raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.9% → 54.1%

Time: 1.4min

Alternatives: 11

Speedup: 26.0×

• could not determine a ground truth

  1. a = 3.6118367995005413e-93
  2. b = 1.2311789701036854e-231
  3. angle = 8.974010764246105e+167
  4. x-scale = 6.247202590781743e+270
  5. y-scale = 9.396061079102307e-138

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: 13.9% 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: 54.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;b\_m \leq 3.7 \cdot 10^{-91}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{x-scale \cdot t\_2}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(\frac{2}{x-scale} \cdot \frac{{\left(t\_1 \cdot a\right)}^{2} + {\left(b\_m \cdot t\_2\right)}^{2}}{t\_2 \cdot \left(t\_1 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin t\_3}{x-scale \cdot \cos t\_3}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3 (* PI (* 0.005555555555555556 angle))))
   (if (<= b_m 3.7e-91)
     (* 180.0 (/ (atan (/ (* y-scale t_1) (* x-scale t_2))) PI))
     (if (<= b_m 3.5e+61)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            y-scale
            (*
             (/ 2.0 x-scale)
             (/
              (+ (pow (* t_1 a) 2.0) (pow (* b_m t_2) 2.0))
              (* t_2 (* t_1 (- (pow b_m 2.0) (pow a 2.0)))))))))
         PI))
       (if (<= b_m 3e+109)
         (*
          180.0
          (/ (atan (* y-scale (/ (sin t_3) (* x-scale (cos t_3))))) PI))
         (*
          180.0
          (pow
           (cbrt
            (/
             (atan
              (*
               (/ y-scale x-scale)
               (/
                1.0
                (tan
                 (*
                  (* 0.005555555555555556 angle)
                  (* (cbrt PI) (pow (cbrt PI) 2.0)))))))
             (- PI)))
           3.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (b_m <= 3.7e-91) {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * t_2))) / ((double) M_PI));
	} else if (b_m <= 3.5e+61) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * ((2.0 / x_45_scale) * ((pow((t_1 * a), 2.0) + pow((b_m * t_2), 2.0)) / (t_2 * (t_1 * (pow(b_m, 2.0) - pow(a, 2.0))))))))) / ((double) M_PI));
	} else if (b_m <= 3e+109) {
		tmp = 180.0 * (atan((y_45_scale * (sin(t_3) / (x_45_scale * cos(t_3))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * pow(cbrt((atan(((y_45_scale / x_45_scale) * (1.0 / tan(((0.005555555555555556 * angle) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))) / -((double) M_PI))), 3.0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (b_m <= 3.7e-91) {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (x_45_scale * t_2))) / Math.PI);
	} else if (b_m <= 3.5e+61) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * ((2.0 / x_45_scale) * ((Math.pow((t_1 * a), 2.0) + Math.pow((b_m * t_2), 2.0)) / (t_2 * (t_1 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))))))))) / Math.PI);
	} else if (b_m <= 3e+109) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.sin(t_3) / (x_45_scale * Math.cos(t_3))))) / Math.PI);
	} else {
		tmp = 180.0 * Math.pow(Math.cbrt((Math.atan(((y_45_scale / x_45_scale) * (1.0 / Math.tan(((0.005555555555555556 * angle) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))) / -Math.PI)), 3.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (b_m <= 3.7e-91)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * t_2))) / pi));
	elseif (b_m <= 3.5e+61)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(2.0 / x_45_scale) * Float64(Float64((Float64(t_1 * a) ^ 2.0) + (Float64(b_m * t_2) ^ 2.0)) / Float64(t_2 * Float64(t_1 * Float64((b_m ^ 2.0) - (a ^ 2.0))))))))) / pi));
	elseif (b_m <= 3e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(sin(t_3) / Float64(x_45_scale * cos(t_3))))) / pi));
	else
		tmp = Float64(180.0 * (cbrt(Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(1.0 / tan(Float64(Float64(0.005555555555555556 * angle) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))))) / Float64(-pi))) ^ 3.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.7e-91], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.5e+61], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(2.0 / x$45$scale), $MachinePrecision] * N[(N[(N[Power[N[(t$95$1 * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b$95$m * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(t$95$1 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3e+109], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Sin[t$95$3], $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[Power[N[Power[N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(1.0 / N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 3.7000000000000002e-91

