b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 44.2%

Time: 1.5min

Alternatives: 5

Speedup: 919.0×

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: 0.1% 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: 44.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 6.5 \cdot 10^{-228}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \sqrt[3]{\left(x-scale\_m \cdot 4\right) \cdot \left(\left(x-scale\_m \cdot 4\right) \cdot \left(x-scale\_m \cdot 4\right)\right)}\\ \mathbf{elif}\;x-scale\_m \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;x-scale\_m \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(\left(y-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 6.5e-228)
   (*
    (* 0.25 a_m)
    (cbrt (* (* x-scale_m 4.0) (* (* x-scale_m 4.0) (* x-scale_m 4.0)))))
   (if (<= x-scale_m 3.2e+39)
     (* x-scale_m a_m)
     (* 0.25 (* b_m (* (* y-scale_m (sqrt 2.0)) (sqrt 8.0)))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 6.5e-228) {
		tmp = (0.25 * a_m) * cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else if (x_45_scale_m <= 3.2e+39) {
		tmp = x_45_scale_m * a_m;
	} else {
		tmp = 0.25 * (b_m * ((y_45_scale_m * sqrt(2.0)) * sqrt(8.0)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 6.5e-228) {
		tmp = (0.25 * a_m) * Math.cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else if (x_45_scale_m <= 3.2e+39) {
		tmp = x_45_scale_m * a_m;
	} else {
		tmp = 0.25 * (b_m * ((y_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 6.5e-228)
		tmp = Float64(Float64(0.25 * a_m) * cbrt(Float64(Float64(x_45_scale_m * 4.0) * Float64(Float64(x_45_scale_m * 4.0) * Float64(x_45_scale_m * 4.0)))));
	elseif (x_45_scale_m <= 3.2e+39)
		tmp = Float64(x_45_scale_m * a_m);
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(Float64(y_45_scale_m * sqrt(2.0)) * sqrt(8.0))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 6.5e-228], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[Power[N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 3.2e+39], N[(x$45$scale$95$m * a$95$m), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(N[(y$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 6.50000000000000029e-228

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 25.4%

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

      1. associate-*r* 25.4%

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

      2. associate-*r* 25.5%

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

   5. Simplified 25.5%

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

      1. associate-*l* 25.4%

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

      2. *-commutative 25.4%

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

      3. add-cbrt-cube 29.9%

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

      4. sqrt-unprod 29.9%

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

      5. metadata-eval 29.9%

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

      6. metadata-eval 29.9%

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

      7. sqrt-unprod 29.9%

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

      8. metadata-eval 29.9%

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

      9. metadata-eval 29.9%

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

      10. sqrt-unprod 29.9%

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

      11. metadata-eval 29.9%

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

      12. metadata-eval 29.9%

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

   7. Applied egg-rr 29.9%

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

   if 6.50000000000000029e-228 < x-scale < 3.19999999999999993e39

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 27.6%

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

      1. associate-*r* 27.6%

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

      2. associate-*r* 27.6%

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

   5. Simplified 27.6%

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

      1. pow1 27.6%

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

      2. associate-*l* 27.6%

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

      3. *-commutative 27.6%

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

      4. sqrt-unprod 27.8%

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

      5. metadata-eval 27.8%

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

      6. metadata-eval 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. unpow1 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in a around 0 27.8%

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

   if 3.19999999999999993e39 < x-scale

   1. Initial program 0.0%

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

   3. Taylor expanded in y-scale around 0 4.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}\right)}{{x-scale}^{2}}}{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)} \]

   4. Taylor expanded in angle around 0 10.6%

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

      1. mul-1-neg 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in angle around 0 28.9%

