a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 56.3%

Time: 53.1s

Alternatives: 6

Speedup: 919.0×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = 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] / y$45$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] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 2.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = 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] / y$45$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] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 56.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;x-scale\_m \leq 6 \cdot 10^{+37}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(y-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b \cdot t\_1, \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale\_m \leq 2.1 \cdot 10^{+183}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\left(x-scale\_m \cdot 0.25\right) \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin t\_0, t\_1 \cdot a\right)\right)\right)}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale\_m \cdot \left(\mathsf{hypot}\left(a \cdot \cos t\_2, b \cdot \sin t\_2\right) \cdot \left(-{16}^{0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (* PI (* 0.005555555555555556 angle))))
   (if (<= x-scale_m 6e+37)
     (*
      0.25
      (*
       (* (* y-scale_m (sqrt 2.0)) (sqrt 8.0))
       (hypot
        (* b t_1)
        (sqrt (* 3.08641975308642e-5 (pow (* PI (* angle a)) 2.0))))))
     (if (<= x-scale_m 2.1e+183)
       (pow
        (cbrt
         (pow
          (* (* x-scale_m 0.25) (* 4.0 (hypot (* b (sin t_0)) (* t_1 a))))
          1.5))
        2.0)
       (*
        -0.25
        (*
         x-scale_m
         (* (hypot (* a (cos t_2)) (* b (sin t_2))) (- (pow 16.0 0.5)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (x_45_scale_m <= 6e+37) {
		tmp = 0.25 * (((y_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((b * t_1), sqrt((3.08641975308642e-5 * pow((((double) M_PI) * (angle * a)), 2.0)))));
	} else if (x_45_scale_m <= 2.1e+183) {
		tmp = pow(cbrt(pow(((x_45_scale_m * 0.25) * (4.0 * hypot((b * sin(t_0)), (t_1 * a)))), 1.5)), 2.0);
	} else {
		tmp = -0.25 * (x_45_scale_m * (hypot((a * cos(t_2)), (b * sin(t_2))) * -pow(16.0, 0.5)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (x_45_scale_m <= 6e+37) {
		tmp = 0.25 * (((y_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * Math.hypot((b * t_1), Math.sqrt((3.08641975308642e-5 * Math.pow((Math.PI * (angle * a)), 2.0)))));
	} else if (x_45_scale_m <= 2.1e+183) {
		tmp = Math.pow(Math.cbrt(Math.pow(((x_45_scale_m * 0.25) * (4.0 * Math.hypot((b * Math.sin(t_0)), (t_1 * a)))), 1.5)), 2.0);
	} else {
		tmp = -0.25 * (x_45_scale_m * (Math.hypot((a * Math.cos(t_2)), (b * Math.sin(t_2))) * -Math.pow(16.0, 0.5)));
	}
	return tmp;
}

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (x_45_scale_m <= 6e+37)
		tmp = Float64(0.25 * Float64(Float64(Float64(y_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(Float64(b * t_1), sqrt(Float64(3.08641975308642e-5 * (Float64(pi * Float64(angle * a)) ^ 2.0))))));
	elseif (x_45_scale_m <= 2.1e+183)
		tmp = cbrt((Float64(Float64(x_45_scale_m * 0.25) * Float64(4.0 * hypot(Float64(b * sin(t_0)), Float64(t_1 * a)))) ^ 1.5)) ^ 2.0;
	else
		tmp = Float64(-0.25 * Float64(x_45_scale_m * Float64(hypot(Float64(a * cos(t_2)), Float64(b * sin(t_2))) * Float64(-(16.0 ^ 0.5)))));
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 6e+37], N[(0.25 * N[(N[(N[(y$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * t$95$1), $MachinePrecision] ^ 2 + N[Sqrt[N[(3.08641975308642e-5 * N[Power[N[(Pi * N[(angle * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 2.1e+183], N[Power[N[Power[N[Power[N[(N[(x$45$scale$95$m * 0.25), $MachinePrecision] * N[(4.0 * N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(t$95$1 * a), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], N[(-0.25 * N[(x$45$scale$95$m * N[(N[Sqrt[N[(a * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * (-N[Power[16.0, 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 6.00000000000000043e37

   1. Initial program 3.2%

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

   3. Simplified 3.7%

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

   5. Taylor expanded in x-scale around 0 11.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
   6. Step-by-step derivation

