raw-angle from scale-rotated-ellipse

Percentage Accurate: 14.0% → 50.6%

Time: 34.8s

Alternatives: 18

Speedup: 49.1×

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: 14.0% 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: 50.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sin t\_1\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_4 := -t\_3\\ t_5 := \cos t\_1\\ t_6 := \cos \left(t\_0 \cdot 2\right)\\ t_7 := \mathsf{fma}\left(\left(0.5 - t\_6 \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(t\_6, 0.5, 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ \mathbf{if}\;\left|b\right| \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_5 \cdot t\_2\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 9000000000:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(y-scale \cdot \frac{\left|t\_7\right| + t\_7}{\left(x-scale \cdot \cos t\_0\right) \cdot \left(\left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right) \cdot \sin t\_0\right)}\right) \cdot -0.5\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_5}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\right)}{x-scale}}{\frac{\sin \left(t\_3 - t\_4\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_4\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (sin t_1))
        (t_3 (* (* PI angle) 0.005555555555555556))
        (t_4 (- t_3))
        (t_5 (cos t_1))
        (t_6 (cos (* t_0 2.0)))
        (t_7
         (fma
          (* (- 0.5 (* t_6 0.5)) a)
          a
          (* (* (fma t_6 0.5 0.5) (fabs b)) (fabs b)))))
   (if (<= (fabs b) 2.3e-151)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
          (* x-scale (* t_5 t_2)))))
       PI))
     (if (<= (fabs b) 9000000000.0)
       (*
        (/
         (atan
          (*
           (*
            y-scale
            (/
             (+ (fabs t_7) t_7)
             (*
              (* x-scale (cos t_0))
              (* (* (- (fabs b) a) (+ (fabs b) a)) (sin t_0)))))
           -0.5))
         PI)
        180.0)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (*
               y-scale
               (+
                (sqrt (pow t_5 4.0))
                (-
                 0.5
                 (*
                  0.5
                  (cos
                   (*
                    2.0
                    (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))))))
              x-scale))
            (/
             (+
              (sin (- t_3 t_4))
              (sin (fma (* 0.005555555555555556 angle) PI t_4)))
             2.0))))
         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 t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = sin(t_1);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_4 = -t_3;
	double t_5 = cos(t_1);
	double t_6 = cos((t_0 * 2.0));
	double t_7 = fma(((0.5 - (t_6 * 0.5)) * a), a, ((fma(t_6, 0.5, 0.5) * fabs(b)) * fabs(b)));
	double tmp;
	if (fabs(b) <= 2.3e-151) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / (x_45_scale * (t_5 * t_2))))) / ((double) M_PI));
	} else if (fabs(b) <= 9000000000.0) {
		tmp = (atan(((y_45_scale * ((fabs(t_7) + t_7) / ((x_45_scale * cos(t_0)) * (((fabs(b) - a) * (fabs(b) + a)) * sin(t_0))))) * -0.5)) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_5, 4.0)) + (0.5 - (0.5 * cos((2.0 * fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)))))))) / x_45_scale)) / ((sin((t_3 - t_4)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_4))) / 2.0)))) / ((double) M_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 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_4 = Float64(-t_3)
	t_5 = cos(t_1)
	t_6 = cos(Float64(t_0 * 2.0))
	t_7 = fma(Float64(Float64(0.5 - Float64(t_6 * 0.5)) * a), a, Float64(Float64(fma(t_6, 0.5, 0.5) * abs(b)) * abs(b)))
	tmp = 0.0
	if (abs(b) <= 2.3e-151)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / Float64(x_45_scale * Float64(t_5 * t_2))))) / pi));
	elseif (abs(b) <= 9000000000.0)
		tmp = Float64(Float64(atan(Float64(Float64(y_45_scale * Float64(Float64(abs(t_7) + t_7) / Float64(Float64(x_45_scale * cos(t_0)) * Float64(Float64(Float64(abs(b) - a) * Float64(abs(b) + a)) * sin(t_0))))) * -0.5)) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_5 ^ 4.0)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))))))) / x_45_scale)) / Float64(Float64(sin(Float64(t_3 - t_4)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_4))) / 2.0)))) / pi));
	end
	return 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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(0.5 - N[(t$95$6 * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(t$95$6 * 0.5 + 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.3e-151], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 9000000000.0], N[(N[(N[ArcTan[N[(N[(y$45$scale * N[(N[(N[Abs[t$95$7], $MachinePrecision] + t$95$7), $MachinePrecision] / N[(N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$5, 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$3 - t$95$4), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.3e-151

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 2.3e-151 < b < 9e9

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   6. Applied rewrites 28.4%

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

   if 9e9 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 45.5%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 45.5%

