raw-angle from scale-rotated-ellipse

Percentage Accurate: 13.7% → 50.5%

Time: 31.0s

Alternatives: 12

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: 13.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 50.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \cos \left(t\_0 \cdot 2\right)\\ t_2 := \mathsf{fma}\left(\left(0.5 - t\_1 \cdot 0.5\right) \cdot a, a, \left(\mathsf{fma}\left(t\_1, 0.5, 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)\\ \mathbf{if}\;\left|b\right| \leq 580000000:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.45 \cdot 10^{+82}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \left(\frac{\left|t\_2\right| + t\_2}{\left(\left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0} \cdot \frac{y-scale}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (cos (* t_0 2.0)))
        (t_2
         (fma
          (* (- 0.5 (* t_1 0.5)) a)
          a
          (* (* (fma t_1 0.5 0.5) (fabs b)) (fabs b)))))
   (if (<= (fabs b) 580000000.0)
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           angle
           (*
            x-scale
            (*
             y-scale
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0)))))))
          PI)))
       PI))
     (if (<= (fabs b) 1.45e+82)
       (*
        180.0
        (/
         (atan
          (*
           -0.5
           (*
            (/
             (+ (fabs t_2) t_2)
             (* (* (* (- (fabs b) a) (+ (fabs b) a)) (sin t_0)) (cos t_0)))
            (/ y-scale x-scale))))
         PI))
       (*
        180.0
        (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = cos((t_0 * 2.0));
	double t_2 = fma(((0.5 - (t_1 * 0.5)) * a), a, ((fma(t_1, 0.5, 0.5) * fabs(b)) * fabs(b)));
	double tmp;
	if (fabs(b) <= 580000000.0) {
		tmp = 180.0 * (atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0))))))) / ((double) M_PI)))) / ((double) M_PI));
	} else if (fabs(b) <= 1.45e+82) {
		tmp = 180.0 * (atan((-0.5 * (((fabs(t_2) + t_2) / ((((fabs(b) - a) * (fabs(b) + a)) * sin(t_0)) * cos(t_0))) * (y_45_scale / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((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 = cos(Float64(t_0 * 2.0))
	t_2 = fma(Float64(Float64(0.5 - Float64(t_1 * 0.5)) * a), a, Float64(Float64(fma(t_1, 0.5, 0.5) * abs(b)) * abs(b)))
	tmp = 0.0
	if (abs(b) <= 580000000.0)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0))))))) / pi))) / pi));
	elseif (abs(b) <= 1.45e+82)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(Float64(abs(t_2) + t_2) / Float64(Float64(Float64(Float64(abs(b) - a) * Float64(abs(b) + a)) * sin(t_0)) * cos(t_0))) * Float64(y_45_scale / x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / 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[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(t$95$1 * 0.5 + 0.5), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 580000000.0], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.45e+82], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(N[(N[Abs[t$95$2], $MachinePrecision] + t$95$2), $MachinePrecision] / N[(N[(N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.8e8

   1. Initial program 13.7%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 20.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

   10. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\pi}}\right)}{\pi} \]

   if 5.8e8 < b < 1.4500000000000001e82

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   6. Applied rewrites 29.2%

