a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 56.8%

Time: 59.1s

Alternatives: 8

Speedup: 919.0×

• 30.7% 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: 2.8% 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: 56.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(angle \cdot 0.005555555555555556\right) \cdot \pi\\ \mathbf{if}\;y-scale\_m \leq 3.6 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos t\_1, b \cdot \sin t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\mathsf{hypot}\left(\left|a \cdot \sin t\_0\right|, \sqrt{2 \cdot {\left(b \cdot \cos t\_0\right)}^{2}}\right) \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

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 (* (* angle 0.005555555555555556) PI)))
   (if (<= y-scale_m 3.6e+80)
     (* 0.25 (* x-scale_m (* 4.0 (hypot (* a (cos t_1)) (* b (sin t_1))))))
     (*
      -0.25
      (*
       (hypot (fabs (* a (sin t_0))) (sqrt (* 2.0 (pow (* b (cos t_0)) 2.0))))
       (* y-scale_m (- (sqrt 8.0))))))))

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 t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (angle * 0.005555555555555556) * ((double) M_PI);
	double tmp;
	if (y_45_scale_m <= 3.6e+80) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * hypot((a * cos(t_1)), (b * sin(t_1)))));
	} else {
		tmp = -0.25 * (hypot(fabs((a * sin(t_0))), sqrt((2.0 * pow((b * cos(t_0)), 2.0)))) * (y_45_scale_m * -sqrt(8.0)));
	}
	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 t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (angle * 0.005555555555555556) * Math.PI;
	double tmp;
	if (y_45_scale_m <= 3.6e+80) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * Math.hypot((a * Math.cos(t_1)), (b * Math.sin(t_1)))));
	} else {
		tmp = -0.25 * (Math.hypot(Math.abs((a * Math.sin(t_0))), Math.sqrt((2.0 * Math.pow((b * Math.cos(t_0)), 2.0)))) * (y_45_scale_m * -Math.sqrt(8.0)));
	}
	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):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = (angle * 0.005555555555555556) * math.pi
	tmp = 0
	if y_45_scale_m <= 3.6e+80:
		tmp = 0.25 * (x_45_scale_m * (4.0 * math.hypot((a * math.cos(t_1)), (b * math.sin(t_1)))))
	else:
		tmp = -0.25 * (math.hypot(math.fabs((a * math.sin(t_0))), math.sqrt((2.0 * math.pow((b * math.cos(t_0)), 2.0)))) * (y_45_scale_m * -math.sqrt(8.0)))
	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)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(angle * 0.005555555555555556) * pi)
	tmp = 0.0
	if (y_45_scale_m <= 3.6e+80)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(4.0 * hypot(Float64(a * cos(t_1)), Float64(b * sin(t_1))))));
	else
		tmp = Float64(-0.25 * Float64(hypot(abs(Float64(a * sin(t_0))), sqrt(Float64(2.0 * (Float64(b * cos(t_0)) ^ 2.0)))) * Float64(y_45_scale_m * Float64(-sqrt(8.0)))));
	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)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = (angle * 0.005555555555555556) * pi;
	tmp = 0.0;
	if (y_45_scale_m <= 3.6e+80)
		tmp = 0.25 * (x_45_scale_m * (4.0 * hypot((a * cos(t_1)), (b * sin(t_1)))));
	else
		tmp = -0.25 * (hypot(abs((a * sin(t_0))), sqrt((2.0 * ((b * cos(t_0)) ^ 2.0)))) * (y_45_scale_m * -sqrt(8.0)));
	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_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.6e+80], N[(0.25 * N[(x$45$scale$95$m * N[(4.0 * N[Sqrt[N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[Sqrt[N[Abs[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + N[Sqrt[N[(2.0 * N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.59999999999999995e80