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

       7. lower-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-cos.f64 N/A

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

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

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

       11. lift-cos.f64 N/A

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

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

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

       13. sqr-sin-a N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-cos.f64 N/A

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

   11. Applied rewrites 45.5%

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

Alternative 2: 50.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := -t\_1\\ t_3 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_4 := \sin t\_3\\ t_5 := \cos t\_3\\ t_6 := \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_4}^{4}} + {t\_4}^{2}\right)}{x-scale \cdot \left(t\_5 \cdot t\_4\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.05 \cdot 10^{+84}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{x-scale \cdot \left(t\_5 \cdot \left(t\_4 \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_5}^{4}} + \frac{\sin \left(t\_6 - t\_1\right) + \sin \left(t\_6 + t\_1\right)}{2}\right)}{x-scale}}{\frac{\sin \left(t\_1 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (- t_1))
        (t_3 (* 0.005555555555555556 (* angle PI)))
        (t_4 (sin t_3))
        (t_5 (cos t_3))
        (t_6 (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))
   (if (<= (fabs b) 4.8e-119)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_4 4.0)) (pow t_4 2.0)))
          (* x-scale (* t_5 t_4)))))
       PI))
     (if (<= (fabs b) 1.05e+84)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* x-scale (* t_5 (* t_4 (- t_0 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (*
               y-scale
               (+
                (sqrt (pow t_5 4.0))
                (/ (+ (sin (- t_6 t_1)) (sin (+ t_6 t_1))) 2.0)))
              x-scale))
            (/
             (+
              (sin (- t_1 t_2))
              (sin (fma (* 0.005555555555555556 angle) PI t_2)))
             2.0))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = -t_1;
	double t_3 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_4 = sin(t_3);
	double t_5 = cos(t_3);
	double t_6 = fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0));
	double tmp;
	if (fabs(b) <= 4.8e-119) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_4, 4.0)) + pow(t_4, 2.0))) / (x_45_scale * (t_5 * t_4))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.05e+84) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (x_45_scale * (t_5 * (t_4 * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_5, 4.0)) + ((sin((t_6 - t_1)) + sin((t_6 + t_1))) / 2.0))) / x_45_scale)) / ((sin((t_1 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(-t_1)
	t_3 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_4 = sin(t_3)
	t_5 = cos(t_3)
	t_6 = fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0))
	tmp = 0.0
	if (abs(b) <= 4.8e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_4 ^ 4.0)) + (t_4 ^ 2.0))) / Float64(x_45_scale * Float64(t_5 * t_4))))) / pi));
	elseif (abs(b) <= 1.05e+84)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(x_45_scale * Float64(t_5 * Float64(t_4 * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_5 ^ 4.0)) + Float64(Float64(sin(Float64(t_6 - t_1)) + sin(Float64(t_6 + t_1))) / 2.0))) / x_45_scale)) / Float64(Float64(sin(Float64(t_1 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-119], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.05e+84], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$5 * N[(t$95$4 * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$5, 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sin[N[(t$95$6 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(t$95$6 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$1 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.8000000000000002e-119

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 4.8000000000000002e-119 < b < 1.0500000000000001e84

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.0500000000000001e84 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 45.5%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 45.5%

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

       7. lower-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-cos.f64 N/A

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

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

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

       11. lift-cos.f64 N/A

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

       12. sin-cos-mult N/A

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

       13. lower-/.f64 N/A

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

   11. Applied rewrites 45.3%

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

Alternative 3: 50.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := \cos t\_2\\ t_4 := \sin t\_2\\ t_5 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_6 := -t\_5\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_4}^{4}} + {t\_4}^{2}\right)}{x-scale \cdot \left(t\_3 \cdot t\_4\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.75 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{x-scale \cdot \left(t\_3 \cdot \left(t\_4 \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + \frac{\cos \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_6\right)\right) + \cos \left(t\_5 - t\_6\right)}{2}\right)}{x-scale}}{t\_1 \cdot t\_4}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0))
        (t_1 (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))
        (t_2 (* 0.005555555555555556 (* angle PI)))
        (t_3 (cos t_2))
        (t_4 (sin t_2))
        (t_5 (* (* PI angle) 0.005555555555555556))
        (t_6 (- t_5)))
   (if (<= (fabs b) 4.8e-119)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_4 4.0)) (pow t_4 2.0)))
          (* x-scale (* t_3 t_4)))))
       PI))
     (if (<= (fabs b) 1.75e+99)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* x-scale (* t_3 (* t_4 (- t_0 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (*
               y-scale
               (+
                (sqrt (pow t_1 4.0))
                (/
                 (+
                  (cos (fma (* 0.005555555555555556 angle) PI t_6))
                  (cos (- t_5 t_6)))
                 2.0)))
              x-scale))
            (* t_1 t_4))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)));
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = cos(t_2);
	double t_4 = sin(t_2);
	double t_5 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_6 = -t_5;
	double tmp;
	if (fabs(b) <= 4.8e-119) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_4, 4.0)) + pow(t_4, 2.0))) / (x_45_scale * (t_3 * t_4))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.75e+99) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (x_45_scale * (t_3 * (t_4 * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + ((cos(fma((0.005555555555555556 * angle), ((double) M_PI), t_6)) + cos((t_5 - t_6))) / 2.0))) / x_45_scale)) / (t_1 * t_4)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = cos(t_2)
	t_4 = sin(t_2)
	t_5 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_6 = Float64(-t_5)
	tmp = 0.0
	if (abs(b) <= 4.8e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_4 ^ 4.0)) + (t_4 ^ 2.0))) / Float64(x_45_scale * Float64(t_3 * t_4))))) / pi));
	elseif (abs(b) <= 1.75e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(x_45_scale * Float64(t_3 * Float64(t_4 * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + Float64(Float64(cos(fma(Float64(0.005555555555555556 * angle), pi, t_6)) + cos(Float64(t_5 - t_6))) / 2.0))) / x_45_scale)) / Float64(t_1 * t_4)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$6 = (-t$95$5)}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-119], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$4, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.75e+99], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$3 * N[(t$95$4 * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$6), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(t$95$5 - t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.8000000000000002e-119

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 4.8000000000000002e-119 < b < 1.7499999999999999e99

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.7499999999999999e99 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 46.0%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. lift-PI.f64 N/A