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

   if 1.4500000000000001e82 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

Alternative 2: 49.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;\left|a\right| \leq 9.12 \cdot 10^{-70}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-0.5 \cdot \frac{y-scale \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{x-scale \cdot \left(t\_1 \cdot \sin t\_0\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (cos t_0)))
   (if (<= (fabs a) 9.12e-70)
     (*
      180.0
      (/
       (atan
        (*
         -0.5
         (/
          (* y-scale (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
          (* x-scale (* t_1 (sin t_0))))))
       PI))
     (*
      180.0
      (/
       (atan
        (*
         90.0
         (/
          (*
           angle
           (*
            x-scale
            (*
             y-scale
             (+
              (sqrt
               (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
              (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0)))))))
          PI)))
       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 tmp;
	if (fabs(a) <= 9.12e-70) {
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / (x_45_scale * (t_1 * sin(t_0)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0))))))) / ((double) M_PI)))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double tmp;
	if (Math.abs(a) <= 9.12e-70) {
		tmp = 180.0 * (Math.atan((-0.5 * ((y_45_scale * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0))) / (x_45_scale * (t_1 * Math.sin(t_0)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (Math.sqrt((9.525986892242036e-10 * (Math.pow(Math.PI, 4.0) / Math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (Math.pow(Math.PI, 2.0) / Math.pow(x_45_scale, 2.0))))))) / Math.PI))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	tmp = 0
	if math.fabs(a) <= 9.12e-70:
		tmp = 180.0 * (math.atan((-0.5 * ((y_45_scale * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0))) / (x_45_scale * (t_1 * math.sin(t_0)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (math.sqrt((9.525986892242036e-10 * (math.pow(math.pi, 4.0) / math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (math.pow(math.pi, 2.0) / math.pow(x_45_scale, 2.0))))))) / math.pi))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	tmp = 0.0
	if (abs(a) <= 9.12e-70)
		tmp = Float64(180.0 * Float64(atan(Float64(-0.5 * Float64(Float64(y_45_scale * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / Float64(x_45_scale * Float64(t_1 * sin(t_0)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0))))))) / pi))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	tmp = 0.0;
	if (abs(a) <= 9.12e-70)
		tmp = 180.0 * (atan((-0.5 * ((y_45_scale * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / (x_45_scale * (t_1 * sin(t_0)))))) / pi);
	else
		tmp = 180.0 * (atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (sqrt((9.525986892242036e-10 * ((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + (3.08641975308642e-5 * ((pi ^ 2.0) / (x_45_scale ^ 2.0))))))) / pi))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 9.12e-70], N[(180.0 * N[(N[ArcTan[N[(-0.5 * N[(N[(y$45$scale * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 9.1200000000000004e-70

   1. Initial program 13.7%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 44.0%

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

   if 9.1200000000000004e-70 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 20.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

   10. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\pi}}\right)}{\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 47.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 3.6e+87)
   (*
    180.0
    (/
     (atan
      (*
       90.0
       (/
        (*
         angle
         (*
          x-scale
          (*
           y-scale
           (+
            (sqrt (* 9.525986892242036e-10 (/ (pow PI 4.0) (pow x-scale 4.0))))
            (* 3.08641975308642e-5 (/ (pow PI 2.0) (pow x-scale 2.0)))))))
        PI)))
     PI))
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 3.6e+87) {
		tmp = 180.0 * (atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (sqrt((9.525986892242036e-10 * (pow(((double) M_PI), 4.0) / pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (pow(((double) M_PI), 2.0) / pow(x_45_scale, 2.0))))))) / ((double) M_PI)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 3.6e+87) {
		tmp = 180.0 * (Math.atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (Math.sqrt((9.525986892242036e-10 * (Math.pow(Math.PI, 4.0) / Math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (Math.pow(Math.PI, 2.0) / Math.pow(x_45_scale, 2.0))))))) / Math.PI))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 3.6e+87:
		tmp = 180.0 * (math.atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (math.sqrt((9.525986892242036e-10 * (math.pow(math.pi, 4.0) / math.pow(x_45_scale, 4.0)))) + (3.08641975308642e-5 * (math.pow(math.pi, 2.0) / math.pow(x_45_scale, 2.0))))))) / math.pi))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 3.6e+87)
		tmp = Float64(180.0 * Float64(atan(Float64(90.0 * Float64(Float64(angle * Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(9.525986892242036e-10 * Float64((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + Float64(3.08641975308642e-5 * Float64((pi ^ 2.0) / (x_45_scale ^ 2.0))))))) / pi))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 3.6e+87)
		tmp = 180.0 * (atan((90.0 * ((angle * (x_45_scale * (y_45_scale * (sqrt((9.525986892242036e-10 * ((pi ^ 4.0) / (x_45_scale ^ 4.0)))) + (3.08641975308642e-5 * ((pi ^ 2.0) / (x_45_scale ^ 2.0))))))) / pi))) / pi);
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.6e+87], N[(180.0 * N[(N[ArcTan[N[(90.0 * N[(N[(angle * N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(9.525986892242036e-10 * N[(N[Power[Pi, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999994e87

