b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 36.6%

Time: 1.6min

Alternatives: 9

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: 36.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\sin t\_0}^{2}\\ t_2 := {\cos t\_0}^{2}\\ \mathbf{if}\;y-scale\_m \leq 2.1 \cdot 10^{-258}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{\frac{t\_2}{{x-scale\_m}^{2}} + -0.5 \cdot \frac{2 \cdot \frac{t\_1 \cdot t\_2}{{x-scale\_m}^{2}}}{t\_1}}\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;0.25 \cdot \left(e^{\mathsf{log1p}\left(a\_m \cdot \left(x-scale\_m \cdot 4\right)\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (sin t_0) 2.0))
        (t_2 (pow (cos t_0) 2.0)))
   (if (<= y-scale_m 2.1e-258)
     (*
      0.25
      (*
       b
       (*
        (* x-scale_m (* y-scale_m (sqrt 8.0)))
        (sqrt
         (+
          (/ t_2 (pow x-scale_m 2.0))
          (* -0.5 (/ (* 2.0 (/ (* t_1 t_2) (pow x-scale_m 2.0))) t_1)))))))
     (if (<= y-scale_m 2.4e-103)
       (* 0.25 (+ (exp (log1p (* a_m (* x-scale_m 4.0)))) -1.0))
       (* a_m x-scale_m)))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, 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 = pow(sin(t_0), 2.0);
	double t_2 = pow(cos(t_0), 2.0);
	double tmp;
	if (y_45_scale_m <= 2.1e-258) {
		tmp = 0.25 * (b * ((x_45_scale_m * (y_45_scale_m * sqrt(8.0))) * sqrt(((t_2 / pow(x_45_scale_m, 2.0)) + (-0.5 * ((2.0 * ((t_1 * t_2) / pow(x_45_scale_m, 2.0))) / t_1))))));
	} else if (y_45_scale_m <= 2.4e-103) {
		tmp = 0.25 * (exp(log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0);
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.pow(Math.sin(t_0), 2.0);
	double t_2 = Math.pow(Math.cos(t_0), 2.0);
	double tmp;
	if (y_45_scale_m <= 2.1e-258) {
		tmp = 0.25 * (b * ((x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))) * Math.sqrt(((t_2 / Math.pow(x_45_scale_m, 2.0)) + (-0.5 * ((2.0 * ((t_1 * t_2) / Math.pow(x_45_scale_m, 2.0))) / t_1))))));
	} else if (y_45_scale_m <= 2.4e-103) {
		tmp = 0.25 * (Math.exp(Math.log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0);
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pow(math.sin(t_0), 2.0)
	t_2 = math.pow(math.cos(t_0), 2.0)
	tmp = 0
	if y_45_scale_m <= 2.1e-258:
		tmp = 0.25 * (b * ((x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))) * math.sqrt(((t_2 / math.pow(x_45_scale_m, 2.0)) + (-0.5 * ((2.0 * ((t_1 * t_2) / math.pow(x_45_scale_m, 2.0))) / t_1))))))
	elif y_45_scale_m <= 2.4e-103:
		tmp = 0.25 * (math.exp(math.log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0)
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0) ^ 2.0
	t_2 = cos(t_0) ^ 2.0
	tmp = 0.0
	if (y_45_scale_m <= 2.1e-258)
		tmp = Float64(0.25 * Float64(b * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))) * sqrt(Float64(Float64(t_2 / (x_45_scale_m ^ 2.0)) + Float64(-0.5 * Float64(Float64(2.0 * Float64(Float64(t_1 * t_2) / (x_45_scale_m ^ 2.0))) / t_1)))))));
	elseif (y_45_scale_m <= 2.4e-103)
		tmp = Float64(0.25 * Float64(exp(log1p(Float64(a_m * Float64(x_45_scale_m * 4.0)))) + -1.0));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, 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[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.1e-258], N[(0.25 * N[(b * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(t$95$2 / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(2.0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.4e-103], N[(0.25 * N[(N[Exp[N[Log[1 + N[(a$95$m * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 2.0999999999999999e-258