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

       18. lower-/.f64 45.8%

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

   11. Applied rewrites 45.8%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. lift-PI.f64 N/A

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

       18. lower-/.f64 45.2%

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

   13. Applied rewrites 45.2%

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

Alternative 4: 50.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_4 := {\left(\left|b\right|\right)}^{2}\\ t_5 := -t\_3\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_1 \cdot t\_2\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_4\right)}{x-scale \cdot \left(t\_1 \cdot \left(t\_2 \cdot \left(t\_4 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\right)}{x-scale}}{\frac{\sin \left(t\_3 - t\_5\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_5\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (* (* PI angle) 0.005555555555555556))
        (t_4 (pow (fabs b) 2.0))
        (t_5 (- t_3)))
   (if (<= (fabs b) 4.8e-119)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
          (* x-scale (* t_1 t_2)))))
       PI))
     (if (<= (fabs b) 1.85e+67)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_4))
            (* x-scale (* t_1 (* t_2 (- t_4 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (*
             -1.0
             (/
              (*
               y-scale
               (+
                (sqrt (pow t_1 4.0))
                (-
                 0.5
                 (*
                  0.5
                  (cos
                   (*
                    2.0
                    (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))))))))
              x-scale))
            (/
             (+
              (sin (- t_3 t_5))
              (sin (fma (* 0.005555555555555556 angle) PI t_5)))
             2.0))))
         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 t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_4 = pow(fabs(b), 2.0);
	double t_5 = -t_3;
	double tmp;
	if (fabs(b) <= 4.8e-119) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / (x_45_scale * (t_1 * t_2))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.85e+67) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_4)) / (x_45_scale * (t_1 * (t_2 * (t_4 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + (0.5 - (0.5 * cos((2.0 * fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0)))))))) / x_45_scale)) / ((sin((t_3 - t_5)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_5))) / 2.0)))) / ((double) M_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 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_4 = abs(b) ^ 2.0
	t_5 = Float64(-t_3)
	tmp = 0.0
	if (abs(b) <= 4.8e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / Float64(x_45_scale * Float64(t_1 * t_2))))) / pi));
	elseif (abs(b) <= 1.85e+67)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_4)) / Float64(x_45_scale * Float64(t_1 * Float64(t_2 * Float64(t_4 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0)))))))) / x_45_scale)) / Float64(Float64(sin(Float64(t_3 - t_5)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_5))) / 2.0)))) / pi));
	end
	return 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[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = (-t$95$3)}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-119], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.85e+67], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * N[(t$95$2 * N[(t$95$4 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$3 - t$95$5), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.8000000000000002e-119

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 4.8000000000000002e-119 < b < 1.8499999999999999e67

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.8499999999999999e67 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 45.5%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 45.5%

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

       7. lower-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-cos.f64 N/A

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

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

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

       11. lift-cos.f64 N/A

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

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

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

       13. sqr-sin-a N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-cos.f64 N/A

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

   11. Applied rewrites 45.5%

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

Alternative 5: 50.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_4 := {\left(\left|b\right|\right)}^{2}\\ t_5 := -t\_3\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{x-scale \cdot \left(t\_1 \cdot t\_2\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_4\right)}{x-scale \cdot \left(t\_1 \cdot \left(t\_2 \cdot \left(t\_4 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-y-scale \cdot \frac{\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) + \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot -0.005555555555555556\right)}^{4}}}{x-scale}}{\frac{\sin \left(t\_3 - t\_5\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_5\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (* (* PI angle) 0.005555555555555556))
        (t_4 (pow (fabs b) 2.0))
        (t_5 (- t_3)))
   (if (<= (fabs b) 4.8e-119)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
          (* x-scale (* t_1 t_2)))))
       PI))
     (if (<= (fabs b) 1.85e+67)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_4))
            (* x-scale (* t_1 (* t_2 (- t_4 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (-
             (*
              y-scale
              (/
               (+
                (+ 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112))))
                (sqrt (pow (cos (* (* PI angle) -0.005555555555555556)) 4.0)))
               x-scale)))
            (/
             (+
              (sin (- t_3 t_5))
              (sin (fma (* 0.005555555555555556 angle) PI t_5)))
             2.0))))
         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 t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_4 = pow(fabs(b), 2.0);
	double t_5 = -t_3;
	double tmp;
	if (fabs(b) <= 4.8e-119) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / (x_45_scale * (t_1 * t_2))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.85e+67) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_4)) / (x_45_scale * (t_1 * (t_2 * (t_4 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * (-(y_45_scale * (((0.5 + (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) + sqrt(pow(cos(((((double) M_PI) * angle) * -0.005555555555555556)), 4.0))) / x_45_scale)) / ((sin((t_3 - t_5)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_5))) / 2.0)))) / ((double) M_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 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_4 = abs(b) ^ 2.0
	t_5 = Float64(-t_3)
	tmp = 0.0
	if (abs(b) <= 4.8e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / Float64(x_45_scale * Float64(t_1 * t_2))))) / pi));
	elseif (abs(b) <= 1.85e+67)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_4)) / Float64(x_45_scale * Float64(t_1 * Float64(t_2 * Float64(t_4 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-Float64(y_45_scale * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) + sqrt((cos(Float64(Float64(pi * angle) * -0.005555555555555556)) ^ 4.0))) / x_45_scale))) / Float64(Float64(sin(Float64(t_3 - t_5)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_5))) / 2.0)))) / pi));
	end
	return 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[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = (-t$95$3)}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-119], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.85e+67], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * N[(t$95$2 * N[(t$95$4 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[((-N[(y$45$scale * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Cos[N[(N[(Pi * angle), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]) / N[(N[(N[Sin[N[(t$95$3 - t$95$5), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.8000000000000002e-119

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 4.8000000000000002e-119 < b < 1.8499999999999999e67

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.8499999999999999e67 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