   1. Initial program 2.2%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in y-scale around 0 27.5%

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

      1. associate-*l* 27.5%

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

      2. distribute-lft-out 27.5%

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

      3. fma-define 27.5%

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

   7. Simplified 27.5%

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

   9. Applied egg-rr 28.6%

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

       1. associate-*r* 28.6%

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

       2. metadata-eval 28.6%

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

   11. Simplified 28.6%

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

       1. sqrt-prod 28.6%

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

       2. metadata-eval 28.6%

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

       3. unpow2 28.6%

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

       4. unpow2 28.6%

          \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \sqrt{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) + \color{blue}{\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}\right)\right) \]

       5. hypot-define 25.7%

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

       6. associate-*r* 25.7%

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

       7. associate-*r* 25.7%

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

   13. Applied egg-rr 25.7%

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

   if 3.59999999999999995e80 < y-scale

   1. Initial program 7.1%

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

   3. Simplified 4.9%

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

   5. Taylor expanded in a around 0 4.9%

      \[\leadsto -0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\color{blue}{\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Taylor expanded in x-scale around 0 65.1%

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

      1. +-commutative 65.1%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]

      2. add-sqr-sqrt 65.1%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{\sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]

      3. add-sqr-sqrt 65.1%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} + \color{blue}{\sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \cdot \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}}\right)\right) \]

      4. hypot-define 65.1%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}, \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)}\right)\right) \]

      5. pow-prod-down 67.4%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}, \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)\right) \]

      6. associate-*r* 67.5%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\sqrt{{\left(a \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}^{2}}, \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)\right) \]

   8. Applied egg-rr 67.2%

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

      1. associate-*r* 67.2%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\sqrt{{\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2}}, \sqrt{2 \cdot {\left(b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]

      2. unpow2 67.2%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}, \sqrt{2 \cdot {\left(b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]

      3. rem-sqrt-square 67.8%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\color{blue}{\left|a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|}, \sqrt{2 \cdot {\left(b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]

      4. *-commutative 67.8%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\left|a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|, \sqrt{2 \cdot {\color{blue}{\left(\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}}^{2}}\right)\right)\right) \]

      5. associate-*r* 67.9%

         \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\left|a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|, \sqrt{2 \cdot {\left(\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot b\right)}^{2}}\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\mathsf{hypot}\left(\left|a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|, \sqrt{2 \cdot {\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.6 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), b \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\mathsf{hypot}\left(\left|a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right|, \sqrt{2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \pi\\ \mathbf{if}\;y-scale\_m \leq 1.26 \cdot 10^{+81}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2 \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

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 (* (* angle 0.005555555555555556) PI)))
   (if (<= y-scale_m 1.26e+81)
     (* 0.25 (* x-scale_m (* 4.0 (hypot (* a (cos t_0)) (* b (sin t_0))))))
     (*
      -0.25
      (*
       (sqrt
        (+
         (* 2.0 (pow b 2.0))
         (*
          (pow a 2.0)
          (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0))))
       (* y-scale_m (- (sqrt 8.0))))))))

