raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.7% → 52.5%

Time: 23.8s

Alternatives: 17

Speedup: 22.2×

Specification

Math

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

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: 13.7% accurate, 1.0× speedup

Math

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

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: 52.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \pi + \left(-\pi\right) \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ t_3 := \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right) + {\left(t\_2 \cdot a\right)}^{2}\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_5 := \sin t\_4\\ t_6 := \sin t\_0\\ \mathbf{if}\;\left|b\right| \leq 5.4 \cdot 10^{-151}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_5}^{4}} + {t\_5}^{2}\right)}{x-scale \cdot \left(\cos t\_4 \cdot t\_5\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.6 \cdot 10^{+76}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{\left|t\_3\right| + t\_3}{\left(\left(\left(\left|b\right| + a\right) \cdot \left(\left|b\right| - a\right)\right) \cdot \sin t\_1\right) \cdot \cos t\_1} \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \sqrt{{t\_6}^{4}}}{\left(t\_2 \cdot t\_6\right) \cdot x-scale}\right) \cdot -0.5\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (+ (* 0.5 PI) (* (- PI) (* 0.005555555555555556 angle))))
       (t_1 (* (* PI angle) 0.005555555555555556))
       (t_2 (sin (* (* 0.005555555555555556 angle) PI)))
       (t_3
        (+
         (* (+ 0.5 (* 0.5 (cos (* 2.0 t_1)))) (* (fabs b) (fabs b)))
         (pow (* t_2 a) 2.0)))
       (t_4 (* 0.005555555555555556 (* angle PI)))
       (t_5 (sin t_4))
       (t_6 (sin t_0)))
  (if (<= (fabs b) 5.4e-151)
    (*
     180.0
     (/
      (atan
       (*
        0.5
        (/
         (* y-scale (+ (sqrt (pow t_5 4.0)) (pow t_5 2.0)))
         (* x-scale (* (cos t_4) t_5)))))
      PI))
    (if (<= (fabs b) 1.6e+76)
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (*
           (/
            (+ (fabs t_3) t_3)
            (*
             (* (* (+ (fabs b) a) (- (fabs b) a)) (sin t_1))
             (cos t_1)))
           (/ y-scale x-scale))))
        PI))
      (/
       (*
        180.0
        (atan
         (*
          (*
           y-scale
           (/
            (+ (- 0.5 (* 0.5 (cos (* 2.0 t_0)))) (sqrt (pow t_6 4.0)))
            (* (* t_2 t_6) x-scale)))
          -0.5)))
       PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * ((double) M_PI)) + (-((double) M_PI) * (0.005555555555555556 * angle));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = sin(((0.005555555555555556 * angle) * ((double) M_PI)));
	double t_3 = ((0.5 + (0.5 * cos((2.0 * t_1)))) * (fabs(b) * fabs(b))) + pow((t_2 * a), 2.0);
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_5 = sin(t_4);
	double t_6 = sin(t_0);
	double tmp;
	if (fabs(b) <= 5.4e-151) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_5, 4.0)) + pow(t_5, 2.0))) / (x_45_scale * (cos(t_4) * t_5))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.6e+76) {
		tmp = 180.0 * (atan((-0.5 * (((fabs(t_3) + t_3) / ((((fabs(b) + a) * (fabs(b) - a)) * sin(t_1)) * cos(t_1))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt(pow(t_6, 4.0))) / ((t_2 * t_6) * x_45_scale))) * -0.5))) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * Math.PI) + (-Math.PI * (0.005555555555555556 * angle));
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double t_2 = Math.sin(((0.005555555555555556 * angle) * Math.PI));
	double t_3 = ((0.5 + (0.5 * Math.cos((2.0 * t_1)))) * (Math.abs(b) * Math.abs(b))) + Math.pow((t_2 * a), 2.0);
	double t_4 = 0.005555555555555556 * (angle * Math.PI);
	double t_5 = Math.sin(t_4);
	double t_6 = Math.sin(t_0);
	double tmp;
	if (Math.abs(b) <= 5.4e-151) {
		tmp = 180.0 * (Math.atan((0.5 * ((y_45_scale * (Math.sqrt(Math.pow(t_5, 4.0)) + Math.pow(t_5, 2.0))) / (x_45_scale * (Math.cos(t_4) * t_5))))) / Math.PI);
	} else if (Math.abs(b) <= 1.6e+76) {
		tmp = 180.0 * (Math.atan((-0.5 * (((Math.abs(t_3) + t_3) / ((((Math.abs(b) + a) * (Math.abs(b) - a)) * Math.sin(t_1)) * Math.cos(t_1))) * (y_45_scale / x_45_scale)))) / Math.PI);
	} else {
		tmp = (180.0 * Math.atan(((y_45_scale * (((0.5 - (0.5 * Math.cos((2.0 * t_0)))) + Math.sqrt(Math.pow(t_6, 4.0))) / ((t_2 * t_6) * x_45_scale))) * -0.5))) / Math.PI;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (0.5 * math.pi) + (-math.pi * (0.005555555555555556 * angle))
	t_1 = (math.pi * angle) * 0.005555555555555556
	t_2 = math.sin(((0.005555555555555556 * angle) * math.pi))
	t_3 = ((0.5 + (0.5 * math.cos((2.0 * t_1)))) * (math.fabs(b) * math.fabs(b))) + math.pow((t_2 * a), 2.0)
	t_4 = 0.005555555555555556 * (angle * math.pi)
	t_5 = math.sin(t_4)
	t_6 = math.sin(t_0)
	tmp = 0
	if math.fabs(b) <= 5.4e-151:
		tmp = 180.0 * (math.atan((0.5 * ((y_45_scale * (math.sqrt(math.pow(t_5, 4.0)) + math.pow(t_5, 2.0))) / (x_45_scale * (math.cos(t_4) * t_5))))) / math.pi)
	elif math.fabs(b) <= 1.6e+76:
		tmp = 180.0 * (math.atan((-0.5 * (((math.fabs(t_3) + t_3) / ((((math.fabs(b) + a) * (math.fabs(b) - a)) * math.sin(t_1)) * math.cos(t_1))) * (y_45_scale / x_45_scale)))) / math.pi)
	else:
		tmp = (180.0 * math.atan(((y_45_scale * (((0.5 - (0.5 * math.cos((2.0 * t_0)))) + math.sqrt(math.pow(t_6, 4.0))) / ((t_2 * t_6) * x_45_scale))) * -0.5))) / math.pi
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.5 * pi) + Float64(Float64(-pi) * Float64(0.005555555555555556 * angle)))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = sin(Float64(Float64(0.005555555555555556 * angle) * pi))
	t_3 = Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_1)))) * Float64(abs(b) * abs(b))) + (Float64(t_2 * a) ^ 2.0))
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_5 = sin(t_4)
	t_6 = sin(t_0)
	tmp = 0.0
	if (abs(b) <= 5.4e-151)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_5 ^ 4.0)) + (t_5 ^ 2.0))) / Float64(x_45_scale * Float64(cos(t_4) * t_5))))) / pi));
	elseif (abs(b) <= 1.6e+76)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(abs(t_3) + t_3) / Float64(Float64(Float64(Float64(abs(b) + a) * Float64(abs(b) - a)) * sin(t_1)) * cos(t_1))) * Float64(y_45_scale / x_45_scale)))) / pi));
	else
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))) + sqrt((t_6 ^ 4.0))) / Float64(Float64(t_2 * t_6) * x_45_scale))) * -0.5))) / pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (0.5 * pi) + (-pi * (0.005555555555555556 * angle));
	t_1 = (pi * angle) * 0.005555555555555556;
	t_2 = sin(((0.005555555555555556 * angle) * pi));
	t_3 = ((0.5 + (0.5 * cos((2.0 * t_1)))) * (abs(b) * abs(b))) + ((t_2 * a) ^ 2.0);
	t_4 = 0.005555555555555556 * (angle * pi);
	t_5 = sin(t_4);
	t_6 = sin(t_0);
	tmp = 0.0;
	if (abs(b) <= 5.4e-151)
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt((t_5 ^ 4.0)) + (t_5 ^ 2.0))) / (x_45_scale * (cos(t_4) * t_5))))) / pi);
	elseif (abs(b) <= 1.6e+76)
		tmp = 180.0 * (atan((-0.5 * (((abs(t_3) + t_3) / ((((abs(b) + a) * (abs(b) - a)) * sin(t_1)) * cos(t_1))) * (y_45_scale / x_45_scale)))) / pi);
	else
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt((t_6 ^ 4.0))) / ((t_2 * t_6) * x_45_scale))) * -0.5))) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] + N[((-Pi) * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$2 * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.4e-151], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$5, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$4], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.6e+76], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(N[Abs[t$95$3], $MachinePrecision] + t$95$3), $MachinePrecision] / N[(N[(N[(N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$6, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$6), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.4000000000000001e-151