   11. Applied rewrites 45.5%

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

Alternative 6: 50.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ t_3 := \sin t\_1\\ t_4 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_5 := -t\_4\\ \mathbf{if}\;\left|b\right| \leq 4.8 \cdot 10^{-119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_3}^{4}} + {t\_3}^{2}\right)}{x-scale \cdot \left(t\_2 \cdot t\_3\right)}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{x-scale \cdot \left(t\_2 \cdot \left(t\_3 \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {t\_2}^{2}\right)}{x-scale}}{\frac{\sin \left(t\_4 - t\_5\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_5\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (cos t_1))
        (t_3 (sin t_1))
        (t_4 (* (* PI angle) 0.005555555555555556))
        (t_5 (- t_4)))
   (if (<= (fabs b) 4.8e-119)
     (*
      180.0
      (/
       (atan
        (*
         0.5
         (/
          (* y-scale (+ (sqrt (pow t_3 4.0)) (pow t_3 2.0)))
          (* x-scale (* t_2 t_3)))))
       PI))
     (if (<= (fabs b) 1.85e+67)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* x-scale (* t_2 (* t_3 (- t_0 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -1.0 (/ (* y-scale (+ 1.0 (pow t_2 2.0))) x-scale))
            (/
             (+
              (sin (- t_4 t_5))
              (sin (fma (* 0.005555555555555556 angle) PI t_5)))
             2.0))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = sin(t_1);
	double t_4 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_5 = -t_4;
	double tmp;
	if (fabs(b) <= 4.8e-119) {
		tmp = 180.0 * (atan((0.5 * ((y_45_scale * (sqrt(pow(t_3, 4.0)) + pow(t_3, 2.0))) / (x_45_scale * (t_2 * t_3))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.85e+67) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (x_45_scale * (t_2 * (t_3 * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + pow(t_2, 2.0))) / x_45_scale)) / ((sin((t_4 - t_5)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_5))) / 2.0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = sin(t_1)
	t_4 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_5 = Float64(-t_4)
	tmp = 0.0
	if (abs(b) <= 4.8e-119)
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_3 ^ 4.0)) + (t_3 ^ 2.0))) / Float64(x_45_scale * Float64(t_2 * t_3))))) / pi));
	elseif (abs(b) <= 1.85e+67)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(x_45_scale * Float64(t_2 * Float64(t_3 * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + (t_2 ^ 2.0))) / x_45_scale)) / Float64(Float64(sin(Float64(t_4 - t_5)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_5))) / 2.0)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $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[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.8e-119], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$3, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.85e+67], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$2 * N[(t$95$3 * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$4 - t$95$5), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.8000000000000002e-119

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   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 38.3%

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

   if 4.8000000000000002e-119 < b < 1.8499999999999999e67

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.8499999999999999e67 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 45.3%

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

Alternative 7: 45.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_3 := -t\_2\\ t_4 := {\left(\left|b\right|\right)}^{2}\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_4\right)}{x-scale \cdot \left(t\_1 \cdot \left(\sin t\_0 \cdot \left(t\_4 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-1 \cdot \frac{y-scale \cdot \left(1 + {t\_1}^{2}\right)}{x-scale}}{\frac{\sin \left(t\_2 - t\_3\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_3\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (* (* PI angle) 0.005555555555555556))
        (t_3 (- t_2))
        (t_4 (pow (fabs b) 2.0)))
   (if (<= (fabs b) 2.8e-144)
     (*
      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) 1.85e+67)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_4))
            (* x-scale (* t_1 (* (sin t_0) (- t_4 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -1.0 (/ (* y-scale (+ 1.0 (pow t_1 2.0))) x-scale))
            (/
             (+
              (sin (- t_2 t_3))
              (sin (fma (* 0.005555555555555556 angle) PI t_3)))
             2.0))))
         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 t_1 = cos(t_0);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_3 = -t_2;
	double t_4 = pow(fabs(b), 2.0);
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		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) <= 1.85e+67) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_4)) / (x_45_scale * (t_1 * (sin(t_0) * (t_4 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-1.0 * ((y_45_scale * (1.0 + pow(t_1, 2.0))) / x_45_scale)) / ((sin((t_2 - t_3)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_3))) / 2.0)))) / ((double) M_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 = cos(t_0)
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_3 = Float64(-t_2)
	t_4 = abs(b) ^ 2.0
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		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) <= 1.85e+67)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_4)) / Float64(x_45_scale * Float64(t_1 * Float64(sin(t_0) * Float64(t_4 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-1.0 * Float64(Float64(y_45_scale * Float64(1.0 + (t_1 ^ 2.0))) / x_45_scale)) / Float64(Float64(sin(Float64(t_2 - t_3)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_3))) / 2.0)))) / pi));
	end
	return 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[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], 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], 1.85e+67], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * N[(N[Sin[t$95$0], $MachinePrecision] * N[(t$95$4 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-1.0 * N[(N[(y$45$scale * N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$2 - t$95$3), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-144

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 1.8499999999999999e67

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.8499999999999999e67 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 45.3%

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

Alternative 8: 45.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := -t\_1\\ t_3 := {\left(\left|b\right|\right)}^{2}\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_3\right)}{x-scale \cdot \left(\cos t\_0 \cdot \left(\sin t\_0 \cdot \left(t\_3 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\frac{\sin \left(t\_1 - t\_2\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_2\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (- t_1))
        (t_3 (pow (fabs b) 2.0)))
   (if (<= (fabs b) 2.8e-144)
     (*
      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) 1.85e+67)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_3))
            (* x-scale (* (cos t_0) (* (sin t_0) (- t_3 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -2.0 (/ y-scale x-scale))
            (/
             (+
              (sin (- t_1 t_2))
              (sin (fma (* 0.005555555555555556 angle) PI t_2)))
             2.0))))
         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 t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = -t_1;
	double t_3 = pow(fabs(b), 2.0);
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		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) <= 1.85e+67) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_3)) / (x_45_scale * (cos(t_0) * (sin(t_0) * (t_3 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / ((sin((t_1 - t_2)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_2))) / 2.0)))) / ((double) M_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(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(-t_1)
	t_3 = abs(b) ^ 2.0
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		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) <= 1.85e+67)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_3)) / Float64(x_45_scale * Float64(cos(t_0) * Float64(sin(t_0) * Float64(t_3 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(Float64(sin(Float64(t_1 - t_2)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_2))) / 2.0)))) / pi));
	end
	return 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[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], 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], 1.85e+67], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(t$95$3 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$1 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-144