   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 b around inf 2.5%

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

      1. associate-*r* 2.6%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in y-scale around 0 17.6%

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

      1. associate-*l* 18.5%

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

   8. Simplified 18.5%

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

   if 2.0999999999999999e-258 < y-scale < 2.4000000000000002e-103

   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 14.3%

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

      1. *-commutative 14.3%

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

   5. Simplified 14.3%

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

      1. expm1-log1p-u 14.0%

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

      2. expm1-undefine 34.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} - 1\right)} \]

      3. sqrt-unprod 34.1%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)} - 1\right) \]

      4. metadata-eval 34.1%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \sqrt{\color{blue}{16}}\right)\right)} - 1\right) \]

      5. metadata-eval 34.1%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \color{blue}{4}\right)\right)} - 1\right) \]

   7. Applied egg-rr 34.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot 4\right)\right)} - 1\right)} \]

   if 2.4000000000000002e-103 < 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 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%

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

      2. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. sqrt-unprod 22.0%

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

      2. metadata-eval 22.0%

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

      3. metadata-eval 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in a around 0 22.0%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.1 \cdot 10^{-258}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + -0.5 \cdot \frac{2 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot 4\right)\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\cos t\_0}^{2}\\ t_2 := \frac{{\sin t\_0}^{2}}{{x-scale\_m}^{2}}\\ \mathbf{if}\;y-scale\_m \leq 1.85 \cdot 10^{-101}:\\ \;\;\;\;\left(\left(0.25 \cdot a\_m\right) \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{t\_2 + -0.5 \cdot \frac{2 \cdot \left(t\_2 \cdot t\_1\right)}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (cos t_0) 2.0))
        (t_2 (/ (pow (sin t_0) 2.0) (pow x-scale_m 2.0))))
   (if (<= y-scale_m 1.85e-101)
     (*
      (* (* 0.25 a_m) (* x-scale_m (* y-scale_m (sqrt 8.0))))
      (sqrt (+ t_2 (* -0.5 (/ (* 2.0 (* t_2 t_1)) t_1)))))
     (* a_m x-scale_m))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, 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 = pow(cos(t_0), 2.0);
	double t_2 = pow(sin(t_0), 2.0) / pow(x_45_scale_m, 2.0);
	double tmp;
	if (y_45_scale_m <= 1.85e-101) {
		tmp = ((0.25 * a_m) * (x_45_scale_m * (y_45_scale_m * sqrt(8.0)))) * sqrt((t_2 + (-0.5 * ((2.0 * (t_2 * t_1)) / t_1))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.pow(Math.cos(t_0), 2.0);
	double t_2 = Math.pow(Math.sin(t_0), 2.0) / Math.pow(x_45_scale_m, 2.0);
	double tmp;
	if (y_45_scale_m <= 1.85e-101) {
		tmp = ((0.25 * a_m) * (x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0)))) * Math.sqrt((t_2 + (-0.5 * ((2.0 * (t_2 * t_1)) / t_1))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pow(math.cos(t_0), 2.0)
	t_2 = math.pow(math.sin(t_0), 2.0) / math.pow(x_45_scale_m, 2.0)
	tmp = 0
	if y_45_scale_m <= 1.85e-101:
		tmp = ((0.25 * a_m) * (x_45_scale_m * (y_45_scale_m * math.sqrt(8.0)))) * math.sqrt((t_2 + (-0.5 * ((2.0 * (t_2 * t_1)) / t_1))))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0) ^ 2.0
	t_2 = Float64((sin(t_0) ^ 2.0) / (x_45_scale_m ^ 2.0))
	tmp = 0.0
	if (y_45_scale_m <= 1.85e-101)
		tmp = Float64(Float64(Float64(0.25 * a_m) * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0)))) * sqrt(Float64(t_2 + Float64(-0.5 * Float64(Float64(2.0 * Float64(t_2 * t_1)) / t_1)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0) ^ 2.0;
	t_2 = (sin(t_0) ^ 2.0) / (x_45_scale_m ^ 2.0);
	tmp = 0.0;
	if (y_45_scale_m <= 1.85e-101)
		tmp = ((0.25 * a_m) * (x_45_scale_m * (y_45_scale_m * sqrt(8.0)))) * sqrt((t_2 + (-0.5 * ((2.0 * (t_2 * t_1)) / t_1))));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, 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[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.85e-101], N[(N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$2 + N[(-0.5 * N[(N[(2.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.85000000000000002e-101