   1. Initial program 11.8%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in x-scale around 0 25.8%

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

   7. Simplified 30.6%

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

   8. Taylor expanded in a around inf 51.2%

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

   if 3.7000000000000002e-91 < b < 3.50000000000000018e61

   1. Initial program 26.3%

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

   3. Simplified 18.9%

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

   5. Taylor expanded in x-scale around 0 42.7%

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

   7. Simplified 50.4%

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

      1. times-frac 50.4%

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

      2. associate-*r* 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. associate-*r* 50.4%

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

   11. Simplified 50.4%

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

   if 3.50000000000000018e61 < b < 3.00000000000000015e109

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in a around inf 16.7%

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

   6. Taylor expanded in y-scale around inf 68.1%

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

      1. associate-*r/ 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x-scale around 0 68.1%

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

       1. associate-/l* 83.6%

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

       2. associate-*r* 83.6%

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

       3. *-commutative 83.6%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if 3.00000000000000015e109 < b

   1. Initial program 4.1%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in x-scale around 0 8.6%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in a around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. times-frac 62.9%

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

   10. Simplified 62.9%

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

       1. add-cube-cbrt 62.2%

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

       2. pow3 62.2%

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

   12. Applied egg-rr 63.0%

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

       1. add-cube-cbrt 68.0%

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

       2. pow2 68.0%

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

   14. Applied egg-rr 68.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-91}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \left(\frac{2}{x-scale} \cdot \frac{{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_3 := \cos t\_0\\ \mathbf{if}\;b\_m \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{x-scale \cdot t\_3}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.35 \cdot 10^{+63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)}^{2} \cdot \left(-y-scale\right)}{x-scale \cdot \left(t\_1 \cdot t\_3\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin t\_2}{x-scale \cdot \cos t\_2}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* PI (* 0.005555555555555556 angle)))
        (t_3 (cos t_0)))
   (if (<= b_m 1.25e-9)
     (* 180.0 (/ (atan (/ (* y-scale t_1) (* x-scale t_3))) PI))
     (if (<= b_m 1.35e+63)
       (*
        180.0
        (/
         (atan
          (/
           (*
            (pow (cos (* angle (* PI (cbrt 1.7146776406035666e-7)))) 2.0)
            (- y-scale))
           (* x-scale (* t_1 t_3))))
         PI))
       (if (<= b_m 4.2e+109)
         (*
          180.0
          (/ (atan (* y-scale (/ (sin t_2) (* x-scale (cos t_2))))) PI))
         (*
          180.0
          (pow
           (cbrt
            (/
             (atan
              (*
               (/ y-scale x-scale)
               (/
                1.0
                (tan
                 (*
                  (* 0.005555555555555556 angle)
                  (* (cbrt PI) (pow (cbrt PI) 2.0)))))))
             (- PI)))
           3.0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_3 = cos(t_0);
	double tmp;
	if (b_m <= 1.25e-9) {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (x_45_scale * t_3))) / ((double) M_PI));
	} else if (b_m <= 1.35e+63) {
		tmp = 180.0 * (atan(((pow(cos((angle * (((double) M_PI) * cbrt(1.7146776406035666e-7)))), 2.0) * -y_45_scale) / (x_45_scale * (t_1 * t_3)))) / ((double) M_PI));
	} else if (b_m <= 4.2e+109) {
		tmp = 180.0 * (atan((y_45_scale * (sin(t_2) / (x_45_scale * cos(t_2))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * pow(cbrt((atan(((y_45_scale / x_45_scale) * (1.0 / tan(((0.005555555555555556 * angle) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))) / -((double) M_PI))), 3.0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double t_3 = Math.cos(t_0);
	double tmp;
	if (b_m <= 1.25e-9) {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (x_45_scale * t_3))) / Math.PI);
	} else if (b_m <= 1.35e+63) {
		tmp = 180.0 * (Math.atan(((Math.pow(Math.cos((angle * (Math.PI * Math.cbrt(1.7146776406035666e-7)))), 2.0) * -y_45_scale) / (x_45_scale * (t_1 * t_3)))) / Math.PI);
	} else if (b_m <= 4.2e+109) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.sin(t_2) / (x_45_scale * Math.cos(t_2))))) / Math.PI);
	} else {
		tmp = 180.0 * Math.pow(Math.cbrt((Math.atan(((y_45_scale / x_45_scale) * (1.0 / Math.tan(((0.005555555555555556 * angle) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))) / -Math.PI)), 3.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_3 = cos(t_0)
	tmp = 0.0
	if (b_m <= 1.25e-9)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(x_45_scale * t_3))) / pi));
	elseif (b_m <= 1.35e+63)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64((cos(Float64(angle * Float64(pi * cbrt(1.7146776406035666e-7)))) ^ 2.0) * Float64(-y_45_scale)) / Float64(x_45_scale * Float64(t_1 * t_3)))) / pi));
	elseif (b_m <= 4.2e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(sin(t_2) / Float64(x_45_scale * cos(t_2))))) / pi));
	else
		tmp = Float64(180.0 * (cbrt(Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(1.0 / tan(Float64(Float64(0.005555555555555556 * angle) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))))) / Float64(-pi))) ^ 3.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 1.25e-9], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(x$45$scale * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.35e+63], N[(180.0 * N[(N[ArcTan[N[(N[(N[Power[N[Cos[N[(angle * N[(Pi * N[Power[1.7146776406035666e-7, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * (-y$45$scale)), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.2e+109], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Sin[t$95$2], $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[Power[N[Power[N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(1.0 / N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.25e-9

