b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 44.3%

Time: 1.5min

Alternatives: 8

Speedup: 919.0×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 0.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 44.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3 \cdot 10^{-37}:\\ \;\;\;\;y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (if (<= b_m 3e-37)
   (* y-scale (* (* (cbrt 0.015625) (sqrt 8.0)) (sqrt 0.0)))
   (* (* 0.25 (* a_m x-scale_m)) 4.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 3e-37) {
		tmp = y_45_scale * ((cbrt(0.015625) * sqrt(8.0)) * sqrt(0.0));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 3e-37) {
		tmp = y_45_scale * ((Math.cbrt(0.015625) * Math.sqrt(8.0)) * Math.sqrt(0.0));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b_m <= 3e-37)
		tmp = Float64(y_45_scale * Float64(Float64(cbrt(0.015625) * sqrt(8.0)) * sqrt(0.0)));
	else
		tmp = Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b$95$m, 3e-37], N[(y$45$scale * N[(N[(N[Power[0.015625, 1/3], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3e-37

   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 b around 0 5.2%

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

      1. associate-*r* 5.2%

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

   5. Simplified 4.0%

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

   6. Taylor expanded in x-scale around inf 1.7%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{x-scale} \cdot \sqrt{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - 0.5 \cdot \frac{{y-scale}^{2} \cdot \left(-2 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}\right)}{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)} \]

   8. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{{\left({\left(0.25 \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot y-scale\right)\right) \cdot \frac{\sqrt{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} + -0.5 \cdot \left({y-scale}^{2} \cdot \frac{\left({a}^{4} \cdot \frac{{\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot 2}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}}\right)}}{x-scale}\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. Taylor expanded in x-scale around 0 26.8%

      \[\leadsto \color{blue}{\left(y-scale \cdot \left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right)\right) \cdot \sqrt{-1 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*l* 26.8%

          \[\leadsto \color{blue}{y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{-1 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)} \]

       2. distribute-lft1-in 26.8%

          \[\leadsto y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{\left(-1 + 1\right) \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]

       3. metadata-eval 26.8%

          \[\leadsto y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{0} \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]

       4. mul0-lft 34.0%

          \[\leadsto y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{0}}\right) \]

   11. Simplified 34.0%

       \[\leadsto \color{blue}{y-scale \cdot \left(\left(\sqrt[3]{0.015625} \cdot \sqrt{8}\right) \cdot \sqrt{0}\right)} \]

   if 3e-37 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 18.2%

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

      1. *-commutative 18.2%

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

   5. Simplified 18.2%

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

      1. associate-*r* 18.1%

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

      2. add-exp-log 16.5%

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

      3. associate-*r* 16.5%

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

      4. sqrt-unprod 16.5%

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

      5. metadata-eval 16.5%

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

      6. metadata-eval 16.5%

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

   7. Applied egg-rr 16.5%

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

      1. rem-exp-log 18.3%

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

      2. metadata-eval 18.3%

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

      3. metadata-eval 18.3%

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

      4. sqrt-unprod 18.2%

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

      5. associate-*r* 18.1%

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

      6. associate-*r* 18.2%

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

      7. sqrt-unprod 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

   9. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 38.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;0.25 \cdot \left(a\_m \cdot \left(e^{\mathsf{log1p}\left(x-scale\_m \cdot 4\right)} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (if (<= b_m 7.6e-56)
   (* 0.25 (* a_m (+ (exp (log1p (* x-scale_m 4.0))) -1.0)))
   (* (* 0.25 (* a_m x-scale_m)) 4.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 7.6e-56) {
		tmp = 0.25 * (a_m * (exp(log1p((x_45_scale_m * 4.0))) + -1.0));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 7.6e-56) {
		tmp = 0.25 * (a_m * (Math.exp(Math.log1p((x_45_scale_m * 4.0))) + -1.0));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b_m <= 7.6e-56:
		tmp = 0.25 * (a_m * (math.exp(math.log1p((x_45_scale_m * 4.0))) + -1.0))
	else:
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b_m <= 7.6e-56)
		tmp = Float64(0.25 * Float64(a_m * Float64(exp(log1p(Float64(x_45_scale_m * 4.0))) + -1.0)));
	else
		tmp = Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b$95$m, 7.6e-56], N[(0.25 * N[(a$95$m * N[(N[Exp[N[Log[1 + N[(x$45$scale$95$m * 4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000004e-56