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 t_0 = (angle * 0.005555555555555556) * ((double) M_PI);
	double tmp;
	if (y_45_scale_m <= 1.26e+81) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * hypot((a * cos(t_0)), (b * sin(t_0)))));
	} else {
		tmp = -0.25 * (sqrt(((2.0 * pow(b, 2.0)) + (pow(a, 2.0) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0)))) * (y_45_scale_m * -sqrt(8.0)));
	}
	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 t_0 = (angle * 0.005555555555555556) * Math.PI;
	double tmp;
	if (y_45_scale_m <= 1.26e+81) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0)))));
	} else {
		tmp = -0.25 * (Math.sqrt(((2.0 * Math.pow(b, 2.0)) + (Math.pow(a, 2.0) * Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 2.0)))) * (y_45_scale_m * -Math.sqrt(8.0)));
	}
	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):
	t_0 = (angle * 0.005555555555555556) * math.pi
	tmp = 0
	if y_45_scale_m <= 1.26e+81:
		tmp = 0.25 * (x_45_scale_m * (4.0 * math.hypot((a * math.cos(t_0)), (b * math.sin(t_0)))))
	else:
		tmp = -0.25 * (math.sqrt(((2.0 * math.pow(b, 2.0)) + (math.pow(a, 2.0) * math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 2.0)))) * (y_45_scale_m * -math.sqrt(8.0)))
	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)
	t_0 = Float64(Float64(angle * 0.005555555555555556) * pi)
	tmp = 0.0
	if (y_45_scale_m <= 1.26e+81)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(4.0 * hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))))));
	else
		tmp = Float64(-0.25 * Float64(sqrt(Float64(Float64(2.0 * (b ^ 2.0)) + Float64((a ^ 2.0) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0)))) * Float64(y_45_scale_m * Float64(-sqrt(8.0)))));
	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)
	t_0 = (angle * 0.005555555555555556) * pi;
	tmp = 0.0;
	if (y_45_scale_m <= 1.26e+81)
		tmp = 0.25 * (x_45_scale_m * (4.0 * hypot((a * cos(t_0)), (b * sin(t_0)))));
	else
		tmp = -0.25 * (sqrt(((2.0 * (b ^ 2.0)) + ((a ^ 2.0) * (sin((0.005555555555555556 * (angle * pi))) ^ 2.0)))) * (y_45_scale_m * -sqrt(8.0)));
	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_] := Block[{t$95$0 = N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.26e+81], N[(0.25 * N[(x$45$scale$95$m * N[(4.0 * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[Sqrt[N[(N[(2.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.25999999999999996e81

   1. Initial program 2.2%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in y-scale around 0 27.5%

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

      1. associate-*l* 27.5%

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

      2. distribute-lft-out 27.5%

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

      3. fma-define 27.5%

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

   7. Simplified 27.5%

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

   9. Applied egg-rr 28.6%

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

       1. associate-*r* 28.6%

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

       2. metadata-eval 28.6%

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

   11. Simplified 28.6%

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

       1. sqrt-prod 28.6%

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

       2. metadata-eval 28.6%

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

       3. unpow2 28.6%

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

       4. unpow2 28.6%

          \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \sqrt{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) + \color{blue}{\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}\right)\right) \]

       5. hypot-define 25.7%

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

       6. associate-*r* 25.7%

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

       7. associate-*r* 25.7%

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

   13. Applied egg-rr 25.7%

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

   if 1.25999999999999996e81 < y-scale

   1. Initial program 7.1%

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

   3. Simplified 4.9%

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

   5. Taylor expanded in a around 0 4.9%

      \[\leadsto -0.25 \cdot \frac{\sqrt{\left(b \cdot \left({a}^{2} \cdot \left(-b\right)\right)\right) \cdot \left(\left(8 \cdot \left(\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} + \mathsf{hypot}\left(\color{blue}{\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}} - \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{{y-scale}^{2}}, \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\left(a \cdot {b}^{2}\right) \cdot \frac{-a}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Taylor expanded in x-scale around 0 65.1%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)} \]