   1. Initial program 12.8%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in x-scale around 0 26.9%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in a around inf 50.6%

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

   if 1.25e-9 < b < 1.35000000000000009e63

   1. Initial program 30.3%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in x-scale around 0 52.2%

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

   7. Simplified 52.2%

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

      1. add-cbrt-cube 41.5%

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

   9. Applied egg-rr 41.5%

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

   10. Taylor expanded in a around 0 53.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot {\cos \left(angle \cdot \left(\pi \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)}^{2}}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}}{\pi} \]

   if 1.35000000000000009e63 < b < 4.2000000000000003e109

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in a around inf 16.7%

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

   6. Taylor expanded in y-scale around inf 68.1%

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

      1. associate-*r/ 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in x-scale around 0 68.1%

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

       1. associate-/l* 83.6%

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

       2. associate-*r* 83.6%

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

       3. *-commutative 83.6%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if 4.2000000000000003e109 < b

   1. Initial program 4.1%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in x-scale around 0 8.6%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in a around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. times-frac 62.9%

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

   10. Simplified 62.9%

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

       1. add-cube-cbrt 62.2%

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

       2. pow3 62.2%

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

   12. Applied egg-rr 63.0%

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

       1. add-cube-cbrt 68.0%

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

       2. pow2 68.0%

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

   14. Applied egg-rr 68.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+63}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)}^{2} \cdot \left(-y-scale\right)}{x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 0.0275:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 0.0275)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale (cos t_0)))) PI))
     (*
      180.0
      (pow
       (cbrt
        (/
         (atan
          (*
           (/ y-scale x-scale)
           (/
            1.0
            (tan (* (* 0.005555555555555556 angle) (cbrt (pow PI 3.0)))))))
         (- PI)))
       3.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 0.0275) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * pow(cbrt((atan(((y_45_scale / x_45_scale) * (1.0 / tan(((0.005555555555555556 * angle) * cbrt(pow(((double) M_PI), 3.0))))))) / -((double) M_PI))), 3.0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 0.0275) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else {
		tmp = 180.0 * Math.pow(Math.cbrt((Math.atan(((y_45_scale / x_45_scale) * (1.0 / Math.tan(((0.005555555555555556 * angle) * Math.cbrt(Math.pow(Math.PI, 3.0))))))) / -Math.PI)), 3.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 0.0275)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * cos(t_0)))) / pi));
	else
		tmp = Float64(180.0 * (cbrt(Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * Float64(1.0 / tan(Float64(Float64(0.005555555555555556 * angle) * cbrt((pi ^ 3.0))))))) / Float64(-pi))) ^ 3.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 0.0275], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[Power[N[Power[N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[(1.0 / N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0275000000000000001