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 37.3%

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

   if 5.4000000000000001e-151 < b < 1.5999999999999999e76

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   8. Applied rewrites 28.0%

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

   10. Applied rewrites 32.4%

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

   if 1.5999999999999999e76 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. mult-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lift-PI.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 44.4%

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

   9. Applied rewrites 44.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.3%

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

   11. Applied rewrites 44.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.5%

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

   13. Applied rewrites 44.5%

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

   15. Applied rewrites 44.7%

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

Alternative 2: 50.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \sin \left(t\_0 + \frac{\pi}{2}\right)\\ t_2 := 0.5 \cdot \pi + \left(-\pi\right) \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_3 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_5 := \sin t\_4\\ t_6 := \sin t\_2\\ \mathbf{if}\;\left|a\right| \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_3\right)\right) + \sqrt{{t\_1}^{4}}}{\left(x-scale \cdot t\_1\right) \cdot \sin t\_3}\right)\right)}{\pi}\\ \mathbf{elif}\;\left|a\right| \leq 1.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_2\right)\right) + \sqrt{{t\_6}^{4}}}{\left(\sin t\_0 \cdot t\_6\right) \cdot x-scale}\right) \cdot -0.5\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_5}^{4}} + {t\_5}^{2}\right)}{x-scale \cdot \left(\cos t\_4 \cdot t\_5\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
       (t_1 (sin (+ t_0 (/ PI 2.0))))
       (t_2 (+ (* 0.5 PI) (* (- PI) (* 0.005555555555555556 angle))))
       (t_3 (* PI (* angle 0.005555555555555556)))
       (t_4 (* 0.005555555555555556 (* angle PI)))
       (t_5 (sin t_4))
       (t_6 (sin t_2)))
  (if (<= (fabs a) 7.8e-212)
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         y-scale
         (/
          (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_3)))) (sqrt (pow t_1 4.0)))
          (* (* x-scale t_1) (sin t_3))))))
      PI))
    (if (<= (fabs a) 1.7e+80)
      (/
       (*
        180.0
        (atan
         (*
          (*
           y-scale
           (/
            (+ (- 0.5 (* 0.5 (cos (* 2.0 t_2)))) (sqrt (pow t_6 4.0)))
            (* (* (sin t_0) t_6) x-scale)))
          -0.5)))
       PI)
      (*
       180.0
       (/
        (atan
         (*
          0.5
          (/
           (* y-scale (+ (sqrt (pow t_5 4.0)) (pow t_5 2.0)))
           (* x-scale (* (cos t_4) t_5)))))
        PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = sin((t_0 + (((double) M_PI) / 2.0)));
	double t_2 = (0.5 * ((double) M_PI)) + (-((double) M_PI) * (0.005555555555555556 * angle));
	double t_3 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_5 = sin(t_4);
	double t_6 = sin(t_2);
	double tmp;
	if (fabs(a) <= 7.8e-212) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_3)))) + sqrt(pow(t_1, 4.0))) / ((x_45_scale * t_1) * sin(t_3)))))) / ((double) M_PI));
	} else if (fabs(a) <= 1.7e+80) {
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_2)))) + sqrt(pow(t_6, 4.0))) / ((sin(t_0) * t_6) * x_45_scale))) * -0.5))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_5, 4.0)) + pow(t_5, 2.0))) / (x_45_scale * (cos(t_4) * t_5))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, 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 + (Math.PI / 2.0)));
	double t_2 = (0.5 * Math.PI) + (-Math.PI * (0.005555555555555556 * angle));
	double t_3 = Math.PI * (angle * 0.005555555555555556);
	double t_4 = 0.005555555555555556 * (angle * Math.PI);
	double t_5 = Math.sin(t_4);
	double t_6 = Math.sin(t_2);
	double tmp;
	if (Math.abs(a) <= 7.8e-212) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * Math.cos((2.0 * t_3)))) + Math.sqrt(Math.pow(t_1, 4.0))) / ((x_45_scale * t_1) * Math.sin(t_3)))))) / Math.PI);
	} else if (Math.abs(a) <= 1.7e+80) {
		tmp = (180.0 * Math.atan(((y_45_scale * (((0.5 - (0.5 * Math.cos((2.0 * t_2)))) + Math.sqrt(Math.pow(t_6, 4.0))) / ((Math.sin(t_0) * t_6) * x_45_scale))) * -0.5))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((y_45_scale * (Math.sqrt(Math.pow(t_5, 4.0)) + Math.pow(t_5, 2.0))) / (x_45_scale * (Math.cos(t_4) * t_5))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (0.005555555555555556 * angle) * math.pi
	t_1 = math.sin((t_0 + (math.pi / 2.0)))
	t_2 = (0.5 * math.pi) + (-math.pi * (0.005555555555555556 * angle))
	t_3 = math.pi * (angle * 0.005555555555555556)
	t_4 = 0.005555555555555556 * (angle * math.pi)
	t_5 = math.sin(t_4)
	t_6 = math.sin(t_2)
	tmp = 0
	if math.fabs(a) <= 7.8e-212:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * math.cos((2.0 * t_3)))) + math.sqrt(math.pow(t_1, 4.0))) / ((x_45_scale * t_1) * math.sin(t_3)))))) / math.pi)
	elif math.fabs(a) <= 1.7e+80:
		tmp = (180.0 * math.atan(((y_45_scale * (((0.5 - (0.5 * math.cos((2.0 * t_2)))) + math.sqrt(math.pow(t_6, 4.0))) / ((math.sin(t_0) * t_6) * x_45_scale))) * -0.5))) / math.pi
	else:
		tmp = 180.0 * (math.atan((0.5 * ((y_45_scale * (math.sqrt(math.pow(t_5, 4.0)) + math.pow(t_5, 2.0))) / (x_45_scale * (math.cos(t_4) * t_5))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = sin(Float64(t_0 + Float64(pi / 2.0)))
	t_2 = Float64(Float64(0.5 * pi) + Float64(Float64(-pi) * Float64(0.005555555555555556 * angle)))
	t_3 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_5 = sin(t_4)
	t_6 = sin(t_2)
	tmp = 0.0
	if (abs(a) <= 7.8e-212)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_3)))) + sqrt((t_1 ^ 4.0))) / Float64(Float64(x_45_scale * t_1) * sin(t_3)))))) / pi));
	elseif (abs(a) <= 1.7e+80)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_2)))) + sqrt((t_6 ^ 4.0))) / Float64(Float64(sin(t_0) * t_6) * x_45_scale))) * -0.5))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_5 ^ 4.0)) + (t_5 ^ 2.0))) / Float64(x_45_scale * Float64(cos(t_4) * t_5))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (0.005555555555555556 * angle) * pi;
	t_1 = sin((t_0 + (pi / 2.0)));
	t_2 = (0.5 * pi) + (-pi * (0.005555555555555556 * angle));
	t_3 = pi * (angle * 0.005555555555555556);
	t_4 = 0.005555555555555556 * (angle * pi);
	t_5 = sin(t_4);
	t_6 = sin(t_2);
	tmp = 0.0;
	if (abs(a) <= 7.8e-212)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_3)))) + sqrt((t_1 ^ 4.0))) / ((x_45_scale * t_1) * sin(t_3)))))) / pi);
	elseif (abs(a) <= 1.7e+80)
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_2)))) + sqrt((t_6 ^ 4.0))) / ((sin(t_0) * t_6) * x_45_scale))) * -0.5))) / pi;
	else
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt((t_5 ^ 4.0)) + (t_5 ^ 2.0))) / (x_45_scale * (cos(t_4) * t_5))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(t$95$0 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * Pi), $MachinePrecision] + N[((-Pi) * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$2], $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7.8e-212], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * t$95$1), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 1.7e+80], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$6, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[t$95$0], $MachinePrecision] * t$95$6), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$5, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$4], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.8e-212