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 1.8499999999999999e67

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 24.8%

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

   7. Applied rewrites 24.8%

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

   if 1.8499999999999999e67 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 45.2%

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

   12. Applied rewrites 45.2%

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

Alternative 9: 45.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 6.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{x-scale}}{\left(\pi \cdot angle\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)) (t_1 (- t_0)))
   (if (<= (fabs b) 2.8e-144)
     (*
      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) 6.4e-37)
       (*
        (/
         (atan
          (/
           (* -90.0 (/ (* (pow (fabs b) 2.0) y-scale) x-scale))
           (* (* PI angle) (* (- (fabs b) a) (+ (fabs b) a)))))
         PI)
        180.0)
       (*
        180.0
        (/
         (atan
          (*
           0.5
           (/
            (* -2.0 (/ y-scale x-scale))
            (/
             (+
              (sin (- t_0 t_1))
              (sin (fma (* 0.005555555555555556 angle) PI t_1)))
             2.0))))
         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 t_1 = -t_0;
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		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) <= 6.4e-37) {
		tmp = (atan(((-90.0 * ((pow(fabs(b), 2.0) * y_45_scale) / x_45_scale)) / ((((double) M_PI) * angle) * ((fabs(b) - a) * (fabs(b) + a))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / ((sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0)))) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		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) <= 6.4e-37)
		tmp = Float64(Float64(atan(Float64(Float64(-90.0 * Float64(Float64((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / Float64(Float64(pi * angle) * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))))) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0)))) / pi));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], 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], 6.4e-37], N[(N[(N[ArcTan[N[(N[(-90.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * angle), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-144

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 6.3999999999999998e-37

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 24.1%

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

   9. Applied rewrites 24.1%

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

   if 6.3999999999999998e-37 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 45.5%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 45.2%

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

   12. Applied rewrites 45.2%

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

Alternative 10: 45.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 5.8 \cdot 10^{-24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{0.5 \cdot \left(-y-scale \cdot \frac{2}{x-scale}\right)}{\sin t\_1 \cdot \cos t\_1}\right)}{\pi} \cdot 180\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0)) (t_1 (* (* PI angle) 0.005555555555555556)))
   (if (<= (fabs b) 2.8e-144)
     (*
      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) 5.8e-24)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* angle (* x-scale (* PI (- t_0 (pow a 2.0))))))))
         PI))
       (*
        (/
         (atan
          (/ (* 0.5 (- (* y-scale (/ 2.0 x-scale)))) (* (sin t_1) (cos t_1))))
         PI)
        180.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		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) <= 5.8e-24) {
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (((double) M_PI) * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = (atan(((0.5 * -(y_45_scale * (2.0 / x_45_scale))) / (sin(t_1) * cos(t_1)))) / ((double) M_PI)) * 180.0;
	}
	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.pow(Math.abs(b), 2.0);
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (Math.abs(b) <= 2.8e-144) {
		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) <= 5.8e-24) {
		tmp = 180.0 * (Math.atan((-90.0 * ((y_45_scale * (Math.sqrt(Math.pow(Math.abs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (Math.PI * (t_0 - Math.pow(a, 2.0)))))))) / Math.PI);
	} else {
		tmp = (Math.atan(((0.5 * -(y_45_scale * (2.0 / x_45_scale))) / (Math.sin(t_1) * Math.cos(t_1)))) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(b), 2.0)
	t_1 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if math.fabs(b) <= 2.8e-144:
		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) <= 5.8e-24:
		tmp = 180.0 * (math.atan((-90.0 * ((y_45_scale * (math.sqrt(math.pow(math.fabs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (math.pi * (t_0 - math.pow(a, 2.0)))))))) / math.pi)
	else:
		tmp = (math.atan(((0.5 * -(y_45_scale * (2.0 / x_45_scale))) / (math.sin(t_1) * math.cos(t_1)))) / math.pi) * 180.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		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) <= 5.8e-24)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(Float64(atan(Float64(Float64(0.5 * Float64(-Float64(y_45_scale * Float64(2.0 / x_45_scale)))) / Float64(sin(t_1) * cos(t_1)))) / pi) * 180.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0;
	t_1 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (abs(b) <= 2.8e-144)
		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) <= 5.8e-24)
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt((abs(b) ^ 4.0)) + t_0)) / (angle * (x_45_scale * (pi * (t_0 - (a ^ 2.0)))))))) / pi);
	else
		tmp = (atan(((0.5 * -(y_45_scale * (2.0 / x_45_scale))) / (sin(t_1) * cos(t_1)))) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], 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], 5.8e-24], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(0.5 * (-N[(y$45$scale * N[(2.0 / x$45$scale), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-144