   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 a around inf 4.3%

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

      1. associate-*r* 4.4%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in y-scale around 0 25.1%

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

      1. associate-*r* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*r* 25.1%

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

   8. Simplified 26.1%

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

   if 1.85000000000000002e-101 < 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 22.1%

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

      1. associate-*r* 22.1%

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

      2. *-commutative 22.1%

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

   5. Simplified 22.1%

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

      1. sqrt-unprod 22.3%

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

      2. metadata-eval 22.3%

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

      3. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

   8. Taylor expanded in a around 0 22.3%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.85 \cdot 10^{-101}:\\ \;\;\;\;\left(\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + -0.5 \cdot \frac{2 \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 4.15 \cdot 10^{-100}:\\ \;\;\;\;0.25 \cdot \log \left({\left(e^{a\_m}\right)}^{\left(x-scale\_m \cdot \sqrt[3]{64}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 4.15e-100)
   (* 0.25 (log (pow (exp a_m) (* x-scale_m (cbrt 64.0)))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.15e-100) {
		tmp = 0.25 * log(pow(exp(a_m), (x_45_scale_m * cbrt(64.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.15e-100) {
		tmp = 0.25 * Math.log(Math.pow(Math.exp(a_m), (x_45_scale_m * Math.cbrt(64.0))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 4.15e-100)
		tmp = Float64(0.25 * log((exp(a_m) ^ Float64(x_45_scale_m * cbrt(64.0)))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 4.15e-100], N[(0.25 * N[Log[N[Power[N[Exp[a$95$m], $MachinePrecision], N[(x$45$scale$95$m * N[Power[64.0, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.1499999999999998e-100

   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 22.7%

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

      1. *-commutative 22.7%

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

   5. Simplified 22.7%

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

      1. add-cbrt-cube 28.2%

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

      2. sqrt-unprod 28.2%

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

      3. metadata-eval 28.2%

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

      4. metadata-eval 28.2%

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

      5. sqrt-unprod 28.2%

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

      6. metadata-eval 28.2%

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

      7. metadata-eval 28.2%

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

      8. sqrt-unprod 28.2%

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

      9. metadata-eval 28.2%

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

      10. metadata-eval 28.2%

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

   7. Applied egg-rr 28.2%

      \[\leadsto 0.25 \cdot \left(a \cdot \color{blue}{\sqrt[3]{\left(\left(x-scale \cdot 4\right) \cdot \left(x-scale \cdot 4\right)\right) \cdot \left(x-scale \cdot 4\right)}}\right) \]
   8. Step-by-step derivation

      1. add-exp-log 23.4%

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

      2. pow3 23.4%

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

      3. unpow-prod-down 23.4%

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

      4. metadata-eval 23.4%

         \[\leadsto 0.25 \cdot \left(a \cdot \sqrt[3]{e^{\log \left({x-scale}^{3} \cdot \color{blue}{64}\right)}}\right) \]

   9. Applied egg-rr 23.4%

      \[\leadsto 0.25 \cdot \left(a \cdot \sqrt[3]{\color{blue}{e^{\log \left({x-scale}^{3} \cdot 64\right)}}}\right) \]
   10. Step-by-step derivation