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.5%

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

   11. Applied rewrites 44.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}}}{\left(x-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   if 7.8e-212 < a < 1.7e80

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. mult-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lift-PI.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 44.4%

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

   9. Applied rewrites 44.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.3%

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

   11. Applied rewrites 44.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.5%

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

   13. Applied rewrites 44.5%

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

   15. Applied rewrites 44.7%

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

   if 1.7e80 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 37.3%

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

Alternative 3: 46.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \pi + \left(-\pi\right) \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_2 := \sin t\_0\\ t_3 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_4 := \sin \left(t\_3 + \frac{\pi}{2}\right)\\ t_5 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|a\right| \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) + \sqrt{{t\_4}^{4}}}{\left(x-scale \cdot t\_4\right) \cdot \sin t\_1}\right)\right)}{\pi}\\ \mathbf{elif}\;\left|a\right| \leq 1.55 \cdot 10^{+144}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\left(y-scale \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \sqrt{{t\_2}^{4}}}{\left(\sin t\_3 \cdot t\_2\right) \cdot x-scale}\right) \cdot -0.5\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{\cos t\_5 \cdot \sin t\_5}\right) \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (+ (* 0.5 PI) (* (- PI) (* 0.005555555555555556 angle))))
       (t_1 (* PI (* angle 0.005555555555555556)))
       (t_2 (sin t_0))
       (t_3 (* (* 0.005555555555555556 angle) PI))
       (t_4 (sin (+ t_3 (/ PI 2.0))))
       (t_5 (* 0.005555555555555556 (* angle PI))))
  (if (<= (fabs a) 7.8e-212)
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         y-scale
         (/
          (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_1)))) (sqrt (pow t_4 4.0)))
          (* (* x-scale t_4) (sin t_1))))))
      PI))
    (if (<= (fabs a) 1.55e+144)
      (/
       (*
        180.0
        (atan
         (*
          (*
           y-scale
           (/
            (+ (- 0.5 (* 0.5 (cos (* 2.0 t_0)))) (sqrt (pow t_2 4.0)))
            (* (* (sin t_3) t_2) x-scale)))
          -0.5)))
       PI)
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (*
           (*
            -1.0
            (/
             (-
              0.5
              (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
             (* (cos t_5) (sin t_5))))
           (/ y-scale x-scale))))
        PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * ((double) M_PI)) + (-((double) M_PI) * (0.005555555555555556 * angle));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_2 = sin(t_0);
	double t_3 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_4 = sin((t_3 + (((double) M_PI) / 2.0)));
	double t_5 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(a) <= 7.8e-212) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt(pow(t_4, 4.0))) / ((x_45_scale * t_4) * sin(t_1)))))) / ((double) M_PI));
	} else if (fabs(a) <= 1.55e+144) {
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt(pow(t_2, 4.0))) / ((sin(t_3) * t_2) * x_45_scale))) * -0.5))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) / (cos(t_5) * sin(t_5)))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * Math.PI) + (-Math.PI * (0.005555555555555556 * angle));
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double t_2 = Math.sin(t_0);
	double t_3 = (0.005555555555555556 * angle) * Math.PI;
	double t_4 = Math.sin((t_3 + (Math.PI / 2.0)));
	double t_5 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(a) <= 7.8e-212) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * Math.cos((2.0 * t_1)))) + Math.sqrt(Math.pow(t_4, 4.0))) / ((x_45_scale * t_4) * Math.sin(t_1)))))) / Math.PI);
	} else if (Math.abs(a) <= 1.55e+144) {
		tmp = (180.0 * Math.atan(((y_45_scale * (((0.5 - (0.5 * Math.cos((2.0 * t_0)))) + Math.sqrt(Math.pow(t_2, 4.0))) / ((Math.sin(t_3) * t_2) * x_45_scale))) * -0.5))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) / (Math.cos(t_5) * Math.sin(t_5)))) * (y_45_scale / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (0.5 * math.pi) + (-math.pi * (0.005555555555555556 * angle))
	t_1 = math.pi * (angle * 0.005555555555555556)
	t_2 = math.sin(t_0)
	t_3 = (0.005555555555555556 * angle) * math.pi
	t_4 = math.sin((t_3 + (math.pi / 2.0)))
	t_5 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(a) <= 7.8e-212:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * math.cos((2.0 * t_1)))) + math.sqrt(math.pow(t_4, 4.0))) / ((x_45_scale * t_4) * math.sin(t_1)))))) / math.pi)
	elif math.fabs(a) <= 1.55e+144:
		tmp = (180.0 * math.atan(((y_45_scale * (((0.5 - (0.5 * math.cos((2.0 * t_0)))) + math.sqrt(math.pow(t_2, 4.0))) / ((math.sin(t_3) * t_2) * x_45_scale))) * -0.5))) / math.pi
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) / (math.cos(t_5) * math.sin(t_5)))) * (y_45_scale / x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.5 * pi) + Float64(Float64(-pi) * Float64(0.005555555555555556 * angle)))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_4 = sin(Float64(t_3 + Float64(pi / 2.0)))
	t_5 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(a) <= 7.8e-212)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_1)))) + sqrt((t_4 ^ 4.0))) / Float64(Float64(x_45_scale * t_4) * sin(t_1)))))) / pi));
	elseif (abs(a) <= 1.55e+144)
		tmp = Float64(Float64(180.0 * atan(Float64(Float64(y_45_scale * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))) + sqrt((t_2 ^ 4.0))) / Float64(Float64(sin(t_3) * t_2) * x_45_scale))) * -0.5))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) / Float64(cos(t_5) * sin(t_5)))) * Float64(y_45_scale / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (0.5 * pi) + (-pi * (0.005555555555555556 * angle));
	t_1 = pi * (angle * 0.005555555555555556);
	t_2 = sin(t_0);
	t_3 = (0.005555555555555556 * angle) * pi;
	t_4 = sin((t_3 + (pi / 2.0)));
	t_5 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(a) <= 7.8e-212)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt((t_4 ^ 4.0))) / ((x_45_scale * t_4) * sin(t_1)))))) / pi);
	elseif (abs(a) <= 1.55e+144)
		tmp = (180.0 * atan(((y_45_scale * (((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt((t_2 ^ 4.0))) / ((sin(t_3) * t_2) * x_45_scale))) * -0.5))) / pi;
	else
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) / (cos(t_5) * sin(t_5)))) * (y_45_scale / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] + N[((-Pi) * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(t$95$3 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7.8e-212], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * t$95$4), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 1.55e+144], N[(N[(180.0 * N[ArcTan[N[(N[(y$45$scale * N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[t$95$3], $MachinePrecision] * t$95$2), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(-1.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$5], $MachinePrecision] * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.8e-212

