 21.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 13.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 46.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \mathsf{PI}\left(\right), \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \mathsf{fma}\left(\frac{1}{180}, \pi, \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. lift-PI.f64 49.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   14. Applied rewrites 49.6%

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

Alternative 7: 44.9% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ \mathbf{if}\;angle \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-2 \cdot \frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \left(angle \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \frac{angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)}{t\_0 \cdot t\_0}, 0.005555555555555556 \cdot \frac{\pi}{t\_0}\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI))))
   (if (<= angle 2.8e+210)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -2.0
          (/
           (* y-scale (sin (* 0.005555555555555556 (* angle PI))))
           (*
            x-scale
            (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          (/ y-scale x-scale)
          (*
           -2.0
           (*
            angle
            (fma
             -3.08641975308642e-5
             (/
              (* angle (* (* PI PI) (sin (fma 0.5 PI (/ PI 2.0)))))
              (* t_0 t_0))
             (* 0.005555555555555556 (/ PI t_0))))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double tmp;
	if (angle <= 2.8e+210) {
		tmp = 180.0 * (atan((-0.5 * (-2.0 * ((y_45_scale * sin((0.005555555555555556 * (angle * ((double) M_PI))))) / (x_45_scale * sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale / x_45_scale) * (-2.0 * (angle * fma(-3.08641975308642e-5, ((angle * ((((double) M_PI) * ((double) M_PI)) * sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))))) / (t_0 * t_0)), (0.005555555555555556 * (((double) M_PI) / t_0)))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.5 * pi))
	tmp = 0.0
	if (angle <= 2.8e+210)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-2.0 * Float64(Float64(y_45_scale * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) / Float64(x_45_scale * sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale / x_45_scale) * Float64(-2.0 * Float64(angle * fma(-3.08641975308642e-5, Float64(Float64(angle * Float64(Float64(pi * pi) * sin(fma(0.5, pi, Float64(pi / 2.0))))) / Float64(t_0 * t_0)), Float64(0.005555555555555556 * Float64(pi / t_0)))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[angle, 2.8e+210], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-2.0 * N[(N[(y$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(-2.0 * N[(angle * N[(-3.08641975308642e-5 * N[(N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.8000000000000002e210

   1. Initial program 14.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 48.5%

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

   if 2.8000000000000002e210 < angle

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 17.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 24.8%

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

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \left(angle \cdot \left(\frac{-1}{32400} \cdot \frac{angle \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \color{blue}{\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)\right)}{\pi} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \left(angle \cdot \left(\frac{-1}{32400} \cdot \frac{angle \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{1}{180} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)\right)}{\pi} \]

       2. lower-fma.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{32400}, \frac{angle \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{\color{blue}{2}}}, \frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right)\right)}{\pi} \]

   11. Applied rewrites 55.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(-2 \cdot \left(angle \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \color{blue}{\frac{angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}}, 0.005555555555555556 \cdot \frac{\pi}{\sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 43.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(0.5 \cdot \pi\right)\\ t_2 := t\_0 \cdot t\_0\\ t_3 := t\_1 \cdot t\_1\\ t_4 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := t\_4 \cdot t\_4\\ t_6 := \left(\pi \cdot \pi\right) \cdot t\_3\\ \mathbf{if}\;angle \leq -8 \cdot 10^{+29}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_5 \cdot \left(t\_0 \cdot t\_1\right), 4 \cdot \left(t\_5 \cdot t\_2\right)\right)}{t\_2}, t\_5\right)}{t\_4 \cdot t\_0}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_6, 0.0001234567901234568 \cdot t\_6\right)}{t\_3}\right)}{t\_1}\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_1 (sin (* 0.5 PI)))
        (t_2 (* t_0 t_0))
        (t_3 (* t_1 t_1))
        (t_4 (sin (* 0.005555555555555556 (* angle PI))))
        (t_5 (* t_4 t_4))
        (t_6 (* (* PI PI) t_3)))
   (if (<= angle -8e+29)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           (/ y-scale x-scale)
           (/
            (fma
             0.5
             (/ (fma -2.0 (* t_5 (* t_0 t_1)) (* 4.0 (* t_5 t_2))) t_2)
             t_5)
            (* t_4 t_0))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           (/ y-scale x-scale)
           (*
            180.0
            (*
             (/ angle PI)
             (/
              (fma
               3.08641975308642e-5
               (* PI PI)
               (*
                0.5
                (/
                 (fma -6.17283950617284e-5 t_6 (* 0.0001234567901234568 t_6))
                 t_3)))
              t_1)))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_1 = sin((0.5 * ((double) M_PI)));
	double t_2 = t_0 * t_0;
	double t_3 = t_1 * t_1;
	double t_4 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = t_4 * t_4;
	double t_6 = (((double) M_PI) * ((double) M_PI)) * t_3;
	double tmp;
	if (angle <= -8e+29) {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (fma(0.5, (fma(-2.0, (t_5 * (t_0 * t_1)), (4.0 * (t_5 * t_2))) / t_2), t_5) / (t_4 * t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (180.0 * ((angle / ((double) M_PI)) * (fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (fma(-6.17283950617284e-5, t_6, (0.0001234567901234568 * t_6)) / t_3))) / t_1))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_1 = sin(Float64(0.5 * pi))
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(t_1 * t_1)
	t_4 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = Float64(t_4 * t_4)
	t_6 = Float64(Float64(pi * pi) * t_3)
	tmp = 0.0
	if (angle <= -8e+29)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(0.5, Float64(fma(-2.0, Float64(t_5 * Float64(t_0 * t_1)), Float64(4.0 * Float64(t_5 * t_2))) / t_2), t_5) / Float64(t_4 * t_0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 * Float64(Float64(angle / pi) * Float64(fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(fma(-6.17283950617284e-5, t_6, Float64(0.0001234567901234568 * t_6)) / t_3))) / t_1))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[angle, -8e+29], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(0.5 * N[(N[(-2.0 * N[(t$95$5 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + t$95$5), $MachinePrecision] / N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 * N[(N[(angle / Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(-6.17283950617284e-5 * t$95$6 + N[(0.0001234567901234568 * t$95$6), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < -7.99999999999999931e29