   7. Taylor expanded in angle around 0 64.1%

      \[\leadsto -0.25 \cdot \left(-1 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot {b}^{2}} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.26 \cdot 10^{+81}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), b \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2 \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}} \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b \cdot \left(y-scale\_m \cdot 4\right)\\ t_1 := \left(angle \cdot 0.005555555555555556\right) \cdot \pi\\ \mathbf{if}\;y-scale\_m \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos t\_1, b \cdot \sin t\_1\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 7 \cdot 10^{+205}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{t\_0 \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* b (* y-scale_m 4.0)))
        (t_1 (* (* angle 0.005555555555555556) PI)))
   (if (<= y-scale_m 4.6e+141)
     (* 0.25 (* x-scale_m (* 4.0 (hypot (* a (cos t_1)) (* b (sin t_1))))))
     (if (<= y-scale_m 7e+205)
       (* y-scale_m b)
       (* 0.25 (cbrt (* t_0 (* t_0 t_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 t_0 = b * (y_45_scale_m * 4.0);
	double t_1 = (angle * 0.005555555555555556) * ((double) M_PI);
	double tmp;
	if (y_45_scale_m <= 4.6e+141) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * hypot((a * cos(t_1)), (b * sin(t_1)))));
	} else if (y_45_scale_m <= 7e+205) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * cbrt((t_0 * (t_0 * t_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 t_0 = b * (y_45_scale_m * 4.0);
	double t_1 = (angle * 0.005555555555555556) * Math.PI;
	double tmp;
	if (y_45_scale_m <= 4.6e+141) {
		tmp = 0.25 * (x_45_scale_m * (4.0 * Math.hypot((a * Math.cos(t_1)), (b * Math.sin(t_1)))));
	} else if (y_45_scale_m <= 7e+205) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * Math.cbrt((t_0 * (t_0 * t_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)
	t_0 = Float64(b * Float64(y_45_scale_m * 4.0))
	t_1 = Float64(Float64(angle * 0.005555555555555556) * pi)
	tmp = 0.0
	if (y_45_scale_m <= 4.6e+141)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(4.0 * hypot(Float64(a * cos(t_1)), Float64(b * sin(t_1))))));
	elseif (y_45_scale_m <= 7e+205)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * cbrt(Float64(t_0 * Float64(t_0 * t_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_] := Block[{t$95$0 = N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 4.6e+141], N[(0.25 * N[(x$45$scale$95$m * N[(4.0 * N[Sqrt[N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 7e+205], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 4.6000000000000003e141

   1. Initial program 2.2%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in y-scale around 0 27.2%

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

      1. associate-*l* 27.2%

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

      2. distribute-lft-out 27.2%

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

      3. fma-define 27.2%

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

   7. Simplified 27.2%

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

   9. Applied egg-rr 28.2%

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

       1. associate-*r* 28.2%

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

       2. metadata-eval 28.2%

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

   11. Simplified 28.2%

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

       1. sqrt-prod 28.2%

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

       2. metadata-eval 28.2%

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

       3. unpow2 28.2%

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

       4. unpow2 28.2%

          \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \sqrt{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) + \color{blue}{\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}}\right)\right) \]

       5. hypot-define 25.5%

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

       6. associate-*r* 25.4%

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

       7. associate-*r* 25.4%

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

   13. Applied egg-rr 25.4%

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

   if 4.6000000000000003e141 < y-scale < 6.9999999999999996e205

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in angle around 0 36.0%

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

      1. *-commutative 36.0%

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

   7. Simplified 36.0%

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

      1. pow1 36.0%

         \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 36.4%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

      3. metadata-eval 36.4%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

      4. metadata-eval 36.4%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 36.4%

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

   11. Simplified 36.4%

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

   12. Taylor expanded in b around 0 36.4%

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

   if 6.9999999999999996e205 < y-scale

   1. Initial program 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in angle around 0 19.1%

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

      1. *-commutative 19.1%

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

   7. Simplified 19.1%

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

      1. add-cbrt-cube 23.3%

         \[\leadsto 0.25 \cdot \color{blue}{\sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)}} \]

      2. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      3. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      4. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      5. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      6. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      7. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      8. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)} \]

      9. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)} \]

      10. metadata-eval 23.3%

          \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)} \]

   9. Applied egg-rr 23.3%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(a \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), b \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{+205}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \sqrt{{a}^{2} \cdot 16}\right)\\ \end{array} \end{array} \]

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 (<= a 2.1e+37)
   (* y-scale_m b)
   (* 0.25 (* x-scale_m (sqrt (* (pow a 2.0) 16.0))))))