      1. pow1/2 11.5%

         \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \color{blue}{{\left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{0.5}}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      2. distribute-lft-out 11.5%

         \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot {\color{blue}{\left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}}^{0.5}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      3. unpow-prod-down 11.5%

         \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \color{blue}{\left({2}^{0.5} \cdot {\left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{0.5}\right)}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      4. pow1/2 11.5%

         \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \left(\color{blue}{\sqrt{2}} \cdot {\left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{0.5}\right)\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   7. Applied egg-rr 12.0%

      \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot a\right)}^{2} + {\left(b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}\right)}^{0.5}\right)}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   8. Taylor expanded in angle around 0 11.9%

      \[\leadsto \left(0.25 \cdot \left(\frac{\sqrt{8}}{{x-scale}^{2} \cdot y-scale} \cdot \left(\sqrt{2} \cdot {\left({\left(\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot a\right)}^{2} + {\left(b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}\right)}^{0.5}\right)\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   9. Taylor expanded in x-scale around 0 19.4%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)} \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{0.25 \cdot \left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}}\right)\right)} \]

   if 6.00000000000000043e37 < x-scale < 2.1e183

   1. Initial program 0.1%

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

   3. Simplified 3.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 62.9%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 62.9%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 62.9%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 62.9%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 62.9%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 62.9%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 62.8%

         \[\leadsto \color{blue}{\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \cdot \sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}} \]

      2. pow2 62.8%

         \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}\right)}^{2}} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)}\right)}^{2}} \]
   10. Step-by-step derivation

   11. Simplified 62.8%

       \[\leadsto \color{blue}{{\left(\sqrt{0.25 \cdot \left(x-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{2}} \]

   13. Applied egg-rr 68.1%

       \[\leadsto {\color{blue}{\left(\sqrt[3]{{\left(\left(0.25 \cdot x-scale\right) \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{1.5}}\right)}}^{2} \]

   if 2.1e183 < x-scale

   1. Initial program 0.1%

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

   3. Simplified 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 72.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 72.0%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 72.0%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 72.0%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 72.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-neg-in 72.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left(\left(-x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]

      2. sqrt-unprod 72.1%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\sqrt{8 \cdot \left(2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}}\right) \]

      3. fma-undefine 72.1%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \color{blue}{\left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      4. pow-prod-down 72.1%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left(\color{blue}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}\right) \]

      5. associate-*r* 72.1%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}^{2} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}\right) \]

      6. pow-prod-down 77.4%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + \color{blue}{{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}}\right)\right)}\right) \]

   9. Applied egg-rr 77.5%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. pow1/2 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{{\left(8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)\right)}^{0.5}}\right) \]

       2. associate-*r* 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot {\color{blue}{\left(\left(8 \cdot 2\right) \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}}^{0.5}\right) \]

       3. unpow-prod-down 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\left({\left(8 \cdot 2\right)}^{0.5} \cdot {\left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)}^{0.5}\right)}\right) \]

       4. metadata-eval 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({\color{blue}{16}}^{0.5} \cdot {\left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)}^{0.5}\right)\right) \]

       5. pow1/2 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \color{blue}{\sqrt{{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}}\right)\right) \]

       6. unpow2 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \sqrt{\color{blue}{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}\right)\right) \]

       7. unpow2 77.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \sqrt{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \color{blue}{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}}\right)\right) \]

       8. hypot-define 83.5%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \color{blue}{\mathsf{hypot}\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}\right)\right) \]