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.5%

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

   11. Applied rewrites 44.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}}}{\left(x-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   if 7.8e-212 < a < 1.5500000000000001e144

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. mult-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lift-PI.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 44.4%

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

   9. Applied rewrites 44.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.3%

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

   11. Applied rewrites 44.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.5%

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

   13. Applied rewrites 44.5%

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

   15. Applied rewrites 44.7%

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

   if 1.5500000000000001e144 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \frac{\color{blue}{y-scale}}{x-scale}\right)\right)}{\pi} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 31.1%

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

Alternative 4: 46.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \pi + \left(-\pi\right) \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_2 := \sin t\_0\\ t_3 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_4 := \sin \left(t\_3 + \frac{\pi}{2}\right)\\ t_5 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|a\right| \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) + \sqrt{{t\_4}^{4}}}{\left(x-scale \cdot t\_4\right) \cdot \sin t\_1}\right)\right)}{\pi}\\ \mathbf{elif}\;\left|a\right| \leq 1.55 \cdot 10^{+144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) + \sqrt{{t\_2}^{4}}\right) \cdot y-scale}{\left(\sin t\_3 \cdot t\_2\right) \cdot x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{\cos t\_5 \cdot \sin t\_5}\right) \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (+ (* 0.5 PI) (* (- PI) (* 0.005555555555555556 angle))))
       (t_1 (* PI (* angle 0.005555555555555556)))
       (t_2 (sin t_0))
       (t_3 (* (* 0.005555555555555556 angle) PI))
       (t_4 (sin (+ t_3 (/ PI 2.0))))
       (t_5 (* 0.005555555555555556 (* angle PI))))
  (if (<= (fabs a) 7.8e-212)
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         y-scale
         (/
          (+ (+ 0.5 (* 0.5 (cos (* 2.0 t_1)))) (sqrt (pow t_4 4.0)))
          (* (* x-scale t_4) (sin t_1))))))
      PI))
    (if (<= (fabs a) 1.55e+144)
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (/
           (*
            (+ (- 0.5 (* 0.5 (cos (* 2.0 t_0)))) (sqrt (pow t_2 4.0)))
            y-scale)
           (* (* (sin t_3) t_2) x-scale))))
        PI))
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (*
           (*
            -1.0
            (/
             (-
              0.5
              (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
             (* (cos t_5) (sin t_5))))
           (/ y-scale x-scale))))
        PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * ((double) M_PI)) + (-((double) M_PI) * (0.005555555555555556 * angle));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_2 = sin(t_0);
	double t_3 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_4 = sin((t_3 + (((double) M_PI) / 2.0)));
	double t_5 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(a) <= 7.8e-212) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt(pow(t_4, 4.0))) / ((x_45_scale * t_4) * sin(t_1)))))) / ((double) M_PI));
	} else if (fabs(a) <= 1.55e+144) {
		tmp = 180.0 * (atan((-0.5 * ((((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt(pow(t_2, 4.0))) * y_45_scale) / ((sin(t_3) * t_2) * x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) / (cos(t_5) * sin(t_5)))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.5 * Math.PI) + (-Math.PI * (0.005555555555555556 * angle));
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double t_2 = Math.sin(t_0);
	double t_3 = (0.005555555555555556 * angle) * Math.PI;
	double t_4 = Math.sin((t_3 + (Math.PI / 2.0)));
	double t_5 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(a) <= 7.8e-212) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * Math.cos((2.0 * t_1)))) + Math.sqrt(Math.pow(t_4, 4.0))) / ((x_45_scale * t_4) * Math.sin(t_1)))))) / Math.PI);
	} else if (Math.abs(a) <= 1.55e+144) {
		tmp = 180.0 * (Math.atan((-0.5 * ((((0.5 - (0.5 * Math.cos((2.0 * t_0)))) + Math.sqrt(Math.pow(t_2, 4.0))) * y_45_scale) / ((Math.sin(t_3) * t_2) * x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) / (Math.cos(t_5) * Math.sin(t_5)))) * (y_45_scale / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (0.5 * math.pi) + (-math.pi * (0.005555555555555556 * angle))
	t_1 = math.pi * (angle * 0.005555555555555556)
	t_2 = math.sin(t_0)
	t_3 = (0.005555555555555556 * angle) * math.pi
	t_4 = math.sin((t_3 + (math.pi / 2.0)))
	t_5 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(a) <= 7.8e-212:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * math.cos((2.0 * t_1)))) + math.sqrt(math.pow(t_4, 4.0))) / ((x_45_scale * t_4) * math.sin(t_1)))))) / math.pi)
	elif math.fabs(a) <= 1.55e+144:
		tmp = 180.0 * (math.atan((-0.5 * ((((0.5 - (0.5 * math.cos((2.0 * t_0)))) + math.sqrt(math.pow(t_2, 4.0))) * y_45_scale) / ((math.sin(t_3) * t_2) * x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) / (math.cos(t_5) * math.sin(t_5)))) * (y_45_scale / x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.5 * pi) + Float64(Float64(-pi) * Float64(0.005555555555555556 * angle)))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_4 = sin(Float64(t_3 + Float64(pi / 2.0)))
	t_5 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(a) <= 7.8e-212)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_1)))) + sqrt((t_4 ^ 4.0))) / Float64(Float64(x_45_scale * t_4) * sin(t_1)))))) / pi));
	elseif (abs(a) <= 1.55e+144)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))) + sqrt((t_2 ^ 4.0))) * y_45_scale) / Float64(Float64(sin(t_3) * t_2) * x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) / Float64(cos(t_5) * sin(t_5)))) * Float64(y_45_scale / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (0.5 * pi) + (-pi * (0.005555555555555556 * angle));
	t_1 = pi * (angle * 0.005555555555555556);
	t_2 = sin(t_0);
	t_3 = (0.005555555555555556 * angle) * pi;
	t_4 = sin((t_3 + (pi / 2.0)));
	t_5 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(a) <= 7.8e-212)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt((t_4 ^ 4.0))) / ((x_45_scale * t_4) * sin(t_1)))))) / pi);
	elseif (abs(a) <= 1.55e+144)
		tmp = 180.0 * (atan((-0.5 * ((((0.5 - (0.5 * cos((2.0 * t_0)))) + sqrt((t_2 ^ 4.0))) * y_45_scale) / ((sin(t_3) * t_2) * x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) / (cos(t_5) * sin(t_5)))) * (y_45_scale / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] + N[((-Pi) * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(t$95$3 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7.8e-212], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * t$95$4), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 1.55e+144], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(N[(N[Sin[t$95$3], $MachinePrecision] * t$95$2), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(-1.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$5], $MachinePrecision] * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.8e-212