   1. Initial program 4.3%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 24.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 16.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 41.2%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lift-PI.f64 44.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   14. Applied rewrites 44.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if -7.99999999999999931e29 < angle

   1. Initial program 18.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 15.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 12.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \frac{angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

   14. Applied rewrites 47.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := t\_4 \cdot t\_4\\ t_6 := \left(\pi \cdot \pi\right) \cdot t\_1\\ \mathbf{if}\;angle \leq -2 \cdot 10^{+78}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_5 \cdot t\_1, 4 \cdot \left(t\_5 \cdot t\_3\right)\right)}{t\_3}, t\_5\right)}{t\_4 \cdot t\_2}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_6, 0.0001234567901234568 \cdot t\_6\right)}{t\_1}\right)}{t\_0}\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI)))
        (t_1 (* t_0 t_0))
        (t_2 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_3 (* t_2 t_2))
        (t_4 (sin (* 0.005555555555555556 (* angle PI))))
        (t_5 (* t_4 t_4))
        (t_6 (* (* PI PI) t_1)))
   (if (<= angle -2e+78)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           (/ y-scale x-scale)
           (/
            (fma 0.5 (/ (fma -2.0 (* t_5 t_1) (* 4.0 (* t_5 t_3))) t_3) t_5)
            (* t_4 t_2))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           (/ y-scale x-scale)
           (*
            180.0
            (*
             (/ angle PI)
             (/
              (fma
               3.08641975308642e-5
               (* PI PI)
               (*
                0.5
                (/
                 (fma -6.17283950617284e-5 t_6 (* 0.0001234567901234568 t_6))
                 t_1)))
              t_0)))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double t_1 = t_0 * t_0;
	double t_2 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_3 = t_2 * t_2;
	double t_4 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_5 = t_4 * t_4;
	double t_6 = (((double) M_PI) * ((double) M_PI)) * t_1;
	double tmp;
	if (angle <= -2e+78) {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (fma(0.5, (fma(-2.0, (t_5 * t_1), (4.0 * (t_5 * t_3))) / t_3), t_5) / (t_4 * t_2)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (180.0 * ((angle / ((double) M_PI)) * (fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (fma(-6.17283950617284e-5, t_6, (0.0001234567901234568 * t_6)) / t_1))) / t_0))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_3 = Float64(t_2 * t_2)
	t_4 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_5 = Float64(t_4 * t_4)
	t_6 = Float64(Float64(pi * pi) * t_1)
	tmp = 0.0
	if (angle <= -2e+78)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(0.5, Float64(fma(-2.0, Float64(t_5 * t_1), Float64(4.0 * Float64(t_5 * t_3))) / t_3), t_5) / Float64(t_4 * t_2)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 * Float64(Float64(angle / pi) * Float64(fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(fma(-6.17283950617284e-5, t_6, Float64(0.0001234567901234568 * t_6)) / t_1))) / t_0))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[angle, -2e+78], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(0.5 * N[(N[(-2.0 * N[(t$95$5 * t$95$1), $MachinePrecision] + N[(4.0 * N[(t$95$5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision] / N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 * N[(N[(angle / Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(-6.17283950617284e-5 * t$95$6 + N[(0.0001234567901234568 * t$95$6), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.00000000000000002e78

   1. Initial program 4.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 24.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 14.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 42.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}, 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       5. lift-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       7. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-2, \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

       8. lift-sin.f64 44.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   14. Applied rewrites 44.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}\right)\right)\right)}{\pi} \]

   if -2.00000000000000002e78 < angle

   1. Initial program 17.3%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 15.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 12.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 39.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \frac{angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)}{\pi} \]
   13. Step-by-step derivation