   11. Applied egg-rr 83.5%

       \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\left({16}^{0.5} \cdot \mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 6 \cdot 10^{+37}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.1 \cdot 10^{+183}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\left(x-scale \cdot 0.25\right) \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)\right)\right)}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(-{16}^{0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 5 \cdot 10^{+108}:\\ \;\;\;\;-0.25 \cdot \left(x-scale\_m \cdot \left(\mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right) \cdot \left(-{16}^{0.5}\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.6 \cdot 10^{+253}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale\_m \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 5e+108)
     (*
      -0.25
      (*
       x-scale_m
       (* (hypot (* a (cos t_0)) (* b (sin t_0))) (- (pow 16.0 0.5)))))
     (if (<= y-scale_m 3.6e+253)
       (* 0.25 (* b (log1p (expm1 (* y-scale_m 4.0)))))
       (* y-scale_m b)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 5e+108) {
		tmp = -0.25 * (x_45_scale_m * (hypot((a * cos(t_0)), (b * sin(t_0))) * -pow(16.0, 0.5)));
	} else if (y_45_scale_m <= 3.6e+253) {
		tmp = 0.25 * (b * log1p(expm1((y_45_scale_m * 4.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 5e+108) {
		tmp = -0.25 * (x_45_scale_m * (Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0))) * -Math.pow(16.0, 0.5)));
	} else if (y_45_scale_m <= 3.6e+253) {
		tmp = 0.25 * (b * Math.log1p(Math.expm1((y_45_scale_m * 4.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if y_45_scale_m <= 5e+108:
		tmp = -0.25 * (x_45_scale_m * (math.hypot((a * math.cos(t_0)), (b * math.sin(t_0))) * -math.pow(16.0, 0.5)))
	elif y_45_scale_m <= 3.6e+253:
		tmp = 0.25 * (b * math.log1p(math.expm1((y_45_scale_m * 4.0))))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 5e+108)
		tmp = Float64(-0.25 * Float64(x_45_scale_m * Float64(hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))) * Float64(-(16.0 ^ 0.5)))));
	elseif (y_45_scale_m <= 3.6e+253)
		tmp = Float64(0.25 * Float64(b * log1p(expm1(Float64(y_45_scale_m * 4.0)))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 5e+108], N[(-0.25 * N[(x$45$scale$95$m * N[(N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * (-N[Power[16.0, 0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.6e+253], N[(0.25 * N[(b * N[Log[1 + N[(Exp[N[(y$45$scale$95$m * 4.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 4.99999999999999991e108

   1. Initial program 3.1%

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

   3. Simplified 2.2%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 22.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 22.0%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 22.0%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 22.0%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 22.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-neg-in 22.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left(\left(-x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]

      2. sqrt-unprod 22.0%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\sqrt{8 \cdot \left(2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}}\right) \]

      3. fma-undefine 22.0%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \color{blue}{\left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)}\right) \]

      4. pow-prod-down 22.0%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left(\color{blue}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}\right) \]

      5. associate-*r* 22.0%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}^{2} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)\right)}\right) \]

      6. pow-prod-down 21.6%

         \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + \color{blue}{{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}}\right)\right)}\right) \]

   9. Applied egg-rr 22.1%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. pow1/2 22.1%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{{\left(8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)\right)}^{0.5}}\right) \]

       2. associate-*r* 22.1%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot {\color{blue}{\left(\left(8 \cdot 2\right) \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}}^{0.5}\right) \]

       3. unpow-prod-down 21.6%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\left({\left(8 \cdot 2\right)}^{0.5} \cdot {\left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)}^{0.5}\right)}\right) \]

       4. metadata-eval 21.6%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({\color{blue}{16}}^{0.5} \cdot {\left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)}^{0.5}\right)\right) \]

       5. pow1/2 21.6%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \color{blue}{\sqrt{{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}}\right)\right) \]

       6. unpow2 21.6%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \sqrt{\color{blue}{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}\right)\right) \]

       7. unpow2 21.6%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \sqrt{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \color{blue}{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}}\right)\right) \]

       8. hypot-define 23.0%

          \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \left({16}^{0.5} \cdot \color{blue}{\mathsf{hypot}\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}\right)\right) \]

   11. Applied egg-rr 23.0%

       \[\leadsto -0.25 \cdot \left(\left(-x-scale\right) \cdot \color{blue}{\left({16}^{0.5} \cdot \mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)}\right) \]

   if 4.99999999999999991e108 < y-scale < 3.6e253

   1. Initial program 0.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in angle around 0 29.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 29.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 29.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 34.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)}\right) \]

      2. *-commutative 34.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale}\right)\right)\right) \]