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 (a <= 2.1e+37) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (x_45_scale_m * sqrt((pow(a, 2.0) * 16.0)));
	}
	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 (a <= 2.1d+37) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * (x_45scale_m * sqrt(((a ** 2.0d0) * 16.0d0)))
    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 (a <= 2.1e+37) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (x_45_scale_m * Math.sqrt((Math.pow(a, 2.0) * 16.0)));
	}
	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 a <= 2.1e+37:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * (x_45_scale_m * math.sqrt((math.pow(a, 2.0) * 16.0)))
	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 (a <= 2.1e+37)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(x_45_scale_m * sqrt(Float64((a ^ 2.0) * 16.0))));
	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 (a <= 2.1e+37)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * (x_45_scale_m * sqrt(((a ^ 2.0) * 16.0)));
	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[a, 2.1e+37], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[N[(N[Power[a, 2.0], $MachinePrecision] * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1000000000000001e37

   1. Initial program 1.8%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in angle around 0 18.5%

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

      1. *-commutative 18.5%

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

   7. Simplified 18.5%

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

      1. pow1 18.5%

         \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 18.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

      3. metadata-eval 18.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

      4. metadata-eval 18.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 18.6%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 18.6%

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

   11. Simplified 18.6%

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

   12. Taylor expanded in b around 0 18.6%

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

   if 2.1000000000000001e37 < a

   1. Initial program 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in y-scale around 0 37.7%

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

      1. associate-*l* 37.8%

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

      2. distribute-lft-out 37.8%

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

      3. fma-define 37.8%

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

   7. Simplified 37.7%

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

   9. Applied egg-rr 39.4%

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

       1. associate-*r* 39.4%

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

       2. metadata-eval 39.4%

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

   11. Simplified 39.4%

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

   12. Taylor expanded in angle around 0 35.8%

       \[\leadsto 0.25 \cdot \left(x-scale \cdot \sqrt{\color{blue}{16 \cdot {a}^{2}}}\right) \]
   13. Step-by-step derivation

       1. *-commutative 35.8%

          \[\leadsto 0.25 \cdot \left(x-scale \cdot \sqrt{\color{blue}{{a}^{2} \cdot 16}}\right) \]

   14. Simplified 35.8%

       \[\leadsto 0.25 \cdot \left(x-scale \cdot \sqrt{\color{blue}{{a}^{2} \cdot 16}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{{a}^{2} \cdot 16}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 22.8% accurate, 21.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;y-scale\_m \leq 1.45 \cdot 10^{+88}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{elif}\;y-scale\_m \leq 2.5 \cdot 10^{+206}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{t\_0 \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* b (* y-scale_m 4.0))))
   (if (<= y-scale_m 1.45e+88)
     (* x-scale_m a)
     (if (<= y-scale_m 2.5e+206)
       (* y-scale_m b)
       (* 0.25 (cbrt (* t_0 (* t_0 t_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 t_0 = b * (y_45_scale_m * 4.0);
	double tmp;
	if (y_45_scale_m <= 1.45e+88) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 2.5e+206) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * cbrt((t_0 * (t_0 * t_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 t_0 = b * (y_45_scale_m * 4.0);
	double tmp;
	if (y_45_scale_m <= 1.45e+88) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 2.5e+206) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * Math.cbrt((t_0 * (t_0 * t_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)
	t_0 = Float64(b * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (y_45_scale_m <= 1.45e+88)
		tmp = Float64(x_45_scale_m * a);
	elseif (y_45_scale_m <= 2.5e+206)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * cbrt(Float64(t_0 * Float64(t_0 * t_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_] := Block[{t$95$0 = N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.45e+88], N[(x$45$scale$95$m * a), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.5e+206], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.45e88

   1. Initial program 2.2%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in y-scale around 0 27.5%

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

      1. associate-*l* 27.5%

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

      2. distribute-lft-out 27.5%

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

      3. fma-define 27.5%

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

   7. Simplified 27.5%

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

   9. Applied egg-rr 28.5%

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

       1. associate-*r* 28.5%

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

       2. metadata-eval 28.5%

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

   11. Simplified 28.5%

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

   12. Taylor expanded in angle around 0 22.6%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   13. Step-by-step derivation