   1. Initial program 13.7%

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

   2. Taylor expanded in y-scale around inf

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

   4. Applied rewrites 20.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)}}\right)}{\pi} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{\frac{1}{1049760000} \cdot \frac{{\mathsf{PI}\left(\right)}^{4}}{{x-scale}^{4}}} + \frac{1}{32400} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)}{\pi} \]

   10. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(90 \cdot \frac{angle \cdot \left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot \frac{{\pi}^{4}}{{x-scale}^{4}}} + 3.08641975308642 \cdot 10^{-5} \cdot \frac{{\pi}^{2}}{{x-scale}^{2}}\right)\right)\right)}{\color{blue}{\pi}}\right)}{\pi} \]

   if 3.59999999999999994e87 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

Alternative 4: 42.2% accurate, 11.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 2.5 \cdot 10^{-72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{\left(\left(\frac{1}{x-scale \cdot x-scale} + \sqrt{{x-scale}^{-4}}\right) \cdot y-scale\right) \cdot x-scale}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{x-scale}}{\left(\pi \cdot angle\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 2.5e-72)
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/
        (*
         (* (+ (/ 1.0 (* x-scale x-scale)) (sqrt (pow x-scale -4.0))) y-scale)
         x-scale)
        (* angle PI))))
     PI))
   (if (<= (fabs b) 1.3e+82)
     (/
      (*
       180.0
       (atan
        (/
         (* -90.0 (/ (* (pow (fabs b) 2.0) y-scale) x-scale))
         (* (* PI angle) (* (- (fabs b) a) (+ (fabs b) a))))))
      PI)
     (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 2.5e-72) {
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt(pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else if (fabs(b) <= 1.3e+82) {
		tmp = (180.0 * 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);
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 2.5e-72) {
		tmp = 180.0 * (Math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + Math.sqrt(Math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * Math.PI)))) / Math.PI);
	} else if (Math.abs(b) <= 1.3e+82) {
		tmp = (180.0 * 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;
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 2.5e-72:
		tmp = 180.0 * (math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + math.sqrt(math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * math.pi)))) / math.pi)
	elif math.fabs(b) <= 1.3e+82:
		tmp = (180.0 * 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
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 2.5e-72)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(Float64(Float64(Float64(1.0 / Float64(x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / Float64(angle * pi)))) / pi));
	elseif (abs(b) <= 1.3e+82)
		tmp = Float64(Float64(180.0 * 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);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 2.5e-72)
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / (angle * pi)))) / pi);
	elseif (abs(b) <= 1.3e+82)
		tmp = (180.0 * atan(((-90.0 * (((abs(b) ^ 2.0) * y_45_scale) / x_45_scale)) / ((pi * angle) * ((abs(b) - a) * (abs(b) + a)))))) / pi;
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 2.5e-72], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(N[(N[(N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[x$45$scale, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.3e+82], N[(N[(180.0 * 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]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.4999999999999998e-72

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 40.4

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 40.4

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

      13. lift-/.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-flip N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval 40.4

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

   9. Applied rewrites 40.4%

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

   if 2.4999999999999998e-72 < b < 1.2999999999999999e82

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   6. Applied rewrites 14.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 23.5