      3. sqrt-unprod 34.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 34.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 34.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(4 \cdot y-scale\right)\right)}\right) \]

   if 3.6e253 < y-scale

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   10. Taylor expanded in b around 0 50.2%

       \[\leadsto \color{blue}{b \cdot y-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 5 \cdot 10^{+108}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(-{16}^{0.5}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.6 \cdot 10^{+253}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 23.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{elif}\;y-scale\_m \leq 1.95 \cdot 10^{+253}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale\_m \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 2.15e+30)
   (* x-scale_m a)
   (if (<= y-scale_m 1.95e+253)
     (* 0.25 (* b (log1p (expm1 (* y-scale_m 4.0)))))
     (* y-scale_m b))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.15e+30) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 1.95e+253) {
		tmp = 0.25 * (b * log1p(expm1((y_45_scale_m * 4.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.15e+30) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 1.95e+253) {
		tmp = 0.25 * (b * Math.log1p(Math.expm1((y_45_scale_m * 4.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.15e+30:
		tmp = x_45_scale_m * a
	elif y_45_scale_m <= 1.95e+253:
		tmp = 0.25 * (b * math.log1p(math.expm1((y_45_scale_m * 4.0))))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.15e+30)
		tmp = Float64(x_45_scale_m * a);
	elseif (y_45_scale_m <= 1.95e+253)
		tmp = Float64(0.25 * Float64(b * log1p(expm1(Float64(y_45_scale_m * 4.0)))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.15e+30], N[(x$45$scale$95$m * a), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.95e+253], N[(0.25 * N[(b * N[Log[1 + N[(Exp[N[(y$45$scale$95$m * 4.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 2.15e30

   1. Initial program 2.8%

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

   3. Simplified 2.3%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 22.8%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 22.8%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 22.7%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 22.7%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 22.7%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 22.7%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 22.4%

         \[\leadsto \color{blue}{\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \cdot \sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}} \]

      2. pow2 22.4%

         \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}\right)}^{2}} \]

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)}\right)}^{2}} \]
   10. Step-by-step derivation

   11. Simplified 22.0%

       \[\leadsto \color{blue}{{\left(\sqrt{0.25 \cdot \left(x-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{2}} \]

   12. Taylor expanded in angle around 0 16.6%

       \[\leadsto \color{blue}{a \cdot x-scale} \]

   if 2.15e30 < y-scale < 1.9500000000000001e253

   1. Initial program 2.1%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in angle around 0 24.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 24.5%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 24.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 29.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)}\right) \]

      2. *-commutative 29.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale}\right)\right)\right) \]

      3. sqrt-unprod 29.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 29.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 29.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 29.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(4 \cdot y-scale\right)\right)}\right) \]

   if 1.9500000000000001e253 < y-scale

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 50.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   10. Taylor expanded in b around 0 50.2%

       \[\leadsto \color{blue}{b \cdot y-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 1.95 \cdot 10^{+253}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-77}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+104}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 4e-77)
   (* x-scale_m a)
   (if (<= b 7.8e+104)
     (* y-scale_m b)
     (* 0.25 (log1p (expm1 (* b (* y-scale_m 4.0))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 4e-77) {
		tmp = x_45_scale_m * a;
	} else if (b <= 7.8e+104) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * log1p(expm1((b * (y_45_scale_m * 4.0))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 4e-77) {
		tmp = x_45_scale_m * a;
	} else if (b <= 7.8e+104) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * Math.log1p(Math.expm1((b * (y_45_scale_m * 4.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 4e-77:
		tmp = x_45_scale_m * a
	elif b <= 7.8e+104:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * math.log1p(math.expm1((b * (y_45_scale_m * 4.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 4e-77)
		tmp = Float64(x_45_scale_m * a);
	elseif (b <= 7.8e+104)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * log1p(expm1(Float64(b * Float64(y_45_scale_m * 4.0)))));
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 4e-77], N[(x$45$scale$95$m * a), $MachinePrecision], If[LessEqual[b, 7.8e+104], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[Log[1 + N[(Exp[N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.9999999999999997e-77

   1. Initial program 0.6%

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

   3. Simplified 0.8%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 21.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 21.6%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 21.6%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 21.6%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 21.6%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 21.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \color{blue}{\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \cdot \sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}} \]

      2. pow2 21.3%

         \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}\right)}^{2}} \]

   9. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)}\right)}^{2}} \]
   10. Step-by-step derivation

   11. Simplified 21.3%

       \[\leadsto \color{blue}{{\left(\sqrt{0.25 \cdot \left(x-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{2}} \]