   14. Applied rewrites 46.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\\ t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_2 := \sin \left(0.5 \cdot \pi\right)\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left(\pi \cdot \pi\right) \cdot t\_3\\ t_5 := \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_4, 0.0001234567901234568 \cdot t\_4\right)\\ t_6 := t\_1 \cdot t\_2\\ t_7 := \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, t\_4, 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\\ t_8 := \mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, t\_6, -2.8577960676726107 \cdot 10^{-8} \cdot t\_6\right)\\ t_9 := t\_0 \cdot t\_2\\ t_10 := \pi \cdot t\_2\\ t_11 := \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot t\_3, 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_7\right)\right)}{t\_3}\\ t_12 := \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{t\_5}{t\_3}\right)\\ t_13 := t\_0 \cdot t\_12\\ t_14 := \frac{y-scale}{x-scale} \cdot \frac{t\_13}{t\_3}\\ t_15 := {t\_2}^{4}\\ t_16 := \frac{t\_1 \cdot t\_0}{t\_2}\\ t_17 := \mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, t\_16, 1.3717421124828533 \cdot 10^{-6} \cdot t\_16\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(t\_0 \cdot t\_5\right)}{t\_3 \cdot t\_2}\\ t_18 := \mathsf{fma}\left(-2, t\_11, 4 \cdot t\_11\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(t\_0 \cdot t\_17\right)}{t\_2}, \frac{t\_5 \cdot t\_7}{t\_15}\right)\\ t_19 := \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4}, 0.5 \cdot t\_18\right)}{t\_10}\\ t_20 := \frac{y-scale}{x-scale} \cdot \frac{t\_17}{t\_10}\\ t_21 := 90 \cdot t\_20 - t\_14\\ t_22 := \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(t\_0 \cdot t\_21\right)}{t\_2}, 32400 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\frac{t\_8}{\pi \cdot \pi} \cdot \frac{t\_12}{t\_3}\right)\right)\right)\\ t_23 := t\_1 \cdot t\_9\\ t_24 := \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, t\_23, -5.7155921353452215 \cdot 10^{-8} \cdot t\_23\right)\\ t_25 := \frac{\mathsf{fma}\left(-3.5281432934229765 \cdot 10^{-12}, {\pi}^{5} \cdot t\_9, 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_24\right)\right)}{t\_3}\\ \mathbf{if}\;y-scale \leq -3.8 \cdot 10^{+22} \lor \neg \left(y-scale \leq 6.1 \cdot 10^{+113}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(180 \cdot \left(\frac{angle}{x-scale} \cdot \frac{y-scale \cdot t\_12}{t\_10}\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(180, \frac{y-scale}{x-scale} \cdot \frac{t\_12}{t\_10}, angle \cdot \left(\mathsf{fma}\left(90, t\_20, angle \cdot \left(\mathsf{fma}\left(180, t\_19, angle \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-2, t\_25, 4 \cdot t\_25\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(t\_0 \cdot t\_18\right)}{t\_2}, \frac{t\_5 \cdot t\_24}{t\_15} + \frac{t\_7 \cdot t\_17}{t\_3}\right)}{t\_10}\right) - \mathsf{fma}\left(-1.02880658436214 \cdot 10^{-5}, \frac{y-scale}{x-scale} \cdot \frac{\left(\pi \cdot \pi\right) \cdot t\_13}{t\_3}, \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(t\_0 \cdot \left(180 \cdot t\_19 + -1 \cdot t\_22\right)\right)}{t\_2}, 180 \cdot \left(\frac{t\_8}{\pi} \cdot \frac{t\_21}{t\_2}\right)\right)\right)\right)\right) - t\_22\right)\right) - t\_14\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (fma 0.5 PI (/ PI 2.0))))
        (t_1 (* (* PI PI) PI))
        (t_2 (sin (* 0.5 PI)))
        (t_3 (* t_2 t_2))
        (t_4 (* (* PI PI) t_3))
        (t_5 (fma -6.17283950617284e-5 t_4 (* 0.0001234567901234568 t_4)))
        (t_6 (* t_1 t_2))
        (t_7
         (fma
          -3.08641975308642e-5
          t_4
          (* 3.08641975308642e-5 (* (* PI PI) (* t_0 t_0)))))
        (t_8 (fma -8.573388203017833e-8 t_6 (* -2.8577960676726107e-8 t_6)))
        (t_9 (* t_0 t_2))
        (t_10 (* PI t_2))
        (t_11
         (/
          (fma
           -3.175328964080679e-10
           (* (pow PI 4.0) t_3)
           (* 3.08641975308642e-5 (* (* PI PI) t_7)))
          t_3))
        (t_12 (fma 3.08641975308642e-5 (* PI PI) (* 0.5 (/ t_5 t_3))))
        (t_13 (* t_0 t_12))
        (t_14 (* (/ y-scale x-scale) (/ t_13 t_3)))
        (t_15 (pow t_2 4.0))
        (t_16 (/ (* t_1 t_0) t_2))
        (t_17
         (-