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 20.6%

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

      1. *-commutative 20.6%

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

   5. Simplified 20.6%

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

      1. expm1-log1p-u 17.3%

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

      2. expm1-undefine 25.1%

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

      3. sqrt-unprod 25.1%

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

      4. metadata-eval 25.1%

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

      5. metadata-eval 25.1%

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

   7. Applied egg-rr 25.1%

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

   if 7.6000000000000004e-56 < 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 17.4%

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

      1. *-commutative 17.4%

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

   5. Simplified 17.4%

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

      1. associate-*r* 17.3%

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

      2. add-exp-log 15.7%

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

      3. associate-*r* 15.7%

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

      4. sqrt-unprod 15.7%

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

      5. metadata-eval 15.7%

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

      6. metadata-eval 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. rem-exp-log 17.5%

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

      2. metadata-eval 17.5%

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

      3. metadata-eval 17.5%

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

      4. sqrt-unprod 17.4%

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

      5. associate-*r* 17.3%

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

      6. associate-*r* 17.3%

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

      7. sqrt-unprod 17.5%

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

      8. metadata-eval 17.5%

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

      9. metadata-eval 17.5%

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

   9. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot 4\right)} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;0.25 \cdot \left(a\_m \cdot \log \left(1 + \mathsf{expm1}\left(x-scale\_m \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (if (<= b_m 3.8e-56)
   (* 0.25 (* a_m (log (+ 1.0 (expm1 (* x-scale_m 4.0))))))
   (* (* 0.25 (* a_m x-scale_m)) 4.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 3.8e-56) {
		tmp = 0.25 * (a_m * log((1.0 + expm1((x_45_scale_m * 4.0)))));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 3.8e-56) {
		tmp = 0.25 * (a_m * Math.log((1.0 + Math.expm1((x_45_scale_m * 4.0)))));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	tmp = 0
	if b_m <= 3.8e-56:
		tmp = 0.25 * (a_m * math.log((1.0 + math.expm1((x_45_scale_m * 4.0)))))
	else:
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.8e-56)
		tmp = Float64(0.25 * Float64(a_m * log(Float64(1.0 + expm1(Float64(x_45_scale_m * 4.0))))));
	else
		tmp = Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b$95$m, 3.8e-56], N[(0.25 * N[(a$95$m * N[Log[N[(1.0 + N[(Exp[N[(x$45$scale$95$m * 4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000002e-56

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 20.6%

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

      1. *-commutative 20.6%

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

   5. Simplified 20.6%

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

      1. log1p-expm1-u 18.0%

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

      2. log1p-undefine 25.8%

         \[\leadsto 0.25 \cdot \left(a \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)}\right) \]

      3. sqrt-unprod 25.8%

         \[\leadsto 0.25 \cdot \left(a \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right)\right) \]

      4. metadata-eval 25.8%

         \[\leadsto 0.25 \cdot \left(a \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \]

      5. metadata-eval 25.8%

         \[\leadsto 0.25 \cdot \left(a \cdot \log \left(1 + \mathsf{expm1}\left(x-scale \cdot \color{blue}{4}\right)\right)\right) \]

   7. Applied egg-rr 25.8%

      \[\leadsto 0.25 \cdot \left(a \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(x-scale \cdot 4\right)\right)}\right) \]

   if 3.8000000000000002e-56 < 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 17.4%

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

      1. *-commutative 17.4%

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

   5. Simplified 17.4%

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

      1. associate-*r* 17.3%

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

      2. add-exp-log 15.7%

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

      3. associate-*r* 15.7%

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

      4. sqrt-unprod 15.7%

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

      5. metadata-eval 15.7%

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

      6. metadata-eval 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. rem-exp-log 17.5%