   12. Taylor expanded in angle around 0 16.5%

       \[\leadsto \color{blue}{a \cdot x-scale} \]

   if 3.9999999999999997e-77 < b < 7.80000000000000033e104

   1. Initial program 10.5%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in angle around 0 9.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 9.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 9.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 9.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 9.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 9.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 9.3%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   10. Taylor expanded in b around 0 9.3%

       \[\leadsto \color{blue}{b \cdot y-scale} \]

   if 7.80000000000000033e104 < b

   1. Initial program 4.9%

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

   3. Simplified 4.8%

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

   5. Taylor expanded in angle around 0 38.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 38.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 38.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 41.9%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)} \]

      2. *-commutative 41.9%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \color{blue}{\left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale\right)}\right)\right) \]

      3. sqrt-unprod 41.9%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 41.9%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 41.9%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 41.9%

      \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(4 \cdot y-scale\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-77}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+104}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.3% accurate, 344.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1.5e-77) (* x-scale_m a) (* y-scale_m b)))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.5e-77) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 1.5d-77) then
        tmp = x_45scale_m * a
    else
        tmp = y_45scale_m * b
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.5e-77) {
		tmp = x_45_scale_m * a;
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.5e-77:
		tmp = x_45_scale_m * a
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.5e-77)
		tmp = Float64(x_45_scale_m * a);
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.5e-77)
		tmp = x_45_scale_m * a;
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.5e-77], N[(x$45$scale$95$m * a), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000008e-77

   1. Initial program 0.6%

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

   3. Simplified 0.8%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 21.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 21.6%

         \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

      2. associate-*l* 21.6%

         \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

      3. distribute-lft-out 21.6%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      4. fma-define 21.6%

         \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   7. Simplified 21.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 21.3%

         \[\leadsto \color{blue}{\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \cdot \sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}} \]

      2. pow2 21.3%

         \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}\right)}^{2}} \]

   9. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)}\right)}^{2}} \]
   10. Step-by-step derivation

   11. Simplified 21.3%

       \[\leadsto \color{blue}{{\left(\sqrt{0.25 \cdot \left(x-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{2}} \]

   12. Taylor expanded in angle around 0 16.5%

       \[\leadsto \color{blue}{a \cdot x-scale} \]

   if 1.50000000000000008e-77 < b

   1. Initial program 7.2%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in angle around 0 26.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 26.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 26.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 26.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 26.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 26.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 26.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   10. Taylor expanded in b around 0 26.8%

       \[\leadsto \color{blue}{b \cdot y-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 17.8% accurate, 919.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ x-scale\_m \cdot a \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m) :precision binary64 (* x-scale_m a))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return x_45_scale_m * a;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = x_45scale_m * a
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return x_45_scale_m * a;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	return x_45_scale_m * a

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	return Float64(x_45_scale_m * a)
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = x_45_scale_m * a;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(x$45$scale$95$m * a), $MachinePrecision]

Derivation

1. Initial program 2.5%

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

3. Simplified 1.8%

   \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}}} \]
4. Add Preprocessing

5. Taylor expanded in y-scale around 0 23.8%

   \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 23.8%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

   2. associate-*l* 23.8%

      \[\leadsto -0.25 \cdot \left(-\color{blue}{x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right) \]

   3. distribute-lft-out 23.8%

      \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

   4. fma-define 23.8%

      \[\leadsto -0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

7. Simplified 23.8%

   \[\leadsto -0.25 \cdot \color{blue}{\left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 23.5%

      \[\leadsto \color{blue}{\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)} \cdot \sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}} \]

   2. pow2 23.5%

      \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(-x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\right)}\right)}^{2}} \]

9. Applied egg-rr 23.6%

   \[\leadsto \color{blue}{{\left(\sqrt{-0.25 \cdot \left(\left(-x-scale\right) \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2} + {\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}\right)\right)}\right)}\right)}^{2}} \]
10. Step-by-step derivation

11. Simplified 23.2%

    \[\leadsto \color{blue}{{\left(\sqrt{0.25 \cdot \left(x-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)}\right)}^{2}} \]

12. Taylor expanded in angle around 0 14.5%

    \[\leadsto \color{blue}{a \cdot x-scale} \]

13. Final simplification 14.5%

    \[\leadsto x-scale \cdot a \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024121 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (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)) (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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))