       1. *-commutative 22.6%

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

   14. Simplified 22.6%

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

   if 1.45e88 < y-scale < 2.5000000000000001e206

   1. Initial program 1.1%

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

   3. Simplified 1.1%

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

   5. Taylor expanded in angle around 0 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. pow1 30.0%

         \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 30.2%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

      3. metadata-eval 30.2%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

      4. metadata-eval 30.2%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 30.2%

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

   11. Simplified 30.2%

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

   12. Taylor expanded in b around 0 30.2%

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

   if 2.5000000000000001e206 < y-scale

   1. Initial program 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in angle around 0 19.1%

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

      1. *-commutative 19.1%

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

   7. Simplified 19.1%

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

      1. add-cbrt-cube 23.3%

         \[\leadsto 0.25 \cdot \color{blue}{\sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)}} \]

      2. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      3. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      4. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      5. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      6. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      7. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]

      8. sqrt-unprod 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)} \]

      9. metadata-eval 23.3%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)} \]

      10. metadata-eval 23.3%

          \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)} \]

   9. Applied egg-rr 23.3%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.45 \cdot 10^{+88}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 2.5 \cdot 10^{+206}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 23.0% accurate, 22.1× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;x-scale\_m \cdot a\\ \mathbf{elif}\;y-scale\_m \leq 9.8 \cdot 10^{+241}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \sqrt[3]{\left(y-scale\_m \cdot 4\right) \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(y-scale\_m \cdot 4\right)\right)}\right)\\ \end{array} \end{array} \]

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 (<= y-scale_m 3.5e+88)
   (* x-scale_m a)
   (if (<= y-scale_m 9.8e+241)
     (* y-scale_m b)
     (*
      0.25
      (*
       b
       (cbrt
        (* (* y-scale_m 4.0) (* (* y-scale_m 4.0) (* 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 (y_45_scale_m <= 3.5e+88) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 9.8e+241) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * cbrt(((y_45_scale_m * 4.0) * ((y_45_scale_m * 4.0) * (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 (y_45_scale_m <= 3.5e+88) {
		tmp = x_45_scale_m * a;
	} else if (y_45_scale_m <= 9.8e+241) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * Math.cbrt(((y_45_scale_m * 4.0) * ((y_45_scale_m * 4.0) * (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 (y_45_scale_m <= 3.5e+88)
		tmp = Float64(x_45_scale_m * a);
	elseif (y_45_scale_m <= 9.8e+241)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(b * cbrt(Float64(Float64(y_45_scale_m * 4.0) * Float64(Float64(y_45_scale_m * 4.0) * 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[y$45$scale$95$m, 3.5e+88], N[(x$45$scale$95$m * a), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 9.8e+241], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(b * N[Power[N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 3.4999999999999998e88

   1. Initial program 2.2%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in y-scale around 0 27.5%

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

      1. associate-*l* 27.5%

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

      2. distribute-lft-out 27.5%

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

      3. fma-define 27.5%

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

   7. Simplified 27.5%

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

   9. Applied egg-rr 28.5%

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

       1. associate-*r* 28.5%

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

       2. metadata-eval 28.5%

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

   11. Simplified 28.5%

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

   12. Taylor expanded in angle around 0 22.6%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   13. Step-by-step derivation

       1. *-commutative 22.6%

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

   14. Simplified 22.6%

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

   if 3.4999999999999998e88 < y-scale < 9.79999999999999943e241

   1. Initial program 4.2%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in angle around 0 28.3%

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

      1. *-commutative 28.3%

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

   7. Simplified 28.3%

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

      1. pow1 28.3%

         \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 28.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

      3. metadata-eval 28.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

      4. metadata-eval 28.6%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 28.6%