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.5%

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

   11. Applied rewrites 44.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}}}{\left(x-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   if 7.8e-212 < a < 1.5500000000000001e144

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. mult-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. mult-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 N/A

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

      22. lift-PI.f64 N/A

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

      23. mult-flip N/A

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

      24. metadata-eval N/A

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

      25. lower-*.f64 44.4%

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

   9. Applied rewrites 44.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.3%

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

   11. Applied rewrites 44.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{4}} + {\sin \left(\left(-\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \pi \cdot 0.5\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. cos-neg-rev N/A

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

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

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

       4. lower-sin.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. mult-flip-rev N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. associate-*l/ N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. mult-flip N/A

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

       20. metadata-eval N/A

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

       21. lower-*.f64 N/A

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

       22. lift-PI.f64 N/A

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

       23. mult-flip N/A

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

       24. metadata-eval N/A

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

       25. lower-*.f64 44.5%

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

   13. Applied rewrites 44.5%

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

   15. Applied rewrites 44.7%

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

   if 1.5500000000000001e144 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \frac{\color{blue}{y-scale}}{x-scale}\right)\right)}{\pi} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 31.1%

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

Alternative 5: 46.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ t_3 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_4 := 0.5 + 0.5 \cdot \cos \left(2 \cdot t\_3\right)\\ t_5 := \sin t\_3\\ \mathbf{if}\;\left|a\right| \leq 2.7 \cdot 10^{-211}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{t\_4 + \sqrt{{t\_0}^{4}}}{\left(x-scale \cdot t\_0\right) \cdot t\_5}\right)\right)}{\pi}\\ \mathbf{elif}\;\left|a\right| \leq 4.5 \cdot 10^{+105}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{t\_4 + \sqrt{{t\_2}^{4}}}{\left(x-scale \cdot t\_2\right) \cdot t\_5}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{t\_2 \cdot \sin t\_1}\right) \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0
        (sin (+ (* (* 0.005555555555555556 angle) PI) (/ PI 2.0))))
       (t_1 (* 0.005555555555555556 (* angle PI)))
       (t_2 (cos t_1))
       (t_3 (* PI (* angle 0.005555555555555556)))
       (t_4 (+ 0.5 (* 0.5 (cos (* 2.0 t_3)))))
       (t_5 (sin t_3)))
  (if (<= (fabs a) 2.7e-211)
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         y-scale
         (/ (+ t_4 (sqrt (pow t_0 4.0))) (* (* x-scale t_0) t_5)))))
      PI))
    (if (<= (fabs a) 4.5e+105)
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (*
           y-scale
           (/ (+ t_4 (sqrt (pow t_2 4.0))) (* (* x-scale t_2) t_5)))))
        PI))
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (*
           (*
            -1.0
            (/
             (-
              0.5
              (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
             (* t_2 (sin t_1))))
           (/ y-scale x-scale))))
        PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((((0.005555555555555556 * angle) * ((double) M_PI)) + (((double) M_PI) / 2.0)));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_4 = 0.5 + (0.5 * cos((2.0 * t_3)));
	double t_5 = sin(t_3);
	double tmp;
	if (fabs(a) <= 2.7e-211) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * ((t_4 + sqrt(pow(t_0, 4.0))) / ((x_45_scale * t_0) * t_5))))) / ((double) M_PI));
	} else if (fabs(a) <= 4.5e+105) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * ((t_4 + sqrt(pow(t_2, 4.0))) / ((x_45_scale * t_2) * t_5))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) / (t_2 * sin(t_1)))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sin((((0.005555555555555556 * angle) * Math.PI) + (Math.PI / 2.0)));
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.cos(t_1);
	double t_3 = Math.PI * (angle * 0.005555555555555556);
	double t_4 = 0.5 + (0.5 * Math.cos((2.0 * t_3)));
	double t_5 = Math.sin(t_3);
	double tmp;
	if (Math.abs(a) <= 2.7e-211) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * ((t_4 + Math.sqrt(Math.pow(t_0, 4.0))) / ((x_45_scale * t_0) * t_5))))) / Math.PI);
	} else if (Math.abs(a) <= 4.5e+105) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * ((t_4 + Math.sqrt(Math.pow(t_2, 4.0))) / ((x_45_scale * t_2) * t_5))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) / (t_2 * Math.sin(t_1)))) * (y_45_scale / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sin((((0.005555555555555556 * angle) * math.pi) + (math.pi / 2.0)))
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.cos(t_1)
	t_3 = math.pi * (angle * 0.005555555555555556)
	t_4 = 0.5 + (0.5 * math.cos((2.0 * t_3)))
	t_5 = math.sin(t_3)
	tmp = 0
	if math.fabs(a) <= 2.7e-211:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * ((t_4 + math.sqrt(math.pow(t_0, 4.0))) / ((x_45_scale * t_0) * t_5))))) / math.pi)
	elif math.fabs(a) <= 4.5e+105:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * ((t_4 + math.sqrt(math.pow(t_2, 4.0))) / ((x_45_scale * t_2) * t_5))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) / (t_2 * math.sin(t_1)))) * (y_45_scale / x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(Float64(Float64(0.005555555555555556 * angle) * pi) + Float64(pi / 2.0)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_4 = Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_3))))
	t_5 = sin(t_3)
	tmp = 0.0
	if (abs(a) <= 2.7e-211)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(t_4 + sqrt((t_0 ^ 4.0))) / Float64(Float64(x_45_scale * t_0) * t_5))))) / pi));
	elseif (abs(a) <= 4.5e+105)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(t_4 + sqrt((t_2 ^ 4.0))) / Float64(Float64(x_45_scale * t_2) * t_5))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) / Float64(t_2 * sin(t_1)))) * Float64(y_45_scale / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin((((0.005555555555555556 * angle) * pi) + (pi / 2.0)));
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = cos(t_1);
	t_3 = pi * (angle * 0.005555555555555556);
	t_4 = 0.5 + (0.5 * cos((2.0 * t_3)));
	t_5 = sin(t_3);
	tmp = 0.0;
	if (abs(a) <= 2.7e-211)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * ((t_4 + sqrt((t_0 ^ 4.0))) / ((x_45_scale * t_0) * t_5))))) / pi);
	elseif (abs(a) <= 4.5e+105)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * ((t_4 + sqrt((t_2 ^ 4.0))) / ((x_45_scale * t_2) * t_5))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) / (t_2 * sin(t_1)))) * (y_45_scale / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.7e-211], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(t$95$4 + N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * t$95$0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 4.5e+105], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(t$95$4 + N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * t$95$2), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(-1.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.6999999999999999e-211