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

      2. metadata-eval 17.5%

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

      3. metadata-eval 17.5%

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

      4. sqrt-unprod 17.4%

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

      5. associate-*r* 17.3%

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

      6. associate-*r* 17.3%

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

      7. sqrt-unprod 17.5%

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

      8. metadata-eval 17.5%

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

      9. metadata-eval 17.5%

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

   9. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 36.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.7 \cdot 10^{-105}:\\ \;\;\;\;0.25 \cdot \left(a\_m \cdot \sqrt[3]{{\left(x-scale\_m \cdot 4\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (if (<= b_m 1.7e-105)
   (* 0.25 (* a_m (cbrt (pow (* x-scale_m 4.0) 3.0))))
   (* (* 0.25 (* a_m x-scale_m)) 4.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 1.7e-105) {
		tmp = 0.25 * (a_m * cbrt(pow((x_45_scale_m * 4.0), 3.0)));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 1.7e-105) {
		tmp = 0.25 * (a_m * Math.cbrt(Math.pow((x_45_scale_m * 4.0), 3.0)));
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.7e-105)
		tmp = Float64(0.25 * Float64(a_m * cbrt((Float64(x_45_scale_m * 4.0) ^ 3.0))));
	else
		tmp = Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b$95$m, 1.7e-105], N[(0.25 * N[(a$95$m * N[Power[N[Power[N[(x$45$scale$95$m * 4.0), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999996e-105

   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 21.0%

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

      1. *-commutative 21.0%

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

   5. Simplified 21.0%

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

      1. add-cbrt-cube 27.2%

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

      2. pow3 27.2%

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

      3. sqrt-unprod 27.3%

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

      4. metadata-eval 27.3%

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

      5. metadata-eval 27.3%

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

   7. Applied egg-rr 27.3%

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

   if 1.69999999999999996e-105 < 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 16.9%

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

      1. *-commutative 16.9%

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

   5. Simplified 16.9%

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

      1. associate-*r* 16.9%

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

      2. add-exp-log 14.1%

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

      3. associate-*r* 14.1%

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

      4. sqrt-unprod 14.1%

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

      5. metadata-eval 14.1%

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

      6. metadata-eval 14.1%

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

   7. Applied egg-rr 14.1%

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

      1. rem-exp-log 17.1%

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

      2. metadata-eval 17.1%

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

      3. metadata-eval 17.1%

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

      4. sqrt-unprod 16.9%

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

      5. associate-*r* 16.9%

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

      6. associate-*r* 16.9%

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

      7. sqrt-unprod 17.1%

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

      8. metadata-eval 17.1%

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

      9. metadata-eval 17.1%

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

   9. Applied egg-rr 17.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 37.4% accurate, 24.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;1 + \mathsf{fma}\left(a\_m, 0.25 \cdot \left(x-scale\_m \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (if (<= b_m 3.8e-56)
   (+ 1.0 (fma a_m (* 0.25 (* x-scale_m 4.0)) -1.0))
   (* (* 0.25 (* a_m x-scale_m)) 4.0)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double tmp;
	if (b_m <= 3.8e-56) {
		tmp = 1.0 + fma(a_m, (0.25 * (x_45_scale_m * 4.0)), -1.0);
	} else {
		tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.0
	if (b_m <= 3.8e-56)
		tmp = Float64(1.0 + fma(a_m, Float64(0.25 * Float64(x_45_scale_m * 4.0)), -1.0));
	else
		tmp = Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := If[LessEqual[b$95$m, 3.8e-56], N[(1.0 + N[(a$95$m * N[(0.25 * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000002e-56

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0 20.6%

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

      1. *-commutative 20.6%

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

   5. Simplified 20.6%

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

      1. pow1 20.6%

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

      2. sqrt-unprod 20.6%

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

      3. metadata-eval 20.6%

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

      4. metadata-eval 20.6%

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

   7. Applied egg-rr 20.6%

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

      1. unpow1 20.6%

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

   9. Simplified 20.6%

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

       1. expm1-log1p-u 20.2%

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

       2. expm1-undefine 26.7%

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

       3. associate-*r* 26.7%

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

   11. Applied egg-rr 26.7%

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

       1. log1p-undefine 26.7%

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

       2. rem-exp-log 27.1%

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

       3. associate-+r- 27.1%

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

       4. associate-*r* 27.1%

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

       5. *-commutative 27.1%

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

       6. associate-*l* 27.1%

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

       7. fma-neg 27.1%

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

       8. metadata-eval 27.1%

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

   13. Simplified 27.1%

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

   if 3.8000000000000002e-56 < 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 17.4%