   1. Initial program 12.8%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in x-scale around 0 26.9%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in a around inf 50.6%

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

   if 0.0275000000000000001 < b

   1. Initial program 9.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x-scale around 0 20.4%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in a around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. times-frac 63.4%

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

   10. Simplified 63.4%

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

       1. add-cube-cbrt 62.7%

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

       2. pow3 62.7%

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

   12. Applied egg-rr 60.4%

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

       1. add-cbrt-cube 65.8%

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

       2. pow3 65.8%

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

   14. Applied egg-rr 65.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0275:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot {\left(\sqrt[3]{\frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)}\right)}{-\pi}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan t\_0}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 9.5e-10)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale (cos t_0)))) PI))
     (* 180.0 (/ (atan (/ y-scale (* x-scale (tan t_0)))) (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 9.5e-10) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((y_45_scale / (x_45_scale * tan(t_0)))) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 9.5e-10) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((y_45_scale / (x_45_scale * Math.tan(t_0)))) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 9.5e-10:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin(t_0)) / (x_45_scale * math.cos(t_0)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((y_45_scale / (x_45_scale * math.tan(t_0)))) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 9.5e-10)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * cos(t_0)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale / Float64(x_45_scale * tan(t_0)))) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 9.5e-10)
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / pi);
	else
		tmp = 180.0 * (atan((y_45_scale / (x_45_scale * tan(t_0)))) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 9.5e-10], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 9.50000000000000028e-10

   1. Initial program 12.8%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in x-scale around 0 26.9%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in a around inf 50.6%

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

   if 9.50000000000000028e-10 < b

   1. Initial program 9.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x-scale around 0 20.4%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in a around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. times-frac 63.4%

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

   10. Simplified 63.4%

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

       1. log1p-expm1-u 63.4%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. log1p-undefine 63.4%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. Applied egg-rr 63.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   13. Step-by-step derivation

       1. atan-neg 63.4%

          \[\leadsto 180 \cdot \frac{\color{blue}{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

       2. clear-num 63.4%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\frac{1}{\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}}\right)}{\pi} \]

       3. associate-*r* 62.3%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\frac{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}\right)}{\pi} \]

       4. log1p-define 62.3%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}}\right)}{\pi} \]

       5. log1p-expm1-u 62.3%

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

       6. associate-*r* 61.1%

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

       7. tan-quot 61.1%

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

   14. Applied egg-rr 63.4%

       \[\leadsto 180 \cdot \frac{\color{blue}{0 - \tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   15. Step-by-step derivation

       1. neg-sub0 63.4%

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

       2. associate-/r* 64.5%

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

   16. Simplified 64.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;b\_m \leq 8 \cdot 10^{-10}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{-\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (if (<= b_m 8e-10)
     (* 180.0 (/ (atan (* y-scale (/ (sin t_0) (* x-scale (cos t_0))))) PI))
     (*
      180.0
      (/
       (atan
        (/ y-scale (* x-scale (tan (* 0.005555555555555556 (* angle PI))))))
       (- PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (b_m <= 8e-10) {
		tmp = 180.0 * (atan((y_45_scale * (sin(t_0) / (x_45_scale * cos(t_0))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * ((double) M_PI))))))) / -((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (b_m <= 8e-10) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.sin(t_0) / (x_45_scale * Math.cos(t_0))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((y_45_scale / (x_45_scale * Math.tan((0.005555555555555556 * (angle * Math.PI)))))) / -Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if b_m <= 8e-10:
		tmp = 180.0 * (math.atan((y_45_scale * (math.sin(t_0) / (x_45_scale * math.cos(t_0))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((y_45_scale / (x_45_scale * math.tan((0.005555555555555556 * (angle * math.pi)))))) / -math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (b_m <= 8e-10)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(sin(t_0) / Float64(x_45_scale * cos(t_0))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale / Float64(x_45_scale * tan(Float64(0.005555555555555556 * Float64(angle * pi)))))) / Float64(-pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = pi * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (b_m <= 8e-10)
		tmp = 180.0 * (atan((y_45_scale * (sin(t_0) / (x_45_scale * cos(t_0))))) / pi);
	else
		tmp = 180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * pi)))))) / -pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 8e-10], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Sin[t$95$0], $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000029e-10