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

   9. Applied rewrites 23.5%

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

   if 1.2999999999999999e82 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

Alternative 5: 41.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.12 \cdot 10^{-174}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{\left(\left(\frac{1}{x-scale \cdot x-scale} + \sqrt{{x-scale}^{-4}}\right) \cdot y-scale\right) \cdot x-scale}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{elif}\;\left|b\right| \leq 2.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(-90 \cdot \frac{{\left(\left|b\right|\right)}^{2} \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(\left(a + \left|b\right|\right) \cdot \left(\left|b\right| - a\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs b) 1.12e-174)
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/
        (*
         (* (+ (/ 1.0 (* x-scale x-scale)) (sqrt (pow x-scale -4.0))) y-scale)
         x-scale)
        (* angle PI))))
     PI))
   (if (<= (fabs b) 2.1e+76)
     (/
      (*
       180.0
       (atan
        (*
         -90.0
         (/
          (* (pow (fabs b) 2.0) y-scale)
          (* angle (* x-scale (* PI (* (+ a (fabs b)) (- (fabs b) a)))))))))
      PI)
     (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 1.12e-174) {
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt(pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * ((double) M_PI))))) / ((double) M_PI));
	} else if (fabs(b) <= 2.1e+76) {
		tmp = (180.0 * 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);
	} else {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 1.12e-174) {
		tmp = 180.0 * (Math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + Math.sqrt(Math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * Math.PI)))) / Math.PI);
	} else if (Math.abs(b) <= 2.1e+76) {
		tmp = (180.0 * 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;
	} else {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 1.12e-174:
		tmp = 180.0 * (math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + math.sqrt(math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * math.pi)))) / math.pi)
	elif math.fabs(b) <= 2.1e+76:
		tmp = (180.0 * 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
	else:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 1.12e-174)
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(Float64(Float64(Float64(1.0 / Float64(x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / Float64(angle * pi)))) / pi));
	elseif (abs(b) <= 2.1e+76)
		tmp = Float64(Float64(180.0 * 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);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 1.12e-174)
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / (angle * pi)))) / pi);
	elseif (abs(b) <= 2.1e+76)
		tmp = (180.0 * atan((-90.0 * (((abs(b) ^ 2.0) * y_45_scale) / (angle * (x_45_scale * (pi * ((a + abs(b)) * (abs(b) - a))))))))) / pi;
	else
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.12e-174], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(N[(N[(N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[x$45$scale, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 2.1e+76], N[(N[(180.0 * 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]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.12e-174

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 40.4

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 40.4

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

      13. lift-/.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-flip N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval 40.4

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

   9. Applied rewrites 40.4%

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

   if 1.12e-174 < b < 2.10000000000000007e76

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   6. Applied rewrites 14.0%

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

   7. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 23.9

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

   9. Applied rewrites 23.9%

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

   if 2.10000000000000007e76 < b

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

Alternative 6: 41.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 9.5 \cdot 10^{-83}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{\left(\left(\frac{1}{x-scale \cdot x-scale} + \sqrt{{x-scale}^{-4}}\right) \cdot y-scale\right) \cdot x-scale}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 9.5e-83)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/
        (*
         (* (+ (/ 1.0 (* x-scale x-scale)) (sqrt (pow x-scale -4.0))) y-scale)
         x-scale)
        (* angle PI))))
     PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 9.5e-83) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt(pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 9.5e-83) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + Math.sqrt(Math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * Math.PI)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 9.5e-83:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + math.sqrt(math.pow(x_45_scale, -4.0))) * y_45_scale) * x_45_scale) / (angle * math.pi)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 9.5e-83)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(Float64(Float64(Float64(1.0 / Float64(x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 9.5e-83)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	else
		tmp = 180.0 * (atan((-90.0 * (((((1.0 / (x_45_scale * x_45_scale)) + sqrt((x_45_scale ^ -4.0))) * y_45_scale) * x_45_scale) / (angle * pi)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 9.5e-83], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(N[(N[(N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[x$45$scale, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.50000000000000051e-83

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

   if 9.50000000000000051e-83 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 40.4

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 40.4

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

      13. lift-/.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-flip N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval 40.4