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

   11. Simplified 28.6%

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

   12. Taylor expanded in b around 0 28.6%

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

   if 9.79999999999999943e241 < y-scale

   1. Initial program 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in angle around 0 14.7%

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

      1. *-commutative 14.7%

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

   7. Simplified 14.7%

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

      1. add-cbrt-cube 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\sqrt[3]{\left(\left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}}\right) \]

      2. sqrt-unprod 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      3. metadata-eval 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      4. metadata-eval 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \color{blue}{4}\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      5. sqrt-unprod 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      6. metadata-eval 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      7. metadata-eval 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \]

      8. sqrt-unprod 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)}\right) \]

      9. metadata-eval 31.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)}\right) \]

      10. metadata-eval 31.6%

          \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)}\right) \]

   9. Applied egg-rr 31.6%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 9.8 \cdot 10^{+241}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \sqrt[3]{\left(y-scale \cdot 4\right) \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 22.8% accurate, 344.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 4000000000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

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 (<= x-scale_m 4000000000000.0) (* y-scale_m b) (* 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) {
	double tmp;
	if (x_45_scale_m <= 4000000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	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 (x_45scale_m <= 4000000000000.0d0) then
        tmp = y_45scale_m * b
    else
        tmp = x_45scale_m * a
    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 (x_45_scale_m <= 4000000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	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 x_45_scale_m <= 4000000000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = x_45_scale_m * a
	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 (x_45_scale_m <= 4000000000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(x_45_scale_m * a);
	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 (x_45_scale_m <= 4000000000000.0)
		tmp = y_45_scale_m * b;
	else
		tmp = x_45_scale_m * a;
	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[x$45$scale$95$m, 4000000000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4e12

   1. Initial program 3.5%

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

   3. Simplified 3.6%

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

   5. Taylor expanded in angle around 0 18.0%

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

      1. *-commutative 18.0%

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

   7. Simplified 18.0%

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

      1. pow1 18.0%

         \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

      2. sqrt-unprod 18.1%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

      3. metadata-eval 18.1%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

      4. metadata-eval 18.1%

         \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   9. Applied egg-rr 18.1%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 18.1%

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

   11. Simplified 18.1%

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

   12. Taylor expanded in b around 0 18.1%

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

   if 4e12 < x-scale

   1. Initial program 1.8%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in y-scale around 0 66.3%

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

      1. associate-*l* 66.3%

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

      2. distribute-lft-out 66.3%

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

      3. fma-define 66.3%

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

   7. Simplified 66.3%

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

   9. Applied egg-rr 69.5%

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

       1. associate-*r* 69.5%

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

       2. metadata-eval 69.5%

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

   11. Simplified 69.5%

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

   12. Taylor expanded in angle around 0 28.7%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   13. Step-by-step derivation

       1. *-commutative 28.7%

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

   14. Simplified 28.7%

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

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4000000000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 17.3% accurate, 919.0× speedup

Math

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

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 (* 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) {
	return y_45_scale_m * b;
}

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 = y_45scale_m * b
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 y_45_scale_m * b;
}

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 y_45_scale_m * b

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(y_45_scale_m * b)
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 = y_45_scale_m * b;
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[(y$45$scale$95$m * b), $MachinePrecision]

Derivation

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 3.2%

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

5. Taylor expanded in angle around 0 15.7%

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

   1. *-commutative 15.7%

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

7. Simplified 15.7%

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

   1. pow1 15.7%

      \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)}^{1}} \]

   2. sqrt-unprod 15.8%

      \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right)}^{1} \]

   3. metadata-eval 15.8%

      \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

   4. metadata-eval 15.8%

      \[\leadsto {\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

9. Applied egg-rr 15.8%

   \[\leadsto \color{blue}{{\left(0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}^{1}} \]
10. Step-by-step derivation

    1. unpow1 15.8%

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

11. Simplified 15.8%

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

12. Taylor expanded in b around 0 15.8%

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

13. Final simplification 15.8%

    \[\leadsto y-scale \cdot b \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))))