   1. Initial program 12.8%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in a around inf 20.0%

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

   6. Taylor expanded in y-scale around inf 39.1%

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

      1. associate-*r/ 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in x-scale around 0 50.6%

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

       1. associate-/l* 49.7%

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

       2. associate-*r* 49.2%

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

       3. *-commutative 49.2%

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

       4. associate-*r* 49.4%

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

       5. *-commutative 49.4%

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

   11. Simplified 49.4%

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

   if 8.00000000000000029e-10 < b

   1. Initial program 9.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x-scale around 0 20.4%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in a around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. times-frac 63.4%

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

   10. Simplified 63.4%

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

       1. log1p-expm1-u 63.4%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

       2. log1p-undefine 63.4%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

   12. Applied egg-rr 63.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]
   13. Step-by-step derivation

       1. atan-neg 63.4%

          \[\leadsto 180 \cdot \frac{\color{blue}{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

       2. clear-num 63.4%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \color{blue}{\frac{1}{\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}}\right)}{\pi} \]

       3. associate-*r* 62.3%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\frac{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}\right)}{\pi} \]

       4. log1p-define 62.3%

          \[\leadsto 180 \cdot \frac{-\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \frac{1}{\frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}}\right)}{\pi} \]

       5. log1p-expm1-u 62.3%

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

       6. associate-*r* 61.1%

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

       7. tan-quot 61.1%

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

   14. Applied egg-rr 63.4%

       \[\leadsto 180 \cdot \frac{\color{blue}{0 - \tan^{-1} \left(\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   15. Step-by-step derivation

       1. neg-sub0 63.4%

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

       2. associate-/r* 64.5%

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

   16. Simplified 64.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-10}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{-\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.35 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.35e+109)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (* y-scale (/ PI x-scale)))))
     PI))
   (*
    -180.0
    (/
     (atan (/ y-scale (* x-scale (tan (* PI (* 0.005555555555555556 angle))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.35e+109) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (((double) M_PI) / x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = -180.0 * (atan((y_45_scale / (x_45_scale * tan((((double) M_PI) * (0.005555555555555556 * angle)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.35e+109) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * (y_45_scale * (Math.PI / x_45_scale))))) / Math.PI);
	} else {
		tmp = -180.0 * (Math.atan((y_45_scale / (x_45_scale * Math.tan((Math.PI * (0.005555555555555556 * angle)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 2.35e+109:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * (y_45_scale * (math.pi / x_45_scale))))) / math.pi)
	else:
		tmp = -180.0 * (math.atan((y_45_scale / (x_45_scale * math.tan((math.pi * (0.005555555555555556 * angle)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 2.35e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(y_45_scale * Float64(pi / x_45_scale))))) / pi));
	else
		tmp = Float64(-180.0 * Float64(atan(Float64(y_45_scale / Float64(x_45_scale * tan(Float64(pi * Float64(0.005555555555555556 * angle)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 2.35e+109)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (pi / x_45_scale))))) / pi);
	else
		tmp = -180.0 * (atan((y_45_scale / (x_45_scale * tan((pi * (0.005555555555555556 * angle)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 2.35e+109], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(-180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.34999999999999999e109

   1. Initial program 13.8%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in a around inf 19.5%

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

   6. Taylor expanded in y-scale around inf 39.2%

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

      1. associate-*r/ 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in angle around 0 43.1%

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

       1. associate-/l* 49.1%

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

       2. associate-/l* 49.1%

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

   11. Simplified 49.1%

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

   if 2.34999999999999999e109 < b

   1. Initial program 4.1%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in x-scale around 0 8.6%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in a around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. times-frac 62.9%