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

   9. Applied rewrites 40.4%

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

Alternative 7: 41.7% accurate, 14.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 1.8 \cdot 10^{-82}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-90 \cdot \frac{x-scale \cdot \left(2 \cdot \frac{y-scale}{{x-scale}^{2}}\right)}{angle \cdot \pi}\right)}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 1.8e-82)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (*
    180.0
    (/
     (atan
      (*
       -90.0
       (/ (* x-scale (* 2.0 (/ y-scale (pow x-scale 2.0)))) (* angle PI))))
     PI))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 1.8e-82) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (2.0 * (y_45_scale / pow(x_45_scale, 2.0)))) / (angle * ((double) M_PI))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 1.8e-82) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((-90.0 * ((x_45_scale * (2.0 * (y_45_scale / Math.pow(x_45_scale, 2.0)))) / (angle * Math.PI)))) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 1.8e-82:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((-90.0 * ((x_45_scale * (2.0 * (y_45_scale / math.pow(x_45_scale, 2.0)))) / (angle * math.pi)))) / math.pi)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.8e-82)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(x_45_scale * Float64(2.0 * Float64(y_45_scale / (x_45_scale ^ 2.0)))) / Float64(angle * pi)))) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 1.8e-82)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	else
		tmp = 180.0 * (atan((-90.0 * ((x_45_scale * (2.0 * (y_45_scale / (x_45_scale ^ 2.0)))) / (angle * pi)))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 1.8e-82], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(x$45$scale * N[(2.0 * N[(y$45$scale / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.79999999999999999e-82

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

   if 1.79999999999999999e-82 < a

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 40.4

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

   10. Applied rewrites 40.4%

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

Alternative 8: 39.5% accurate, 25.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* -90.0 (/ (* 2.0 (/ y-scale x-scale)) (* angle PI)))) PI)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-90.0 * ((2.0 * (y_45_scale / x_45_scale)) / (angle * ((double) M_PI))))) / ((double) M_PI));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-90.0 * ((2.0 * (y_45_scale / x_45_scale)) / (angle * Math.PI)))) / Math.PI);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-90.0 * ((2.0 * (y_45_scale / x_45_scale)) / (angle * math.pi)))) / math.pi)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-90.0 * Float64(Float64(2.0 * Float64(y_45_scale / x_45_scale)) / Float64(angle * pi)))) / pi))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-90.0 * ((2.0 * (y_45_scale / x_45_scale)) / (angle * pi)))) / pi);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-90.0 * N[(N[(2.0 * N[(y$45$scale / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.7%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.3%

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

5. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

7. Applied rewrites 40.4%

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

8. Taylor expanded in x-scale around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 39.5

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

10. Applied rewrites 39.5%

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

Alternative 9: 38.3% accurate, 24.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 3.5 \cdot 10^{+116}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 3.5e+116)
   (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* x-scale PI))))) PI))
   (* 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 <= 3.5e+116) {
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI)))))) / ((double) M_PI));
	} 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 <= 3.5e+116) {
		tmp = 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * Math.PI))))) / Math.PI);
	} 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 <= 3.5e+116:
		tmp = 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (x_45_scale * math.pi))))) / math.pi)
	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 <= 3.5e+116)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi))))) / pi));
	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 <= 3.5e+116)
		tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (x_45_scale * pi))))) / pi);
	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, 3.5e+116], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.49999999999999997e116

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in x-scale around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 38.1

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

   10. Applied rewrites 38.1%

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

   if 3.49999999999999997e116 < x-scale

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   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 7.2%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.9%

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

Alternative 10: 35.7% accurate, 24.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 3.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(-90 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 3.5e+116)
   (/ (* 180.0 (atan (* -90.0 (/ y-scale (* angle (* x-scale PI)))))) PI)
   (* 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 <= 3.5e+116) {
		tmp = (180.0 * atan((-90.0 * (y_45_scale / (angle * (x_45_scale * ((double) M_PI))))))) / ((double) M_PI);
	} 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 <= 3.5e+116) {
		tmp = (180.0 * Math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * Math.PI)))))) / Math.PI;
	} 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 <= 3.5e+116:
		tmp = (180.0 * math.atan((-90.0 * (y_45_scale / (angle * (x_45_scale * math.pi)))))) / math.pi
	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 <= 3.5e+116)
		tmp = Float64(Float64(180.0 * atan(Float64(-90.0 * Float64(y_45_scale / Float64(angle * Float64(x_45_scale * pi)))))) / pi);
	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 <= 3.5e+116)
		tmp = (180.0 * atan((-90.0 * (y_45_scale / (angle * (x_45_scale * pi)))))) / pi;
	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, 3.5e+116], N[(N[(180.0 * N[ArcTan[N[(-90.0 * N[(y$45$scale / N[(angle * N[(x$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.49999999999999997e116