          (fma -6.858710562414266e-7 t_16 (* 1.3717421124828533e-6 t_16))
          (* 0.011111111111111112 (/ (* PI (* t_0 t_5)) (* t_3 t_2)))))
        (t_18
         (-
          (fma -2.0 t_11 (* 4.0 t_11))
          (fma
           0.011111111111111112
           (/ (* PI (* t_0 t_17)) t_2)
           (/ (* t_5 t_7) t_15))))
        (t_19
         (*
          (/ y-scale x-scale)
          (/ (fma -3.175328964080679e-10 (pow PI 4.0) (* 0.5 t_18)) t_10)))
        (t_20 (* (/ y-scale x-scale) (/ t_17 t_10)))
        (t_21 (- (* 90.0 t_20) t_14))
        (t_22
         (fma
          0.005555555555555556
          (/ (* PI (* t_0 t_21)) t_2)
          (*
           32400.0
           (* (/ y-scale x-scale) (* (/ t_8 (* PI PI)) (/ t_12 t_3))))))
        (t_23 (* t_1 t_9))
        (t_24
         (fma -1.7146776406035666e-7 t_23 (* -5.7155921353452215e-8 t_23)))
        (t_25
         (/
          (fma
           -3.5281432934229765e-12
           (* (pow PI 5.0) t_9)
           (* 3.08641975308642e-5 (* (* PI PI) t_24)))
          t_3)))
   (if (or (<= y-scale -3.8e+22) (not (<= y-scale 6.1e+113)))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (* -1.0 (* 180.0 (* (/ angle x-scale) (/ (* y-scale t_12) t_10))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (*
          -1.0
          (*
           angle
           (fma
            180.0
            (* (/ y-scale x-scale) (/ t_12 t_10))
            (*
             angle
             (-
              (fma
               90.0
               t_20
               (*
                angle
                (-
                 (fma
                  180.0
                  t_19
                  (*
                   angle
                   (-
                    (*
                     90.0
                     (*
                      (/ y-scale x-scale)
                      (/
                       (-
                        (fma -2.0 t_25 (* 4.0 t_25))
                        (fma
                         0.011111111111111112
                         (/ (* PI (* t_0 t_18)) t_2)
                         (+ (/ (* t_5 t_24) t_15) (/ (* t_7 t_17) t_3))))
                       t_10)))
                    (fma
                     -1.02880658436214e-5
                     (* (/ y-scale x-scale) (/ (* (* PI PI) t_13) t_3))
                     (fma
                      0.005555555555555556
                      (/ (* PI (* t_0 (+ (* 180.0 t_19) (* -1.0 t_22)))) t_2)
                      (* 180.0 (* (/ t_8 PI) (/ t_21 t_2))))))))
                 t_22)))
              t_14)))))))
       PI)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0)));
	double t_1 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_2 = sin((0.5 * ((double) M_PI)));
	double t_3 = t_2 * t_2;
	double t_4 = (((double) M_PI) * ((double) M_PI)) * t_3;
	double t_5 = fma(-6.17283950617284e-5, t_4, (0.0001234567901234568 * t_4));
	double t_6 = t_1 * t_2;
	double t_7 = fma(-3.08641975308642e-5, t_4, (3.08641975308642e-5 * ((((double) M_PI) * ((double) M_PI)) * (t_0 * t_0))));
	double t_8 = fma(-8.573388203017833e-8, t_6, (-2.8577960676726107e-8 * t_6));
	double t_9 = t_0 * t_2;
	double t_10 = ((double) M_PI) * t_2;
	double t_11 = fma(-3.175328964080679e-10, (pow(((double) M_PI), 4.0) * t_3), (3.08641975308642e-5 * ((((double) M_PI) * ((double) M_PI)) * t_7))) / t_3;
	double t_12 = fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (t_5 / t_3)));
	double t_13 = t_0 * t_12;
	double t_14 = (y_45_scale / x_45_scale) * (t_13 / t_3);
	double t_15 = pow(t_2, 4.0);
	double t_16 = (t_1 * t_0) / t_2;
	double t_17 = fma(-6.858710562414266e-7, t_16, (1.3717421124828533e-6 * t_16)) - (0.011111111111111112 * ((((double) M_PI) * (t_0 * t_5)) / (t_3 * t_2)));
	double t_18 = fma(-2.0, t_11, (4.0 * t_11)) - fma(0.011111111111111112, ((((double) M_PI) * (t_0 * t_17)) / t_2), ((t_5 * t_7) / t_15));
	double t_19 = (y_45_scale / x_45_scale) * (fma(-3.175328964080679e-10, pow(((double) M_PI), 4.0), (0.5 * t_18)) / t_10);
	double t_20 = (y_45_scale / x_45_scale) * (t_17 / t_10);
	double t_21 = (90.0 * t_20) - t_14;
	double t_22 = fma(0.005555555555555556, ((((double) M_PI) * (t_0 * t_21)) / t_2), (32400.0 * ((y_45_scale / x_45_scale) * ((t_8 / (((double) M_PI) * ((double) M_PI))) * (t_12 / t_3)))));