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

   10. Simplified 62.9%

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

       1. add-cube-cbrt 62.2%

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

       2. pow3 62.2%

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

   12. Applied egg-rr 63.0%

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

   13. Taylor expanded in y-scale around 0 66.5%

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

       1. associate-*r/ 66.5%

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

       2. rem-cube-cbrt 66.5%

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

       3. associate-/r* 62.9%

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

       4. *-commutative 62.9%

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

       5. neg-mul-1 62.9%

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

       6. associate-/l* 62.9%

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

       7. distribute-frac-neg 62.9%

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

       8. distribute-rgt-neg-out 62.9%

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

       9. distribute-lft-neg-in 62.9%

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

       10. metadata-eval 62.9%

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

   15. Simplified 67.4%

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

Alternative 7: 53.1% accurate, 24.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.9 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\frac{y-scale}{angle}}{\pi \cdot x-scale}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 3.9e+109)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (* y-scale (/ PI x-scale)))))
     PI))
   (* 180.0 (/ (atan (* -180.0 (/ (/ y-scale angle) (* PI x-scale)))) PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.9e+109) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (((double) M_PI) / x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * ((y_45_scale / angle) / (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 3.9e+109) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * (y_45_scale * (Math.PI / x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * ((y_45_scale / angle) / (Math.PI * x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 3.9e+109:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * (y_45_scale * (math.pi / x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * ((y_45_scale / angle) / (math.pi * x_45_scale)))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.9e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(y_45_scale * Float64(pi / x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(Float64(y_45_scale / angle) / Float64(pi * x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 3.9e+109)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * (y_45_scale * (pi / x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * ((y_45_scale / angle) / (pi * x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 3.9e+109], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(y$45$scale * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(N[(y$45$scale / angle), $MachinePrecision] / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.90000000000000016e109

   1. Initial program 13.8%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in a around inf 19.5%

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

   6. Taylor expanded in y-scale around inf 39.2%

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

      1. associate-*r/ 39.2%

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

   8. Simplified 39.2%

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

   9. Taylor expanded in angle around 0 43.1%

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

       1. associate-/l* 49.1%

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

       2. associate-/l* 49.1%

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

   11. Simplified 49.1%

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

   if 3.90000000000000016e109 < b

   1. Initial program 4.1%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in angle around 0 4.4%

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

      1. associate-*r/ 4.4%

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

      2. distribute-lft-out-- 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in a around 0 61.8%

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

      1. associate-/r* 61.8%

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

   10. Simplified 61.8%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \left(y-scale \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\frac{y-scale}{angle}}{\pi \cdot x-scale}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.9% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 11.9%

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

3. Simplified 11.8%

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

5. Taylor expanded in x-scale around 0 25.2%

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

7. Simplified 29.7%

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

8. Taylor expanded in a around 0 46.5%

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

   1. mul-1-neg 46.5%

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

   2. times-frac 47.5%

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

10. Simplified 47.5%

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

    1. log1p-expm1-u 47.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

    2. log1p-undefine 47.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

12. Applied egg-rr 47.5%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale}{x-scale} \cdot \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi} \]

13. Taylor expanded in angle around 0 41.0%

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

    1. *-commutative 41.0%

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

    2. associate-/r* 42.1%

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

    3. *-commutative 42.1%

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

15. Simplified 42.1%

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

Alternative 9: 37.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 11.9%

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

3. Simplified 11.8%

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

5. Taylor expanded in angle around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. distribute-lft-out-- 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in a around 0 41.0%

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

   1. associate-/r* 41.1%

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

10. Simplified 41.1%

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

11. Final simplification 41.1%

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

Alternative 10: 37.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 11.9%

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

3. Simplified 11.8%

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

5. Taylor expanded in angle around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. distribute-lft-out-- 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in a around 0 41.0%

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

9. Final simplification 41.0%

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

Alternative 11: 11.9% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 11.9%

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

3. Simplified 11.8%

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

5. Taylor expanded in angle around 0 9.1%

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

   1. associate-*r/ 9.1%

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

   2. distribute-lft-out-- 9.1%

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

7. Simplified 9.1%

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

8. Taylor expanded in a around inf 13.9%

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

Reproduce

herbie shell --seed 2024180 
(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)))