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   6. Applied rewrites 14.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 34.8

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

   9. Applied rewrites 34.8%

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

   if 3.49999999999999997e116 < x-scale

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   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 7.2%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.9%

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

Alternative 11: 22.5% accurate, 22.3× speedup

Math

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} 0}{\pi}\\ \mathbf{if}\;x-scale \leq -4.3 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale \leq 7.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(-90 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ (atan 0.0) PI))))
   (if (<= x-scale -4.3e+89)
     t_0
     (if (<= x-scale 7.1e+49)
       (/ (* 180.0 (atan (* -90.0 (/ x-scale (* angle (* y-scale PI)))))) 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(0.0) / ((double) M_PI));
	double tmp;
	if (x_45_scale <= -4.3e+89) {
		tmp = t_0;
	} else if (x_45_scale <= 7.1e+49) {
		tmp = (180.0 * atan((-90.0 * (x_45_scale / (angle * (y_45_scale * ((double) M_PI))))))) / ((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(0.0) / Math.PI);
	double tmp;
	if (x_45_scale <= -4.3e+89) {
		tmp = t_0;
	} else if (x_45_scale <= 7.1e+49) {
		tmp = (180.0 * Math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * Math.PI)))))) / 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(0.0) / math.pi)
	tmp = 0
	if x_45_scale <= -4.3e+89:
		tmp = t_0
	elif x_45_scale <= 7.1e+49:
		tmp = (180.0 * math.atan((-90.0 * (x_45_scale / (angle * (y_45_scale * math.pi)))))) / 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(0.0) / pi))
	tmp = 0.0
	if (x_45_scale <= -4.3e+89)
		tmp = t_0;
	elseif (x_45_scale <= 7.1e+49)
		tmp = Float64(Float64(180.0 * atan(Float64(-90.0 * Float64(x_45_scale / Float64(angle * Float64(y_45_scale * pi)))))) / 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(0.0) / pi);
	tmp = 0.0;
	if (x_45_scale <= -4.3e+89)
		tmp = t_0;
	elseif (x_45_scale <= 7.1e+49)
		tmp = (180.0 * atan((-90.0 * (x_45_scale / (angle * (y_45_scale * pi)))))) / 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[0.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -4.3e+89], t$95$0, If[LessEqual[x$45$scale, 7.1e+49], N[(N[(180.0 * N[ArcTan[N[(-90.0 * N[(x$45$scale / N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -4.3000000000000002e89 or 7.09999999999999971e49 < x-scale

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
   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 7.2%

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

   8. Taylor expanded in y-scale around 0

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

   10. Applied rewrites 18.9%

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

   if -4.3000000000000002e89 < x-scale < 7.09999999999999971e49

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 12.3%

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

   6. Applied rewrites 14.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 12.3

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

   9. Applied rewrites 12.3%

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

Alternative 12: 18.9% 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 13.7%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 12.3%

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

5. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-90 \cdot \color{blue}{\frac{x-scale \cdot \left(y-scale \cdot \left(\frac{1}{{y-scale}^{2}} - \sqrt{\frac{1}{{y-scale}^{4}}}\right)\right)}{angle \cdot \mathsf{PI}\left(\right)}}\right)}{\pi} \]
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 7.2%

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

8. Taylor expanded in y-scale around 0

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

10. Applied rewrites 18.9%

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

Reproduce

herbie shell --seed 2025178 
(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)))