	double t_23 = t_1 * t_9;
	double t_24 = fma(-1.7146776406035666e-7, t_23, (-5.7155921353452215e-8 * t_23));
	double t_25 = fma(-3.5281432934229765e-12, (pow(((double) M_PI), 5.0) * t_9), (3.08641975308642e-5 * ((((double) M_PI) * ((double) M_PI)) * t_24))) / t_3;
	double tmp;
	if ((y_45_scale <= -3.8e+22) || !(y_45_scale <= 6.1e+113)) {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * (180.0 * ((angle / x_45_scale) * ((y_45_scale * t_12) / t_10)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (-1.0 * (angle * fma(180.0, ((y_45_scale / x_45_scale) * (t_12 / t_10)), (angle * (fma(90.0, t_20, (angle * (fma(180.0, t_19, (angle * ((90.0 * ((y_45_scale / x_45_scale) * ((fma(-2.0, t_25, (4.0 * t_25)) - fma(0.011111111111111112, ((((double) M_PI) * (t_0 * t_18)) / t_2), (((t_5 * t_24) / t_15) + ((t_7 * t_17) / t_3)))) / t_10))) - fma(-1.02880658436214e-5, ((y_45_scale / x_45_scale) * (((((double) M_PI) * ((double) M_PI)) * t_13) / t_3)), fma(0.005555555555555556, ((((double) M_PI) * (t_0 * ((180.0 * t_19) + (-1.0 * t_22)))) / t_2), (180.0 * ((t_8 / ((double) M_PI)) * (t_21 / t_2)))))))) - t_22))) - t_14))))))) / ((double) M_PI));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(fma(0.5, pi, Float64(pi / 2.0)))
	t_1 = Float64(Float64(pi * pi) * pi)
	t_2 = sin(Float64(0.5 * pi))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(Float64(pi * pi) * t_3)
	t_5 = fma(-6.17283950617284e-5, t_4, Float64(0.0001234567901234568 * t_4))
	t_6 = Float64(t_1 * t_2)
	t_7 = fma(-3.08641975308642e-5, t_4, Float64(3.08641975308642e-5 * Float64(Float64(pi * pi) * Float64(t_0 * t_0))))
	t_8 = fma(-8.573388203017833e-8, t_6, Float64(-2.8577960676726107e-8 * t_6))
	t_9 = Float64(t_0 * t_2)
	t_10 = Float64(pi * t_2)
	t_11 = Float64(fma(-3.175328964080679e-10, Float64((pi ^ 4.0) * t_3), Float64(3.08641975308642e-5 * Float64(Float64(pi * pi) * t_7))) / t_3)
	t_12 = fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(t_5 / t_3)))
	t_13 = Float64(t_0 * t_12)
	t_14 = Float64(Float64(y_45_scale / x_45_scale) * Float64(t_13 / t_3))
	t_15 = t_2 ^ 4.0
	t_16 = Float64(Float64(t_1 * t_0) / t_2)
	t_17 = Float64(fma(-6.858710562414266e-7, t_16, Float64(1.3717421124828533e-6 * t_16)) - Float64(0.011111111111111112 * Float64(Float64(pi * Float64(t_0 * t_5)) / Float64(t_3 * t_2))))
	t_18 = Float64(fma(-2.0, t_11, Float64(4.0 * t_11)) - fma(0.011111111111111112, Float64(Float64(pi * Float64(t_0 * t_17)) / t_2), Float64(Float64(t_5 * t_7) / t_15)))
	t_19 = Float64(Float64(y_45_scale / x_45_scale) * Float64(fma(-3.175328964080679e-10, (pi ^ 4.0), Float64(0.5 * t_18)) / t_10))
	t_20 = Float64(Float64(y_45_scale / x_45_scale) * Float64(t_17 / t_10))
	t_21 = Float64(Float64(90.0 * t_20) - t_14)
	t_22 = fma(0.005555555555555556, Float64(Float64(pi * Float64(t_0 * t_21)) / t_2), Float64(32400.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(Float64(t_8 / Float64(pi * pi)) * Float64(t_12 / t_3)))))
	t_23 = Float64(t_1 * t_9)
	t_24 = fma(-1.7146776406035666e-7, t_23, Float64(-5.7155921353452215e-8 * t_23))
	t_25 = Float64(fma(-3.5281432934229765e-12, Float64((pi ^ 5.0) * t_9), Float64(3.08641975308642e-5 * Float64(Float64(pi * pi) * t_24))) / t_3)
	tmp = 0.0
	if ((y_45_scale <= -3.8e+22) || !(y_45_scale <= 6.1e+113))
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(180.0 * Float64(Float64(angle / x_45_scale) * Float64(Float64(y_45_scale * t_12) / t_10)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(angle * fma(180.0, Float64(Float64(y_45_scale / x_45_scale) * Float64(t_12 / t_10)), Float64(angle * Float64(fma(90.0, t_20, Float64(angle * Float64(fma(180.0, t_19, Float64(angle * Float64(Float64(90.