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 44.5%

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

   11. Applied rewrites 44.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi + \frac{\pi}{2}\right)}^{4}}}{\left(x-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 43.8%

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

   13. Applied rewrites 43.8%

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

   if 2.6999999999999999e-211 < a < 4.5000000000000001e105

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

   10. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{\left(x-scale \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-cos.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 44.4%

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

   12. Applied rewrites 44.4%

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

   13. Taylor expanded in angle around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{\left(x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   14. Step-by-step derivation

       1. lower-cos.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 44.6%

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

   15. Applied rewrites 44.6%

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

   if 4.5000000000000001e105 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \frac{\color{blue}{y-scale}}{x-scale}\right)\right)}{\pi} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 31.1%

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

Alternative 6: 46.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\left|a\right| \leq 4.5 \cdot 10^{+105}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) + \sqrt{{\cos t\_1}^{4}}}{x-scale \cdot \sin t\_1}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{\cos t\_0 \cdot \sin t\_0}\right) \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
       (t_1 (* PI (* angle 0.005555555555555556))))
  (if (<= (fabs a) 4.5e+105)
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         y-scale
         (/
          (+
           (+ 0.5 (* 0.5 (cos (* 2.0 t_1))))
           (sqrt (pow (cos t_1) 4.0)))
          (* x-scale (sin t_1))))))
      PI))
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         (*
          -1.0
          (/
           (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
           (* (cos t_0) (sin t_0))))
         (/ y-scale x-scale))))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (fabs(a) <= 4.5e+105) {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt(pow(cos(t_1), 4.0))) / (x_45_scale * sin(t_1)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) / (cos(t_0) * sin(t_0)))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (Math.abs(a) <= 4.5e+105) {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * Math.cos((2.0 * t_1)))) + Math.sqrt(Math.pow(Math.cos(t_1), 4.0))) / (x_45_scale * Math.sin(t_1)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) / (Math.cos(t_0) * Math.sin(t_0)))) * (y_45_scale / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if math.fabs(a) <= 4.5e+105:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * math.cos((2.0 * t_1)))) + math.sqrt(math.pow(math.cos(t_1), 4.0))) / (x_45_scale * math.sin(t_1)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) / (math.cos(t_0) * math.sin(t_0)))) * (y_45_scale / x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (abs(a) <= 4.5e+105)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_1)))) + sqrt((cos(t_1) ^ 4.0))) / Float64(x_45_scale * sin(t_1)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) / Float64(cos(t_0) * sin(t_0)))) * Float64(y_45_scale / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (abs(a) <= 4.5e+105)
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (((0.5 + (0.5 * cos((2.0 * t_1)))) + sqrt((cos(t_1) ^ 4.0))) / (x_45_scale * sin(t_1)))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) / (cos(t_0) * sin(t_0)))) * (y_45_scale / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 4.5e+105], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[t$95$1], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(-1.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.5000000000000001e105

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) + \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}}{x-scale \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}{\pi} \]
   11. Step-by-step derivation

   12. Applied rewrites 43.5%

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

   if 4.5000000000000001e105 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \frac{\color{blue}{y-scale}}{x-scale}\right)\right)}{\pi} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 31.1%

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

Alternative 7: 46.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0 \cdot \sin t\_0\\ \mathbf{if}\;\left|a\right| \leq 1.6 \cdot 10^{+144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{t\_1}\right) \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
       (t_1 (* (cos t_0) (sin t_0))))
  (if (<= (fabs a) 1.6e+144)
    (*
     180.0
     (/ (atan (* -0.5 (/ (* y-scale 2.0) (* x-scale t_1)))) PI))
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (*
         (*
          -1.0
          (/
           (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
           t_1))
         (/ y-scale x-scale))))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0) * sin(t_0);
	double tmp;
	if (fabs(a) <= 1.6e+144) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * t_1)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) / t_1)) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0) * Math.sin(t_0);
	double tmp;
	if (Math.abs(a) <= 1.6e+144) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * t_1)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) / t_1)) * (y_45_scale / x_45_scale)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0) * math.sin(t_0)
	tmp = 0
	if math.fabs(a) <= 1.6e+144:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * t_1)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) / t_1)) * (y_45_scale / x_45_scale)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(cos(t_0) * sin(t_0))
	tmp = 0.0
	if (abs(a) <= 1.6e+144)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * t_1)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) / t_1)) * Float64(y_45_scale / x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0) * sin(t_0);
	tmp = 0.0;
	if (abs(a) <= 1.6e+144)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * t_1)))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((-1.0 * ((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) / t_1)) * (y_45_scale / x_45_scale)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.6e+144], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(-1.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6e144

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.0%

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

   if 1.6e144 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \frac{\color{blue}{y-scale}}{x-scale}\right)\right)}{\pi} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 31.1%