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 5.7999999999999997e-24

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 23.0%

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

   if 5.7999999999999997e-24 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

   9. Applied rewrites 45.6%

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

   10. Taylor expanded in angle around 0

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

       1. lower-/.f64 45.3%

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

   12. Applied rewrites 45.3%

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

Alternative 11: 44.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 5.8 \cdot 10^{-24}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_0\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(t\_0 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.5 \cdot \frac{-2 \cdot \frac{y-scale}{x-scale}}{\cos t\_1 \cdot \sin t\_1}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0)) (t_1 (* 0.005555555555555556 (* angle PI))))
   (if (<= (fabs b) 2.8e-144)
     (*
      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) 5.8e-24)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_0))
            (* angle (* x-scale (* PI (- t_0 (pow a 2.0))))))))
         PI))
       (*
        180.0
        (/
         (atan
          (* 0.5 (/ (* -2.0 (/ y-scale x-scale)) (* (cos t_1) (sin t_1)))))
         PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		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) <= 5.8e-24) {
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (((double) M_PI) * (t_0 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_1) * sin(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 = Math.pow(Math.abs(b), 2.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (Math.abs(b) <= 2.8e-144) {
		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) <= 5.8e-24) {
		tmp = 180.0 * (Math.atan((-90.0 * ((y_45_scale * (Math.sqrt(Math.pow(Math.abs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (Math.PI * (t_0 - Math.pow(a, 2.0)))))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (Math.cos(t_1) * Math.sin(t_1))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(b), 2.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if math.fabs(b) <= 2.8e-144:
		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) <= 5.8e-24:
		tmp = 180.0 * (math.atan((-90.0 * ((y_45_scale * (math.sqrt(math.pow(math.fabs(b), 4.0)) + t_0)) / (angle * (x_45_scale * (math.pi * (t_0 - math.pow(a, 2.0)))))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (math.cos(t_1) * math.sin(t_1))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		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) <= 5.8e-24)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_0)) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(t_0 - (a ^ 2.0)))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(0.5 * Float64(Float64(-2.0 * Float64(y_45_scale / x_45_scale)) / Float64(cos(t_1) * sin(t_1))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0;
	t_1 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (abs(b) <= 2.8e-144)
		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) <= 5.8e-24)
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt((abs(b) ^ 4.0)) + t_0)) / (angle * (x_45_scale * (pi * (t_0 - (a ^ 2.0)))))))) / pi);
	else
		tmp = 180.0 * (atan((0.5 * ((-2.0 * (y_45_scale / x_45_scale)) / (cos(t_1) * sin(t_1))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], 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], 5.8e-24], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(t$95$0 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(0.5 * N[(N[(-2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-144

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 5.7999999999999997e-24

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 23.0%

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

   if 5.7999999999999997e-24 < b

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 24.0%

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

   5. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 45.3%

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

   10. Applied rewrites 45.3%

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

Alternative 12: 41.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ t_1 := {\left(\left|b\right|\right)}^{2}\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|b\right| \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{y-scale \cdot \left(\sqrt{{\left(\left|b\right|\right)}^{4}} + t\_1\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(t\_1 - {a}^{2}\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          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)))
        (t_1 (pow (fabs b) 2.0)))
   (if (<= (fabs b) 2.8e-144)
     t_0
     (if (<= (fabs b) 1.35e+25)
       (*
        180.0
        (/
         (atan
          (*
           -90.0
           (/
            (* y-scale (+ (sqrt (pow (fabs b) 4.0)) t_1))
            (* angle (* x-scale (* PI (- t_1 (pow a 2.0))))))))
         PI))
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 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));
	double t_1 = pow(fabs(b), 2.0);
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		tmp = t_0;
	} else if (fabs(b) <= 1.35e+25) {
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt(pow(fabs(b), 4.0)) + t_1)) / (angle * (x_45_scale * (((double) M_PI) * (t_1 - pow(a, 2.0)))))))) / ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 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);
	double t_1 = Math.pow(Math.abs(b), 2.0);
	double tmp;
	if (Math.abs(b) <= 2.8e-144) {
		tmp = t_0;
	} else if (Math.abs(b) <= 1.35e+25) {
		tmp = 180.0 * (Math.atan((-90.0 * ((y_45_scale * (Math.sqrt(Math.pow(Math.abs(b), 4.0)) + t_1)) / (angle * (x_45_scale * (Math.PI * (t_1 - Math.pow(a, 2.0)))))))) / Math.PI);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 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)
	t_1 = math.pow(math.fabs(b), 2.0)
	tmp = 0
	if math.fabs(b) <= 2.8e-144:
		tmp = t_0
	elif math.fabs(b) <= 1.35e+25:
		tmp = 180.0 * (math.atan((-90.0 * ((y_45_scale * (math.sqrt(math.pow(math.fabs(b), 4.0)) + t_1)) / (angle * (x_45_scale * (math.pi * (t_1 - math.pow(a, 2.0)))))))) / math.pi)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 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))
	t_1 = abs(b) ^ 2.0
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		tmp = t_0;
	elseif (abs(b) <= 1.35e+25)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(y_45_scale * Float64(sqrt((abs(b) ^ 4.0)) + t_1)) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(t_1 - (a ^ 2.0)))))))) / pi));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 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);
	t_1 = abs(b) ^ 2.0;
	tmp = 0.0;
	if (abs(b) <= 2.8e-144)
		tmp = t_0;
	elseif (abs(b) <= 1.35e+25)
		tmp = 180.0 * (atan((-90.0 * ((y_45_scale * (sqrt((abs(b) ^ 4.0)) + t_1)) / (angle * (x_45_scale * (pi * (t_1 - (a ^ 2.0)))))))) / pi);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.8e-144], t$95$0, If[LessEqual[N[Abs[b], $MachinePrecision], 1.35e+25], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(t$95$1 - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e-144 or 1.35e25 < b

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 1.35e25

   1. Initial program 14.0%

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

   2. 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 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 23.0%