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(Float64(fma(-2.0, t_25, Float64(4.0 * t_25)) - fma(0.011111111111111112, Float64(Float64(pi * Float64(t_0 * t_18)) / t_2), Float64(Float64(Float64(t_5 * t_24) / t_15) + Float64(Float64(t_7 * t_17) / t_3)))) / t_10))) - fma(-1.02880658436214e-5, Float64(Float64(y_45_scale / x_45_scale) * Float64(Float64(Float64(pi * pi) * t_13) / t_3)), fma(0.005555555555555556, Float64(Float64(pi * Float64(t_0 * Float64(Float64(180.0 * t_19) + Float64(-1.0 * t_22)))) / t_2), Float64(180.0 * Float64(Float64(t_8 / pi) * Float64(t_21 / t_2)))))))) - t_22))) - t_14))))))) / pi));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(-6.17283950617284e-5 * t$95$4 + N[(0.0001234567901234568 * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(-3.08641975308642e-5 * t$95$4 + N[(3.08641975308642e-5 * N[(N[(Pi * Pi), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(-8.573388203017833e-8 * t$95$6 + N[(-2.8577960676726107e-8 * t$95$6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$0 * t$95$2), $MachinePrecision]}, Block[{t$95$10 = N[(Pi * t$95$2), $MachinePrecision]}, Block[{t$95$11 = N[(N[(-3.175328964080679e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(Pi * Pi), $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$12 = N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(t$95$5 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[(t$95$0 * t$95$12), $MachinePrecision]}, Block[{t$95$14 = N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$13 / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[Power[t$95$2, 4.0], $MachinePrecision]}, Block[{t$95$16 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$17 = N[(N[(-6.858710562414266e-7 * t$95$16 + N[(1.3717421124828533e-6 * t$95$16), $MachinePrecision]), $MachinePrecision] - N[(0.011111111111111112 * N[(N[(Pi * N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$18 = N[(N[(-2.0 * t$95$11 + N[(4.0 * t$95$11), $MachinePrecision]), $MachinePrecision] - N[(0.011111111111111112 * N[(N[(Pi * N[(t$95$0 * t$95$17), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(t$95$5 * t$95$7), $MachinePrecision] / t$95$15), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$19 = N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(-3.175328964080679e-10 * N[Power[Pi, 4.0], $MachinePrecision] + N[(0.5 * t$95$18), $MachinePrecision]), $MachinePrecision] / t$95$10), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$20 = N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$17 / t$95$10), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$21 = N[(N[(90.0 * t$95$20), $MachinePrecision] - t$95$14), $MachinePrecision]}, Block[{t$95$22 = N[(0.005555555555555556 * N[(N[(Pi * N[(t$95$0 * t$95$21), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(32400.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(t$95$8 / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$12 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$23 = N[(t$95$1 * t$95$9), $MachinePrecision]}, Block[{t$95$24 = N[(-1.7146776406035666e-7 * t$95$23 + N[(-5.7155921353452215e-8 * t$95$23), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$25 = N[(N[(-3.5281432934229765e-12 * N[(N[Power[Pi, 5.0], $MachinePrecision] * t$95$9), $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(Pi * Pi), $MachinePrecision] * t$95$24), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[Or[LessEqual[y$45$scale, -3.8e+22], N[Not[LessEqual[y$45$scale, 6.1e+113]], $MachinePrecision]], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(180.0 * N[(N[(angle / x$45$scale), $MachinePrecision] * N[(N[(y$45$scale * t$95$12), $MachinePrecision] / t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(angle * N[(180.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(t$95$12 / t$95$10), $MachinePrecision]), $MachinePrecision] + N[(angle * N[(N[(90.0 * t$95$20 + N[(angle * N[(N[(180.0 * t$95$19 + N[(angle * N[(N[(90.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(N[(-2.0 * t$95$25 + N[(4.0 * t$95$25), $MachinePrecision]), $MachinePrecision] - N[(0.011111111111111112 * N[(N[(Pi * N[(t$95$0 * t$95$18), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(N[(t$95$5 * t$95$24), $MachinePrecision] / t$95$15), $MachinePrecision] + N[(N[(t$95$7 * t$95$17), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.02880658436214e-5 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * t$95$13), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * N[(N[(Pi * N[(t$95$0 * N[(N[(180.0 * t$95$19), $MachinePrecision] + N[(-1.0 * t$95$22), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(180.0 * N[(N[(t$95$8 / Pi), $MachinePrecision] * N[(t$95$21 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$22), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$14), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < -3.8000000000000004e22 or 6.09999999999999996e113 < y-scale