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

Alternative 8: 46.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;\left|b\right| \leq 3.2 \cdot 10^{-83}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 8 \cdot 10^{+57}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{1}^{4}} + {1}^{2}\right)}{x-scale \cdot \left(1 \cdot t\_1\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot t\_1\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
  (if (<= (fabs b) 3.2e-83)
    (*
     180.0
     (/
      (atan
       (*
        -90.0
        (/
         (*
          x-scale
          (*
           y-scale
           (+
            (sqrt (/ 1.0 (pow x-scale 4.0)))
            (/ 1.0 (pow x-scale 2.0)))))
         (* angle PI))))
      PI))
    (if (<= (fabs b) 8e+57)
      (*
       180.0
       (/
        (atan
         (*
          -0.5
          (/
           (* y-scale (+ (sqrt (pow 1.0 4.0)) (pow 1.0 2.0)))
           (* x-scale (* 1.0 t_1)))))
        PI))
      (*
       180.0
       (/
        (atan
         (* -0.5 (/ (* y-scale 2.0) (* x-scale (* (cos t_0) t_1)))))
        PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (fabs(b) <= 3.2e-83) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else if (fabs(b) <= 8e+57) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(1.0, 4.0)) + pow(1.0, 2.0))) / (x_45_scale * (1.0 * t_1))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * t_1))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (Math.abs(b) <= 3.2e-83) {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	} else if (Math.abs(b) <= 8e+57) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (Math.sqrt(Math.pow(1.0, 4.0)) + Math.pow(1.0, 2.0))) / (x_45_scale * (1.0 * t_1))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * t_1))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	tmp = 0
	if math.fabs(b) <= 3.2e-83:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	elif math.fabs(b) <= 8e+57:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (math.sqrt(math.pow(1.0, 4.0)) + math.pow(1.0, 2.0))) / (x_45_scale * (1.0 * t_1))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * t_1))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (abs(b) <= 3.2e-83)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	elseif (abs(b) <= 8e+57)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((1.0 ^ 4.0)) + (1.0 ^ 2.0))) / Float64(x_45_scale * Float64(1.0 * t_1))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * Float64(cos(t_0) * t_1))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (abs(b) <= 3.2e-83)
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	elseif (abs(b) <= 8e+57)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt((1.0 ^ 4.0)) + (1.0 ^ 2.0))) / (x_45_scale * (1.0 * t_1))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * t_1))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 3.2e-83], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 8e+57], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[1.0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[1.0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.2000000000000001e-83

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   if 3.2000000000000001e-83 < b < 8.0000000000000004e57

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{1}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

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

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{1}^{4}} + {1}^{2}\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.0%

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

   14. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{1}^{4}} + {1}^{2}\right)}{x-scale \cdot \left(1 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\pi} \]
   15. Step-by-step derivation

   16. Applied rewrites 43.9%

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

   if 8.0000000000000004e57 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.0%

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

Alternative 9: 45.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\\ \mathbf{if}\;\left|a\right| \leq 1.6 \cdot 10^{+144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{t\_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{t\_1}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
       (t_1 (* x-scale (* (cos t_0) (sin t_0)))))
  (if (<= (fabs a) 1.6e+144)
    (* 180.0 (/ (atan (* -0.5 (/ (* y-scale 2.0) t_1))) PI))
    (*
     180.0
     (/
      (atan
       (*
        0.5
        (/
         (*
          y-scale
          (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
         t_1)))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	double tmp;
	if (fabs(a) <= 1.6e+144) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))))) / t_1))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = x_45_scale * (Math.cos(t_0) * Math.sin(t_0));
	double tmp;
	if (Math.abs(a) <= 1.6e+144) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((y_45_scale * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))))) / t_1))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = x_45_scale * (math.cos(t_0) * math.sin(t_0))
	tmp = 0
	if math.fabs(a) <= 1.6e+144:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((y_45_scale * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))) / t_1))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))
	tmp = 0.0
	if (abs(a) <= 1.6e+144)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / t_1))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))) / t_1))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = x_45_scale * (cos(t_0) * sin(t_0));
	tmp = 0.0;
	if (abs(a) <= 1.6e+144)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / t_1))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi)))))) / t_1))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.6e+144], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6e144

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.0%

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

   if 1.6e144 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 30.5%

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

Alternative 10: 45.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 3.9 \cdot 10^{-87}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
  (if (<= (fabs b) 3.9e-87)
    (*
     180.0
     (/
      (atan
       (*
        -90.0
        (/
         (*
          x-scale
          (*
           y-scale
           (+
            (sqrt (/ 1.0 (pow x-scale 4.0)))
            (/ 1.0 (pow x-scale 2.0)))))
         (* angle PI))))
      PI))
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (/ (* y-scale 2.0) (* x-scale (* (cos t_0) (sin t_0))))))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 3.9e-87) {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / pow(x_45_scale, 4.0))) + (1.0 / pow(x_45_scale, 2.0))))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(b) <= 3.9e-87) {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (y_45_scale * (Math.sqrt((1.0 / Math.pow(x_45_scale, 4.0))) + (1.0 / Math.pow(x_45_scale, 2.0))))) / (angle * Math.PI)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(b) <= 3.9e-87:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (y_45_scale * (math.sqrt((1.0 / math.pow(x_45_scale, 4.0))) + (1.0 / math.pow(x_45_scale, 2.0))))) / (angle * math.pi)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * math.sin(t_0)))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 3.9e-87)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(1.0 / (x_45_scale ^ 4.0))) + Float64(1.0 / (x_45_scale ^ 2.0))))) / Float64(angle * pi)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(b) <= 3.9e-87)
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (y_45_scale * (sqrt((1.0 / (x_45_scale ^ 4.0))) + (1.0 / (x_45_scale ^ 2.0))))) / (angle * pi)))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 3.9e-87], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(1.0 / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8999999999999998e-87

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   if 3.8999999999999998e-87 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.0%

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

Alternative 11: 44.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 1.55 \cdot 10^{-81}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(x-scale \cdot \left(-1 \cdot \frac{y-scale}{angle \cdot \left({x-scale}^{2} \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot 2}{x-scale \cdot \left(\cos t\_0 \cdot \sin t\_0\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
  (if (<= (fabs b) 1.55e-81)
    (*
     180.0
     (/
      (atan
       (*
        90.0
        (*
         x-scale
         (* -1.0 (/ y-scale (* angle (* (pow x-scale 2.0) PI)))))))
      PI))
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (/ (* y-scale 2.0) (* x-scale (* (cos t_0) (sin t_0))))))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 1.55e-81) {
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (pow(x_45_scale, 2.0) * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(b) <= 1.55e-81) {
		tmp = 180.0 * (Math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (Math.pow(x_45_scale, 2.0) * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (Math.cos(t_0) * Math.sin(t_0)))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(b) <= 1.55e-81:
		tmp = 180.0 * (math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (math.pow(x_45_scale, 2.0) * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (math.cos(t_0) * math.sin(t_0)))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 1.55e-81)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(x_45_scale * Float64(-1.0 * Float64(y_45_scale / Float64(angle * Float64((x_45_scale ^ 2.0) * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * 2.0) / Float64(x_45_scale * Float64(cos(t_0) * sin(t_0)))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(b) <= 1.55e-81)
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * ((x_45_scale ^ 2.0) * pi))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * 2.0) / (x_45_scale * (cos(t_0) * sin(t_0)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e-81], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(x$45$scale * N[(-1.0 * N[(y$45$scale / N[(angle * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * 2.0), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5499999999999999e-81