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

Alternative 13: 40.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{{x-scale}^{4}}} + \frac{1}{{x-scale}^{2}}\right)\right)}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{if}\;\left|b\right| \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|b\right| \leq 2.85 \cdot 10^{+141}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{x-scale}}{\left(\pi \cdot angle\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          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) 2.8e-144)
     t_0
     (if (<= (fabs b) 2.85e+141)
       (*
        (/
         (atan
          (/
           (* -90.0 (/ (* (pow (fabs b) 2.0) y-scale) x-scale))
           (* (* PI angle) (* (- (fabs b) a) (+ (fabs b) a)))))
         PI)
        180.0)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 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));
	double tmp;
	if (fabs(b) <= 2.8e-144) {
		tmp = t_0;
	} else if (fabs(b) <= 2.85e+141) {
		tmp = (atan(((-90.0 * ((pow(fabs(b), 2.0) * y_45_scale) / x_45_scale)) / ((((double) M_PI) * angle) * ((fabs(b) - a) * (fabs(b) + a))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 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);
	double tmp;
	if (Math.abs(b) <= 2.8e-144) {
		tmp = t_0;
	} else if (Math.abs(b) <= 2.85e+141) {
		tmp = (Math.atan(((-90.0 * ((Math.pow(Math.abs(b), 2.0) * y_45_scale) / x_45_scale)) / ((Math.PI * angle) * ((Math.abs(b) - a) * (Math.abs(b) + a))))) / Math.PI) * 180.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 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)
	tmp = 0
	if math.fabs(b) <= 2.8e-144:
		tmp = t_0
	elif math.fabs(b) <= 2.85e+141:
		tmp = (math.atan(((-90.0 * ((math.pow(math.fabs(b), 2.0) * y_45_scale) / x_45_scale)) / ((math.pi * angle) * ((math.fabs(b) - a) * (math.fabs(b) + a))))) / math.pi) * 180.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 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))
	tmp = 0.0
	if (abs(b) <= 2.8e-144)
		tmp = t_0;
	elseif (abs(b) <= 2.85e+141)
		tmp = Float64(Float64(atan(Float64(Float64(-90.0 * Float64(Float64((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / Float64(Float64(pi * angle) * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))))) / pi) * 180.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 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);
	tmp = 0.0;
	if (abs(b) <= 2.8e-144)
		tmp = t_0;
	elseif (abs(b) <= 2.85e+141)
		tmp = (atan(((-90.0 * (((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / ((pi * angle) * ((abs(b) - a) * (abs(b) + a))))) / pi) * 180.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = 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], 2.8e-144], t$95$0, If[LessEqual[N[Abs[b], $MachinePrecision], 2.85e+141], N[(N[(N[ArcTan[N[(N[(-90.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * angle), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e-144 or 2.85e141 < b

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   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 41.0%

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

   if 2.8e-144 < b < 2.85e141

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 24.1%

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

   9. Applied rewrites 24.1%

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

Alternative 14: 37.9% accurate, 11.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 6 \cdot 10^{-146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.22 \cdot 10^{+57}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{x-scale}}{\left(\pi \cdot angle\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \cdot 180\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 6e-146)
   (* 180.0 (/ (atan 0.0) PI))
   (if (<= (fabs b) 1.22e+57)
     (*
      (/
       (atan
        (/
         (* -90.0 (/ (* (pow (fabs b) 2.0) y-scale) x-scale))
         (* (* PI angle) (* (- (fabs b) a) (+ (fabs b) a)))))
       PI)
      180.0)
     (* (/ (atan (* -90.0 (/ y-scale (* angle (* x-scale PI))))) PI) 180.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 6e-146) {
		tmp = 180.0 * (atan(0.0) / ((double) M_PI));
	} else if (fabs(b) <= 1.22e+57) {
		tmp = (atan(((-90.0 * ((pow(fabs(b), 2.0) * y_45_scale) / x_45_scale)) / ((((double) M_PI) * angle) * ((fabs(b) - a) * (fabs(b) + a))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI)) * 180.0;
	}
	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) <= 6e-146) {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	} else if (Math.abs(b) <= 1.22e+57) {
		tmp = (Math.atan(((-90.0 * ((Math.pow(Math.abs(b), 2.0) * y_45_scale) / x_45_scale)) / ((Math.PI * angle) * ((Math.abs(b) - a) * (Math.abs(b) + a))))) / Math.PI) * 180.0;
	} else {
		tmp = (Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 6e-146:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	elif math.fabs(b) <= 1.22e+57:
		tmp = (math.atan(((-90.0 * ((math.pow(math.fabs(b), 2.0) * y_45_scale) / x_45_scale)) / ((math.pi * angle) * ((math.fabs(b) - a) * (math.fabs(b) + a))))) / math.pi) * 180.0
	else:
		tmp = (math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi) * 180.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 6e-146)
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	elseif (abs(b) <= 1.22e+57)
		tmp = Float64(Float64(atan(Float64(Float64(-90.0 * Float64(Float64((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / Float64(Float64(pi * angle) * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))))) / pi) * 180.0);
	else
		tmp = Float64(Float64(atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi) * 180.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 6e-146)
		tmp = 180.0 * (atan(0.0) / pi);
	elseif (abs(b) <= 1.22e+57)
		tmp = (atan(((-90.0 * (((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / ((pi * angle) * ((abs(b) - a) * (abs(b) + a))))) / pi) * 180.0;
	else
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 6e-146], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.22e+57], N[(N[(N[ArcTan[N[(N[(-90.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * angle), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(N[(N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.0000000000000004e-146