   1. Initial program 25.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 24.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 9.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 38.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(180 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)\right)}{\pi} \]

   14. Applied rewrites 46.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(180 \cdot \left(\frac{angle}{x-scale} \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)}{\pi} \]

   if -3.8000000000000004e22 < y-scale < 6.09999999999999996e113

   1. Initial program 8.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 11.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 15.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

   9. Taylor expanded in y-scale around 0

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

   11. Applied rewrites 40.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

   12. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(angle \cdot \left(180 \cdot \frac{y-scale \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + angle \cdot \color{blue}{\left(\left(90 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + angle \cdot \left(\left(180 \cdot \frac{y-scale \cdot \left(\frac{-1}{3149280000} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{2} \cdot \left(\left(-2 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + 4 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right) - \left(\frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{\left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)\right)\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} + angle \cdot \left(90 \cdot \frac{y-scale \cdot \left(\left(-2 \cdot \frac{\frac{-1}{283435200000} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{5832000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + 4 \cdot \frac{\frac{-1}{283435200000} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{5832000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right) - \left(\frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(-2 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + 4 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right) - \left(\frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{\left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\frac{\left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}} + \frac{\left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} - \left(\frac{-1}{97200} \cdot \frac{y-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right)}{x-scale \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(180 \cdot \frac{y-scale \cdot \left(\frac{-1}{3149280000} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{2} \cdot \left(\left(-2 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + 4 \cdot \frac{\frac{-1}{3149280000} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right) - \left(\frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{\left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)\right)\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} - \left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(90 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} - \frac{y-scale \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + 32400 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}\right)\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + 180 \cdot \frac{\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(90 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} - \frac{y-scale \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) - \left(\frac{1}{180} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(90 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{1458000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{729000} \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right) - \frac{1}{90} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{3}}\right)}{x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} - \frac{y-scale \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} + 32400 \cdot \frac{y-scale \cdot \left(\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}\right)\right)\right) - \frac{y-scale \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}\right)\right)\right)\right)}{\pi} \]

   14. Applied rewrites 45.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(180, \frac{y-scale}{x-scale} \cdot \color{blue}{\frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}, angle \cdot \left(\mathsf{fma}\left(90, \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}, angle \cdot \left(\mathsf{fma}\left(180, \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4}, 0.5 \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}, angle \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.5281432934229765 \cdot 10^{-12}, {\pi}^{5} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.5281432934229765 \cdot 10^{-12}, {\pi}^{5} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}} + \frac{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(-1.02880658436214 \cdot 10^{-5}, \frac{y-scale}{x-scale} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(180 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4}, 0.5 \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 32400 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi \cdot \pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 180 \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi} \cdot \frac{90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 32400 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi \cdot \pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -3.8 \cdot 10^{+22} \lor \neg \left(y-scale \leq 6.1 \cdot 10^{+113}\right):\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(180 \cdot \left(\frac{angle}{x-scale} \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(angle \cdot \mathsf{fma}\left(180, \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}, angle \cdot \left(\mathsf{fma}\left(90, \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}, angle \cdot \left(\mathsf{fma}\left(180, \frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4}, 0.5 \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}, angle \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.5281432934229765 \cdot 10^{-12}, {\pi}^{5} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.5281432934229765 \cdot 10^{-12}, {\pi}^{5} \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), -5.7155921353452215 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}} + \frac{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(-1.02880658436214 \cdot 10^{-5}, \frac{y-scale}{x-scale} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(180 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4}, 0.5 \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}, 4 \cdot \frac{\mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10}, {\pi}^{4} \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \mathsf{fma}\left(0.011111111111111112, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right)}{{\sin \left(0.5 \cdot \pi\right)}^{4}}\right)\right)\right)}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) + -1 \cdot \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 32400 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi \cdot \pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 180 \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi} \cdot \frac{90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.005555555555555556, \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \left(90 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(-6.858710562414266 \cdot 10^{-7}, \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 1.3717421124828533 \cdot 10^{-6} \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}\right) - 0.011111111111111112 \cdot \frac{\pi \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\right)}{\left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)}}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)}{\sin \left(0.5 \cdot \pi\right)}, 32400 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(\frac{\mathsf{fma}\left(-8.573388203017833 \cdot 10^{-8}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)}{\pi \cdot \pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right) - \frac{y-scale}{x-scale} \cdot \frac{\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)\right)\right)\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.2% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\pi \cdot \pi\right) \cdot t\_1\\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_2, 0.0001234567901234568 \cdot t\_2\right)}{t\_1}\right)}{t\_0}\right)\right)\right)\right)\right)}{\pi} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI))) (t_1 (* t_0 t_0)) (t_2 (* (* PI PI) t_1)))
   (*
    180.0
    (/
     (atan
      (*
       -0.5
       (*
        -1.0
        (*
         (/ y-scale x-scale)
         (*
          180.0
          (*
           (/ angle PI)
           (/
            (fma
             3.08641975308642e-5
             (* PI PI)
             (*
              0.5
              (/
               (fma -6.17283950617284e-5 t_2 (* 0.0001234567901234568 t_2))
               t_1)))
            t_0)))))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double t_1 = t_0 * t_0;
	double t_2 = (((double) M_PI) * ((double) M_PI)) * t_1;
	return 180.0 * (atan((-0.5 * (-1.0 * ((y_45_scale / x_45_scale) * (180.0 * ((angle / ((double) M_PI)) * (fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (fma(-6.17283950617284e-5, t_2, (0.0001234567901234568 * t_2)) / t_1))) / t_0))))))) / ((double) M_PI));
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(pi * pi) * t_1)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(Float64(y_45_scale / x_45_scale) * Float64(180.0 * Float64(Float64(angle / pi) * Float64(fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(fma(-6.17283950617284e-5, t_2, Float64(0.0001234567901234568 * t_2)) / t_1))) / t_0))))))) / pi))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(180.0 * N[(N[(angle / Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(-6.17283950617284e-5 * t$95$2 + N[(0.0001234567901234568 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 16.9%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