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 12.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-PI.f64 38.0%

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

   9. Applied rewrites 38.0%

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

   if 1.5499999999999999e-81 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 44.0%

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

Alternative 12: 44.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\left|b\right| \leq 1.55 \cdot 10^{-81}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(x-scale \cdot \left(-1 \cdot \frac{y-scale}{angle \cdot \left({x-scale}^{2} \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{2}{\left(x-scale \cdot \cos t\_0\right) \cdot \sin t\_0}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* PI (* angle 0.005555555555555556))))
  (if (<= (fabs b) 1.55e-81)
    (*
     180.0
     (/
      (atan
       (*
        90.0
        (*
         x-scale
         (* -1.0 (/ y-scale (* angle (* (pow x-scale 2.0) PI)))))))
      PI))
    (*
     180.0
     (/
      (atan
       (*
        -0.5
        (* y-scale (/ 2.0 (* (* x-scale (cos t_0)) (sin t_0))))))
      PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (fabs(b) <= 1.55e-81) {
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (pow(x_45_scale, 2.0) * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 / ((x_45_scale * cos(t_0)) * sin(t_0)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (Math.abs(b) <= 1.55e-81) {
		tmp = 180.0 * (Math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (Math.pow(x_45_scale, 2.0) * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (2.0 / ((x_45_scale * Math.cos(t_0)) * Math.sin(t_0)))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if math.fabs(b) <= 1.55e-81:
		tmp = 180.0 * (math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (math.pow(x_45_scale, 2.0) * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (2.0 / ((x_45_scale * math.cos(t_0)) * math.sin(t_0)))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (abs(b) <= 1.55e-81)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(x_45_scale * Float64(-1.0 * Float64(y_45_scale / Float64(angle * Float64((x_45_scale ^ 2.0) * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(2.0 / Float64(Float64(x_45_scale * cos(t_0)) * sin(t_0)))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (abs(b) <= 1.55e-81)
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * ((x_45_scale ^ 2.0) * pi))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (2.0 / ((x_45_scale * cos(t_0)) * sin(t_0)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e-81], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(x$45$scale * N[(-1.0 * N[(y$45$scale / N[(angle * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(2.0 / N[(N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5499999999999999e-81

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 12.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-PI.f64 38.0%

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

   9. Applied rewrites 38.0%

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

   if 1.5499999999999999e-81 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 44.1%

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

Alternative 13: 40.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \left(x-scale \cdot \left(-1 \cdot \frac{y-scale}{angle \cdot \left({x-scale}^{2} \cdot \pi\right)}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (if (<= (fabs b) 4.5e-56)
  (*
   180.0
   (/
    (atan
     (*
      90.0
      (*
       x-scale
       (* -1.0 (/ y-scale (* angle (* (pow x-scale 2.0) PI)))))))
    PI))
  (*
   180.0
   (/
    (atan (* -0.5 (* y-scale (/ 360.0 (* angle (* x-scale PI))))))
    PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 4.5e-56) {
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (pow(x_45_scale, 2.0) * ((double) M_PI)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 4.5e-56) {
		tmp = 180.0 * (Math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (Math.pow(x_45_scale, 2.0) * Math.PI))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 4.5e-56:
		tmp = 180.0 * (math.atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * (math.pow(x_45_scale, 2.0) * math.pi))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * math.pi)))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 4.5e-56)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(x_45_scale * Float64(-1.0 * Float64(y_45_scale / Float64(angle * Float64((x_45_scale ^ 2.0) * pi))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(360.0 / Float64(angle * Float64(x_45_scale * pi)))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 4.5e-56)
		tmp = 180.0 * (atan((90.0 * (x_45_scale * (-1.0 * (y_45_scale / (angle * ((x_45_scale ^ 2.0) * pi))))))) / pi);
	else
		tmp = 180.0 * (atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * pi)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 4.5e-56], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(x$45$scale * N[(-1.0 * N[(y$45$scale / N[(angle * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(360.0 / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000001e-56

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 12.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-PI.f64 38.0%

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

   9. Applied rewrites 38.0%

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

   if 4.5000000000000001e-56 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.4%

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

   9. Applied rewrites 44.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

       2. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

       4. lower-PI.f64 38.1%

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

   12. Applied rewrites 38.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.1% accurate, 21.5× speedup

Math

\[180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 180.0
 (/
  (atan (* -0.5 (* y-scale (/ 360.0 (* angle (* x-scale PI))))))
  PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * math.pi)))))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(y_45_scale * Float64(360.0 / Float64(angle * Float64(x_45_scale * pi)))))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-0.5 * (y_45_scale * (360.0 / (angle * (x_45_scale * pi)))))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(y$45$scale * N[(360.0 / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.7%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 24.7%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 44.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 44.4%

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

9. Applied rewrites 44.5%

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

10. Taylor expanded in angle around 0

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

    2. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

    3. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

    4. lower-PI.f64 38.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

12. Applied rewrites 38.1%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(y-scale \cdot \frac{360}{angle \cdot \color{blue}{\left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
13. Add Preprocessing

Alternative 15: 38.1% accurate, 21.5× speedup

Math

\[180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 180.0
 (/
  (atan (* -0.5 (* 360.0 (/ y-scale (* angle (* x-scale PI))))))
  PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(360.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-0.5 * (360.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(360.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.7%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 24.7%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 44.4%

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

8. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\pi} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\pi} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-1}{2} \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}{\pi} \]

   5. lower-PI.f64 38.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)\right)}{\pi} \]

10. Applied rewrites 38.1%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(360 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \pi\right)}}\right)\right)}{\pi} \]
11. Add Preprocessing

Alternative 16: 34.5% accurate, 22.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (* 180.0 (/ (atan (* -90.0 (/ y-scale (* angle (* x-scale PI))))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.7%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 12.7%

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

7. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 12.2%

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

9. Applied rewrites 12.2%

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

10. Taylor expanded in b around inf

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 34.5%

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

12. Applied rewrites 34.5%

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

Alternative 17: 12.2% accurate, 22.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (* 180.0 (/ (atan (* -90.0 (/ x-scale (* angle (* y-scale PI))))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-90.0 * (x_45_scale / (angle * (y_45_scale * ((double) M_PI)))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * Math.PI))))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * math.pi))))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(x_45_scale / Float64(angle * Float64(y_45_scale * pi))))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-90.0 * (x_45_scale / (angle * (y_45_scale * pi))))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(x$45$scale / N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.7%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 12.7%

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

7. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 12.2%

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

9. Applied rewrites 12.2%

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

Reproduce

herbie shell --seed 2025258 
(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)))