       1. add-log-exp 23.8%

          \[\leadsto 0.25 \cdot \color{blue}{\log \left(e^{a \cdot \sqrt[3]{e^{\log \left({x-scale}^{3} \cdot 64\right)}}}\right)} \]

       2. exp-prod 35.3%

          \[\leadsto 0.25 \cdot \log \color{blue}{\left({\left(e^{a}\right)}^{\left(\sqrt[3]{e^{\log \left({x-scale}^{3} \cdot 64\right)}}\right)}\right)} \]

       3. rem-exp-log 35.6%

          \[\leadsto 0.25 \cdot \log \left({\left(e^{a}\right)}^{\left(\sqrt[3]{\color{blue}{{x-scale}^{3} \cdot 64}}\right)}\right) \]

       4. cbrt-prod 35.6%

          \[\leadsto 0.25 \cdot \log \left({\left(e^{a}\right)}^{\color{blue}{\left(\sqrt[3]{{x-scale}^{3}} \cdot \sqrt[3]{64}\right)}}\right) \]

       5. unpow3 35.6%

          \[\leadsto 0.25 \cdot \log \left({\left(e^{a}\right)}^{\left(\sqrt[3]{\color{blue}{\left(x-scale \cdot x-scale\right) \cdot x-scale}} \cdot \sqrt[3]{64}\right)}\right) \]

       6. add-cbrt-cube 28.6%

          \[\leadsto 0.25 \cdot \log \left({\left(e^{a}\right)}^{\left(\color{blue}{x-scale} \cdot \sqrt[3]{64}\right)}\right) \]

   11. Applied egg-rr 28.6%

       \[\leadsto 0.25 \cdot \color{blue}{\log \left({\left(e^{a}\right)}^{\left(x-scale \cdot \sqrt[3]{64}\right)}\right)} \]

   if 4.1499999999999998e-100 < 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 22.1%

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

      1. associate-*r* 22.1%

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

      2. *-commutative 22.1%

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

   5. Simplified 22.1%

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

      1. sqrt-unprod 22.3%

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

      2. metadata-eval 22.3%

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

      3. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

   8. Taylor expanded in a around 0 22.3%

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

Alternative 4: 36.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 7.5 \cdot 10^{-103}:\\ \;\;\;\;0.25 \cdot \left(e^{\mathsf{log1p}\left(a\_m \cdot \left(x-scale\_m \cdot 4\right)\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 7.5e-103)
   (* 0.25 (+ (exp (log1p (* a_m (* x-scale_m 4.0)))) -1.0))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 7.5e-103) {
		tmp = 0.25 * (exp(log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0);
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 7.5e-103) {
		tmp = 0.25 * (Math.exp(Math.log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0);
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 7.5e-103:
		tmp = 0.25 * (math.exp(math.log1p((a_m * (x_45_scale_m * 4.0)))) + -1.0)
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 7.5e-103)
		tmp = Float64(0.25 * Float64(exp(log1p(Float64(a_m * Float64(x_45_scale_m * 4.0)))) + -1.0));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 7.5e-103], N[(0.25 * N[(N[Exp[N[Log[1 + N[(a$95$m * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.5e-103

   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 22.8%

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

      1. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. expm1-log1p-u 22.4%

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

      2. expm1-undefine 31.7%

         \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} - 1\right)} \]

      3. sqrt-unprod 31.7%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)} - 1\right) \]

      4. metadata-eval 31.7%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \sqrt{\color{blue}{16}}\right)\right)} - 1\right) \]

      5. metadata-eval 31.7%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot \color{blue}{4}\right)\right)} - 1\right) \]

   7. Applied egg-rr 31.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot 4\right)\right)} - 1\right)} \]

   if 7.5e-103 < 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 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%