6. Taylor expanded in x-scale around inf

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

8. Applied rewrites 13.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

9. Taylor expanded in y-scale around 0

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

11. Applied rewrites 39.7%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

12. Taylor expanded in angle around 0

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \frac{angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right)\right)}{\pi} \]
13. Step-by-step derivation

14. Applied rewrites 44.6%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \left(180 \cdot \left(\frac{angle}{\pi} \cdot \frac{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)\right)}{\pi} \]
15. Add Preprocessing

Alternative 12: 39.8% accurate, N/A× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\pi \cdot \pi\right) \cdot t\_1\\ 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(180 \cdot \left(\frac{angle}{x-scale} \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, t\_2, 0.0001234567901234568 \cdot t\_2\right)}{t\_1}\right)}{\pi \cdot t\_0}\right)\right)\right)\right)}{\pi} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI))) (t_1 (* t_0 t_0)) (t_2 (* (* PI PI) t_1)))
   (*
    180.0
    (/
     (atan
      (*
       -0.5
       (*
        -1.0
        (*
         180.0
         (*
          (/ angle x-scale)
          (/
           (*
            y-scale
            (fma
             3.08641975308642e-5
             (* PI PI)
             (*
              0.5
              (/
               (fma -6.17283950617284e-5 t_2 (* 0.0001234567901234568 t_2))
               t_1))))
           (* PI t_0)))))))
     PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double t_1 = t_0 * t_0;
	double t_2 = (((double) M_PI) * ((double) M_PI)) * t_1;
	return 180.0 * (atan((-0.5 * (-1.0 * (180.0 * ((angle / x_45_scale) * ((y_45_scale * fma(3.08641975308642e-5, (((double) M_PI) * ((double) M_PI)), (0.5 * (fma(-6.17283950617284e-5, t_2, (0.0001234567901234568 * t_2)) / t_1)))) / (((double) M_PI) * t_0))))))) / ((double) M_PI));
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(pi * pi) * t_1)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(-1.0 * Float64(180.0 * Float64(Float64(angle / x_45_scale) * Float64(Float64(y_45_scale * fma(3.08641975308642e-5, Float64(pi * pi), Float64(0.5 * Float64(fma(-6.17283950617284e-5, t_2, Float64(0.0001234567901234568 * t_2)) / t_1)))) / Float64(pi * t_0))))))) / pi))
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * Pi), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(-1.0 * N[(180.0 * N[(N[(angle / x$45$scale), $MachinePrecision] * N[(N[(y$45$scale * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision] + N[(0.5 * N[(N[(-6.17283950617284e-5 * t$95$2 + N[(0.0001234567901234568 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 16.9%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-0.5 \cdot \left(\frac{x-scale}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)} \cdot \frac{y-scale \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale} - \left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot x-scale}\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)}}{\pi} \]

6. Taylor expanded in x-scale around inf

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

8. Applied rewrites 13.1%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \color{blue}{\frac{y-scale \cdot \mathsf{fma}\left(0.5, \frac{\left(y-scale \cdot y-scale\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}}\right)\right)}{\pi} \]

9. Taylor expanded in y-scale around 0

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

11. Applied rewrites 39.7%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(\frac{y-scale}{x-scale} \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right), 4 \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}}\right)\right)\right)}{\pi} \]

12. Taylor expanded in angle around 0

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(-1 \cdot \left(180 \cdot \frac{angle \cdot \left(y-scale \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{2} \cdot \frac{\frac{-1}{16200} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{8100} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)}{{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)}{x-scale \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)\right)}{\pi} \]

14. Applied rewrites 41.0%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(-1 \cdot \left(180 \cdot \left(\frac{angle}{x-scale} \cdot \frac{y-scale \cdot \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \pi \cdot \pi, 0.5 \cdot \frac{\mathsf{fma}\left(-6.17283950617284 \cdot 10^{-5}, \left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right), 0.0001234567901234568 \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}{\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)}\right)}{\color{blue}{\pi \cdot \sin \left(0.5 \cdot \pi\right)}}\right)\right)\right)\right)}{\pi} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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)))