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

      1. *-commutative 17.4%

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

   5. Simplified 17.4%

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

      1. associate-*r* 17.3%

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

      2. add-exp-log 15.7%

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

      3. associate-*r* 15.7%

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

      4. sqrt-unprod 15.7%

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

      5. metadata-eval 15.7%

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

      6. metadata-eval 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. rem-exp-log 17.5%

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

      2. metadata-eval 17.5%

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

      3. metadata-eval 17.5%

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

      4. sqrt-unprod 17.4%

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

      5. associate-*r* 17.3%

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

      6. associate-*r* 17.3%

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

      7. sqrt-unprod 17.5%

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

      8. metadata-eval 17.5%

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

      9. metadata-eval 17.5%

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

   9. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;1 + \mathsf{fma}\left(a, 0.25 \cdot \left(x-scale \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(a \cdot x-scale\right)\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.8% accurate, 393.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \left(0.25 \cdot \left(a\_m \cdot x-scale\_m\right)\right) \cdot 4 \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (* (* 0.25 (* a_m x-scale_m)) 4.0))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * (a_m * x_45_scale_m)) * 4.0;
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (0.25d0 * (a_m * x_45scale_m)) * 4.0d0
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	return (0.25 * (a_m * x_45_scale_m)) * 4.0;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	return (0.25 * (a_m * x_45_scale_m)) * 4.0

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	return Float64(Float64(0.25 * Float64(a_m * x_45_scale_m)) * 4.0)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = (0.25 * (a_m * x_45_scale_m)) * 4.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(N[(0.25 * N[(a$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

3. Taylor expanded in angle around 0 19.5%

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

   1. *-commutative 19.5%

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

5. Simplified 19.5%

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

   1. associate-*r* 19.4%

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

   2. add-exp-log 17.5%

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

   3. associate-*r* 17.5%

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

   4. sqrt-unprod 17.5%

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

   5. metadata-eval 17.5%

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

   6. metadata-eval 17.5%

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

7. Applied egg-rr 17.5%

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

   1. rem-exp-log 19.5%

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

   2. metadata-eval 19.5%

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

   3. metadata-eval 19.5%

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

   4. sqrt-unprod 19.5%

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

   5. associate-*r* 19.4%

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

   6. associate-*r* 19.4%

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

   7. sqrt-unprod 19.5%

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

   8. metadata-eval 19.5%

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

   9. metadata-eval 19.5%

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

9. Applied egg-rr 19.5%

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

Alternative 7: 32.8% accurate, 393.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ 0.25 \cdot \left(a\_m \cdot \left(x-scale\_m \cdot 4\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (* 0.25 (* a_m (* x-scale_m 4.0))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * (a_m * (x_45_scale_m * 4.0));
}

Fortran

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45scale)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * (a_m * (x_45scale_m * 4.0d0))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	return 0.25 * (a_m * (x_45_scale_m * 4.0));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	return 0.25 * (a_m * (x_45_scale_m * 4.0))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	return Float64(0.25 * Float64(a_m * Float64(x_45_scale_m * 4.0)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	tmp = 0.25 * (a_m * (x_45_scale_m * 4.0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(0.25 * N[(a$95$m * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

3. Taylor expanded in angle around 0 19.5%

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

   1. *-commutative 19.5%

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

5. Simplified 19.5%

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

   1. pow1 19.5%

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

   2. sqrt-unprod 19.5%

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

   3. metadata-eval 19.5%

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

   4. metadata-eval 19.5%

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

7. Applied egg-rr 19.5%

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

   1. unpow1 19.5%

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

9. Simplified 19.5%

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

Alternative 8: 32.8% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Taylor expanded in angle around 0 19.5%

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

   1. *-commutative 19.5%

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

5. Simplified 19.5%

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

   1. pow1 19.5%

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

   2. sqrt-unprod 19.5%

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

   3. metadata-eval 19.5%

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

   4. metadata-eval 19.5%

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

7. Applied egg-rr 19.5%

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

   1. unpow1 19.5%

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

9. Simplified 19.5%

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

10. Taylor expanded in a around 0 19.5%

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

Reproduce

herbie shell --seed 2024110 
(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))))