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

      2. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. sqrt-unprod 22.0%

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

      2. metadata-eval 22.0%

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

      3. metadata-eval 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in a around 0 22.0%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.5 \cdot 10^{-103}:\\ \;\;\;\;0.25 \cdot \left(e^{\mathsf{log1p}\left(a \cdot \left(x-scale \cdot 4\right)\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 4.2 \cdot 10^{-103}:\\ \;\;\;\;e^{\mathsf{log1p}\left(a\_m \cdot x-scale\_m\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 4.2e-103)
   (+ (exp (log1p (* a_m x-scale_m))) -1.0)
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.2e-103) {
		tmp = exp(log1p((a_m * x_45_scale_m))) + -1.0;
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 4.2e-103) {
		tmp = Math.exp(Math.log1p((a_m * x_45_scale_m))) + -1.0;
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 4.2e-103:
		tmp = math.exp(math.log1p((a_m * x_45_scale_m))) + -1.0
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 4.2e-103)
		tmp = Float64(exp(log1p(Float64(a_m * x_45_scale_m))) + -1.0);
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 4.2e-103], N[(N[Exp[N[Log[1 + N[(a$95$m * x$45$scale$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.20000000000000009e-103

   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 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-unprod 22.9%

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

      2. metadata-eval 22.9%

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

      3. metadata-eval 22.9%

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

   7. Applied egg-rr 22.9%

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

   8. Taylor expanded in a around 0 22.9%

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

      1. expm1-log1p-u 22.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot x-scale\right)\right)} \]

      2. expm1-undefine 31.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot x-scale\right)} - 1} \]

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot x-scale\right)} - 1} \]

   if 4.20000000000000009e-103 < 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 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%

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

      2. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. sqrt-unprod 22.0%

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

      2. metadata-eval 22.0%

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

      3. metadata-eval 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in a around 0 22.0%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4.2 \cdot 10^{-103}:\\ \;\;\;\;e^{\mathsf{log1p}\left(a \cdot x-scale\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 9 \cdot 10^{-103}:\\ \;\;\;\;\log \left(e^{a\_m \cdot x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 9e-103) (log (exp (* a_m x-scale_m))) (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 9e-103) {
		tmp = log(exp((a_m * x_45_scale_m)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Fortran

a_m = abs(a)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    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 (y_45scale_m <= 9d-103) then
        tmp = log(exp((a_m * x_45scale_m)))
    else
        tmp = a_m * x_45scale_m
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 9e-103) {
		tmp = Math.log(Math.exp((a_m * x_45_scale_m)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 9e-103:
		tmp = math.log(math.exp((a_m * x_45_scale_m)))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 9e-103)
		tmp = log(exp(Float64(a_m * x_45_scale_m)));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 9e-103)
		tmp = log(exp((a_m * x_45_scale_m)));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 9e-103], N[Log[N[Exp[N[(a$95$m * x$45$scale$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9e-103

   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 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-unprod 22.9%

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

      2. metadata-eval 22.9%

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

      3. metadata-eval 22.9%

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

   7. Applied egg-rr 22.9%

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

   8. Taylor expanded in a around 0 22.9%

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

      1. add-log-exp 30.7%

         \[\leadsto \color{blue}{\log \left(e^{a \cdot x-scale}\right)} \]

   10. Applied egg-rr 30.7%

       \[\leadsto \color{blue}{\log \left(e^{a \cdot x-scale}\right)} \]

   if 9e-103 < 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 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%

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

      2. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. sqrt-unprod 22.0%

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

      2. metadata-eval 22.0%

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

      3. metadata-eval 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in a around 0 22.0%

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

Alternative 7: 32.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(a\_m \cdot \sqrt[3]{\left(x-scale\_m \cdot 4\right) \cdot \left(\left(x-scale\_m \cdot 4\right) \cdot \left(x-scale\_m \cdot 4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 3.8e-42)
   (*
    0.25
    (*
     a_m
     (cbrt (* (* x-scale_m 4.0) (* (* x-scale_m 4.0) (* x-scale_m 4.0))))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.8e-42) {
		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 = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.8e-42) {
		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 = a_m * x_45_scale_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 3.8e-42)
		tmp = Float64(0.25 * Float64(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(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 3.8e-42], N[(0.25 * N[(a$95$m * 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]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000017e-42