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 6.9%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]

   if 6.0000000000000004e-146 < b < 1.22e57

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 24.1%

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

   9. Applied rewrites 24.1%

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

   if 1.22e57 < b

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 35.3%

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

   9. Applied rewrites 35.3%

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

Alternative 15: 37.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 4.6 \cdot 10^{-146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{\tan^{-1} \left(-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(\left(a + \left|b\right|\right) \cdot \left(\left|b\right| - a\right)\right)\right)\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \cdot 180\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 4.6e-146)
   (* 180.0 (/ (atan 0.0) PI))
   (if (<= (fabs b) 1.05e+61)
     (*
      (/
       (atan
        (*
         -90.0
         (/
          (* (pow (fabs b) 2.0) y-scale)
          (* angle (* x-scale (* PI (* (+ a (fabs b)) (- (fabs b) a))))))))
       PI)
      180.0)
     (* (/ (atan (* -90.0 (/ y-scale (* angle (* x-scale PI))))) PI) 180.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 4.6e-146) {
		tmp = 180.0 * (atan(0.0) / ((double) M_PI));
	} else if (fabs(b) <= 1.05e+61) {
		tmp = (atan((-90.0 * ((pow(fabs(b), 2.0) * y_45_scale) / (angle * (x_45_scale * (((double) M_PI) * ((a + fabs(b)) * (fabs(b) - a)))))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI)) * 180.0;
	}
	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) <= 4.6e-146) {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	} else if (Math.abs(b) <= 1.05e+61) {
		tmp = (Math.atan((-90.0 * ((Math.pow(Math.abs(b), 2.0) * y_45_scale) / (angle * (x_45_scale * (Math.PI * ((a + Math.abs(b)) * (Math.abs(b) - a)))))))) / Math.PI) * 180.0;
	} else {
		tmp = (Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 4.6e-146:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	elif math.fabs(b) <= 1.05e+61:
		tmp = (math.atan((-90.0 * ((math.pow(math.fabs(b), 2.0) * y_45_scale) / (angle * (x_45_scale * (math.pi * ((a + math.fabs(b)) * (math.fabs(b) - a)))))))) / math.pi) * 180.0
	else:
		tmp = (math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi) * 180.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 4.6e-146)
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	elseif (abs(b) <= 1.05e+61)
		tmp = Float64(Float64(atan(Float64(-90.0 * Float64(Float64((abs(b) ^ 2.0) * y_45_scale) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(Float64(a + abs(b)) * Float64(abs(b) - a)))))))) / pi) * 180.0);
	else
		tmp = Float64(Float64(atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi) * 180.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 4.6e-146)
		tmp = 180.0 * (atan(0.0) / pi);
	elseif (abs(b) <= 1.05e+61)
		tmp = (atan((-90.0 * (((abs(b) ^ 2.0) * y_45_scale) / (angle * (x_45_scale * (pi * ((a + abs(b)) * (abs(b) - a)))))))) / pi) * 180.0;
	else
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 4.6e-146], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.05e+61], N[(N[(N[ArcTan[N[(-90.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * y$45$scale), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(N[(N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.6000000000000001e-146

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 6.9%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]

   if 4.6000000000000001e-146 < b < 1.0500000000000001e61

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 24.2%

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

   9. Applied rewrites 24.2%

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

   if 1.0500000000000001e61 < b

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 35.3%

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

   9. Applied rewrites 35.3%

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

Alternative 16: 35.6% accurate, 24.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 1.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{\tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 1.8e+60)
   (* (/ (atan (* -90.0 (/ y-scale (* angle (* x-scale PI))))) PI) 180.0)
   (* 180.0 (/ (atan 0.0) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 1.8e+60) {
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan(0.0) / ((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 tmp;
	if (x_45_scale <= 1.8e+60) {
		tmp = (Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI) * 180.0;
	} else {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 1.8e+60:
		tmp = (math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi) * 180.0
	else:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 1.8e+60)
		tmp = Float64(Float64(atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 1.8e+60)
		tmp = (atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi) * 180.0;
	else
		tmp = 180.0 * (atan(0.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 1.8e+60], N[(N[(N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.7999999999999998e60

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 35.3%

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

   9. Applied rewrites 35.3%

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

   if 1.7999999999999998e60 < x-scale

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 6.9%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 20.1% accurate, 24.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\tan^{-1} \left(-90 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}{\pi} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 4.5e-63)
   (* (/ (atan (* -90.0 (/ x-scale (* angle (* y-scale PI))))) PI) 180.0)
   (* 180.0 (/ (atan 0.0) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 4.5e-63) {
		tmp = (atan((-90.0 * (x_45_scale / (angle * (y_45_scale * ((double) M_PI)))))) / ((double) M_PI)) * 180.0;
	} else {
		tmp = 180.0 * (atan(0.0) / ((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 tmp;
	if (Math.abs(a) <= 4.5e-63) {
		tmp = (Math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * Math.PI))))) / Math.PI) * 180.0;
	} else {
		tmp = 180.0 * (Math.atan(0.0) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 4.5e-63:
		tmp = (math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * math.pi))))) / math.pi) * 180.0
	else:
		tmp = 180.0 * (math.atan(0.0) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 4.5e-63)
		tmp = Float64(Float64(atan(Float64(-90.0 * Float64(x_45_scale / Float64(angle * Float64(y_45_scale * pi))))) / pi) * 180.0);
	else
		tmp = Float64(180.0 * Float64(atan(0.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 4.5e-63)
		tmp = (atan((-90.0 * (x_45_scale / (angle * (y_45_scale * pi))))) / pi) * 180.0;
	else
		tmp = 180.0 * (atan(0.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 4.5e-63], N[(N[(N[ArcTan[N[(-90.0 * N[(x$45$scale / N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.5e-63

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   6. Applied rewrites 14.4%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 12.1%

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

   9. Applied rewrites 12.1%

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

   if 4.5e-63 < a

   1. Initial program 14.0%

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

   2. 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 12.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 6.9%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} 0}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 18.6% accurate, 49.1× speedup

Math

\[180 \cdot \frac{\tan^{-1} 0}{\pi} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan 0.0) PI)))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.0%

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

2. 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 12.8%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

7. Applied rewrites 6.9%

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

8. Taylor expanded in y-scale around 0

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

10. Applied rewrites 18.6%

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

Reproduce

herbie shell --seed 2025187 
(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)))