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

      1. associate-*r* 28.9%

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

   9. Simplified 28.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 6.5 \cdot 10^{-228}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \sqrt[3]{\left(x-scale \cdot 4\right) \cdot \left(\left(x-scale \cdot 4\right) \cdot \left(x-scale \cdot 4\right)\right)}\\ \mathbf{elif}\;x-scale \leq 3.2 \cdot 10^{+39}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 44.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.65 \cdot 10^{-226}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \sqrt[3]{\left(x-scale\_m \cdot 4\right) \cdot \left(\left(x-scale\_m \cdot 4\right) \cdot \left(x-scale\_m \cdot 4\right)\right)}\\ \mathbf{elif}\;x-scale\_m \leq 9 \cdot 10^{+39}:\\ \;\;\;\;x-scale\_m \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 1.65e-226)
   (*
    (* 0.25 a_m)
    (cbrt (* (* x-scale_m 4.0) (* (* x-scale_m 4.0) (* x-scale_m 4.0)))))
   (if (<= x-scale_m 9e+39)
     (* x-scale_m a_m)
     (* 0.25 (* b_m (* y-scale_m (* (sqrt 2.0) (sqrt 8.0))))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.65e-226) {
		tmp = (0.25 * a_m) * cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else if (x_45_scale_m <= 9e+39) {
		tmp = x_45_scale_m * a_m;
	} else {
		tmp = 0.25 * (b_m * (y_45_scale_m * (sqrt(2.0) * sqrt(8.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.65e-226) {
		tmp = (0.25 * a_m) * Math.cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else if (x_45_scale_m <= 9e+39) {
		tmp = x_45_scale_m * a_m;
	} else {
		tmp = 0.25 * (b_m * (y_45_scale_m * (Math.sqrt(2.0) * Math.sqrt(8.0))));
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 1.65e-226)
		tmp = Float64(Float64(0.25 * a_m) * cbrt(Float64(Float64(x_45_scale_m * 4.0) * Float64(Float64(x_45_scale_m * 4.0) * Float64(x_45_scale_m * 4.0)))));
	elseif (x_45_scale_m <= 9e+39)
		tmp = Float64(x_45_scale_m * a_m);
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * Float64(sqrt(2.0) * sqrt(8.0)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 1.65e-226], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[Power[N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 9e+39], N[(x$45$scale$95$m * a$95$m), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 1.65e-226

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 25.4%

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

      1. associate-*r* 25.4%

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

      2. associate-*r* 25.5%

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

   5. Simplified 25.5%

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

      1. associate-*l* 25.4%

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

      2. *-commutative 25.4%

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

      3. add-cbrt-cube 29.9%

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

      4. sqrt-unprod 29.9%

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

      5. metadata-eval 29.9%

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

      6. metadata-eval 29.9%

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

      7. sqrt-unprod 29.9%

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

      8. metadata-eval 29.9%

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

      9. metadata-eval 29.9%

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

      10. sqrt-unprod 29.9%

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

      11. metadata-eval 29.9%

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

      12. metadata-eval 29.9%

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

   7. Applied egg-rr 29.9%

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

   if 1.65e-226 < x-scale < 8.99999999999999991e39

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 27.6%

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

      1. associate-*r* 27.6%

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

      2. associate-*r* 27.6%

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

   5. Simplified 27.6%

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

      1. pow1 27.6%

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

      2. associate-*l* 27.6%

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

      3. *-commutative 27.6%

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

      4. sqrt-unprod 27.8%

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

      5. metadata-eval 27.8%

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

      6. metadata-eval 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. unpow1 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in a around 0 27.8%

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

   if 8.99999999999999991e39 < x-scale

   1. Initial program 0.0%

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

   3. Taylor expanded in y-scale around 0 4.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}\right)}{{x-scale}^{2}}}{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)} \]

   4. Taylor expanded in angle around 0 10.6%

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

      1. mul-1-neg 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in angle around 0 28.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.65 \cdot 10^{-226}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \sqrt[3]{\left(x-scale \cdot 4\right) \cdot \left(\left(x-scale \cdot 4\right) \cdot \left(x-scale \cdot 4\right)\right)}\\ \mathbf{elif}\;x-scale \leq 9 \cdot 10^{+39}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 8 \cdot 10^{-118}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \sqrt[3]{\left(x-scale\_m \cdot 4\right) \cdot \left(\left(x-scale\_m \cdot 4\right) \cdot \left(x-scale\_m \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 8e-118)
   (*
    (* 0.25 a_m)
    (cbrt (* (* x-scale_m 4.0) (* (* x-scale_m 4.0) (* x-scale_m 4.0)))))
   (* x-scale_m a_m)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 8e-118) {
		tmp = (0.25 * a_m) * cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 8e-118) {
		tmp = (0.25 * a_m) * Math.cbrt(((x_45_scale_m * 4.0) * ((x_45_scale_m * 4.0) * (x_45_scale_m * 4.0))));
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 8e-118)
		tmp = Float64(Float64(0.25 * a_m) * cbrt(Float64(Float64(x_45_scale_m * 4.0) * Float64(Float64(x_45_scale_m * 4.0) * Float64(x_45_scale_m * 4.0)))));
	else
		tmp = Float64(x_45_scale_m * a_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 8e-118], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[Power[N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999988e-118

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. associate-*r* 21.3%

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

   5. Simplified 21.3%

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

      1. associate-*l* 21.3%

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

      2. *-commutative 21.3%

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

      3. add-cbrt-cube 25.3%

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

      4. sqrt-unprod 25.3%

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

      5. metadata-eval 25.3%

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

      6. metadata-eval 25.3%

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

      7. sqrt-unprod 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. sqrt-unprod 25.3%