   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 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. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. add-cbrt-cube 25.4%

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

      2. sqrt-unprod 25.5%

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

      3. metadata-eval 25.5%

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

      4. metadata-eval 25.5%

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

      5. sqrt-unprod 25.5%

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

      6. metadata-eval 25.5%

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

      7. metadata-eval 25.5%

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

      8. sqrt-unprod 25.5%

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

      9. metadata-eval 25.5%

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

      10. metadata-eval 25.5%

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

   7. Applied egg-rr 25.5%

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

   if 3.80000000000000017e-42 < b

   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 24.1%

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

      1. associate-*r* 24.1%

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

      2. *-commutative 24.1%

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

   5. Simplified 24.1%

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

      1. sqrt-unprod 24.2%

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

      2. metadata-eval 24.2%

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

      3. metadata-eval 24.2%

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

   7. Applied egg-rr 24.2%

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

   8. Taylor expanded in a around 0 24.2%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \sqrt[3]{\left(x-scale \cdot 4\right) \cdot \left(\left(x-scale \cdot 4\right) \cdot \left(x-scale \cdot 4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.2 \cdot 10^{-103}:\\ \;\;\;\;\sqrt[3]{\left(a\_m \cdot x-scale\_m\right) \cdot \left(\left(a\_m \cdot x-scale\_m\right) \cdot \left(a\_m \cdot x-scale\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.2e-103)
   (cbrt (* (* a_m x-scale_m) (* (* a_m x-scale_m) (* a_m x-scale_m))))
   (* a_m x-scale_m)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.2e-103) {
		tmp = cbrt(((a_m * x_45_scale_m) * ((a_m * x_45_scale_m) * (a_m * x_45_scale_m))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.2e-103) {
		tmp = Math.cbrt(((a_m * x_45_scale_m) * ((a_m * x_45_scale_m) * (a_m * x_45_scale_m))));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.2e-103)
		tmp = cbrt(Float64(Float64(a_m * x_45_scale_m) * Float64(Float64(a_m * x_45_scale_m) * Float64(a_m * x_45_scale_m))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.2e-103], N[Power[N[(N[(a$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(a$95$m * x$45$scale$95$m), $MachinePrecision] * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.2000000000000001e-103

   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 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-unprod 22.9%

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

      2. metadata-eval 22.9%

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

      3. metadata-eval 22.9%

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

   7. Applied egg-rr 22.9%

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

   8. Taylor expanded in a around 0 22.9%

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

      1. add-cbrt-cube 29.6%

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

   10. Applied egg-rr 29.6%

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

   if 1.2000000000000001e-103 < 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 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%

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

      2. *-commutative 21.9%

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

   5. Simplified 21.9%

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

      1. sqrt-unprod 22.0%

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

      2. metadata-eval 22.0%

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

      3. metadata-eval 22.0%

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

   7. Applied egg-rr 22.0%

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

   8. Taylor expanded in a around 0 22.0%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.2 \cdot 10^{-103}:\\ \;\;\;\;\sqrt[3]{\left(a \cdot x-scale\right) \cdot \left(\left(a \cdot x-scale\right) \cdot \left(a \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.7% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

a_m = Math.abs(a);
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, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 22.5%

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

   1. associate-*r* 22.5%

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

   2. *-commutative 22.5%

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

5. Simplified 22.5%

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

   1. sqrt-unprod 22.6%

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

   2. metadata-eval 22.6%

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

   3. metadata-eval 22.6%

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

7. Applied egg-rr 22.6%

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

8. Taylor expanded in a around 0 22.6%

   \[\leadsto \color{blue}{a \cdot x-scale} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024121 
(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))))