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

      11. metadata-eval 25.3%

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

      12. metadata-eval 25.3%

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

   7. Applied egg-rr 25.3%

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

   if 7.99999999999999988e-118 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 23.6%

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

      1. associate-*r* 23.6%

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

      2. associate-*r* 23.6%

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

   5. Simplified 23.6%

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

      1. pow1 23.6%

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

      2. associate-*l* 23.6%

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

      3. *-commutative 23.6%

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

      4. sqrt-unprod 23.8%

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

      5. metadata-eval 23.8%

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

      6. metadata-eval 23.8%

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

   7. Applied egg-rr 23.8%

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

      1. unpow1 23.8%

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

   9. Simplified 23.8%

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

   10. Taylor expanded in a around 0 23.8%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-118}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \sqrt[3]{\left(x-scale \cdot 4\right) \cdot \left(\left(x-scale \cdot 4\right) \cdot \left(x-scale \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.6% accurate, 23.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;\left(x-scale\_m \cdot 4\right) \cdot \sqrt[3]{\left(0.25 \cdot a\_m\right) \cdot \left(\left(0.25 \cdot a\_m\right) \cdot \left(0.25 \cdot a\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 4.5e-126)
   (* (* x-scale_m 4.0) (cbrt (* (* 0.25 a_m) (* (* 0.25 a_m) (* 0.25 a_m)))))
   (* x-scale_m a_m)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.5e-126) {
		tmp = (x_45_scale_m * 4.0) * cbrt(((0.25 * a_m) * ((0.25 * a_m) * (0.25 * a_m))));
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.5e-126) {
		tmp = (x_45_scale_m * 4.0) * Math.cbrt(((0.25 * a_m) * ((0.25 * a_m) * (0.25 * a_m))));
	} else {
		tmp = x_45_scale_m * a_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 4.5e-126)
		tmp = Float64(Float64(x_45_scale_m * 4.0) * cbrt(Float64(Float64(0.25 * a_m) * Float64(Float64(0.25 * a_m) * Float64(0.25 * a_m)))));
	else
		tmp = Float64(x_45_scale_m * a_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 4.5e-126], N[(N[(x$45$scale$95$m * 4.0), $MachinePrecision] * N[Power[N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(0.25 * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * a$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.50000000000000025e-126

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 20.7%

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

      1. associate-*r* 20.7%

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

      2. associate-*r* 20.7%

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

   5. Simplified 20.7%

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

      1. pow1 20.7%

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

      2. associate-*l* 20.7%

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

      3. *-commutative 20.7%

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

      4. sqrt-unprod 20.9%

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

      5. metadata-eval 20.9%

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

      6. metadata-eval 20.9%

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

   7. Applied egg-rr 20.9%

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

      1. unpow1 20.9%

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

   9. Simplified 20.9%

      \[\leadsto \left(0.25 \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot 4\right)} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 21.0%

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

   11. Applied egg-rr 21.0%

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

   if 4.50000000000000025e-126 < y-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 24.3%

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

      1. associate-*r* 24.3%

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

      2. associate-*r* 24.3%

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

   5. Simplified 24.3%

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

      1. pow1 24.3%

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

      2. associate-*l* 24.3%

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

      3. *-commutative 24.3%

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

      4. sqrt-unprod 24.4%

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

      5. metadata-eval 24.4%

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

      6. metadata-eval 24.4%

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

   7. Applied egg-rr 24.4%

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

      1. unpow1 24.4%

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

   9. Simplified 24.4%

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

   10. Taylor expanded in a around 0 24.4%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;\left(x-scale \cdot 4\right) \cdot \sqrt[3]{\left(0.25 \cdot a\right) \cdot \left(\left(0.25 \cdot a\right) \cdot \left(0.25 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.8% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Taylor expanded in angle around 0 21.9%

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

   1. associate-*r* 21.9%

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

   2. associate-*r* 22.0%

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

5. Simplified 22.0%

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

   1. pow1 22.0%

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

   2. associate-*l* 21.9%

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

   3. *-commutative 21.9%

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

   4. sqrt-unprod 22.1%

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

   5. metadata-eval 22.1%

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

   6. metadata-eval 22.1%

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

7. Applied egg-rr 22.1%

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

   1. unpow1 22.1%

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

9. Simplified 22.1%

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

10. Taylor expanded in a around 0 22.1%

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

11. Final simplification 22.1%

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

Reproduce

herbie shell --seed 2024136 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))