Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.7% → 81.4%

Time: 16.9s

Alternatives: 7

Speedup: 19.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 81.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{-1} \cdot \frac{\frac{x\_m}{y} \cdot 0.5}{\frac{y}{x\_m}} + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x\_m \leq 5.1 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x\_m}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x\_m}^{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x\_m} \cdot \frac{y}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 9e-156)
   (log1p (+ (* (exp -1.0) (/ (* (/ x_m y) 0.5) (/ y x_m))) (expm1 -1.0)))
   (if (<= x_m 5.1e+115)
     (log1p
      (expm1
       (/
        (fma y (* y -4.0) (pow x_m 2.0))
        (fma 4.0 (pow y 2.0) (pow x_m 2.0)))))
     (+ 1.0 (* -8.0 (* (/ y x_m) (/ y x_m)))))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 9e-156) {
		tmp = log1p(((exp(-1.0) * (((x_m / y) * 0.5) / (y / x_m))) + expm1(-1.0)));
	} else if (x_m <= 5.1e+115) {
		tmp = log1p(expm1((fma(y, (y * -4.0), pow(x_m, 2.0)) / fma(4.0, pow(y, 2.0), pow(x_m, 2.0)))));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 9e-156)
		tmp = log1p(Float64(Float64(exp(-1.0) * Float64(Float64(Float64(x_m / y) * 0.5) / Float64(y / x_m))) + expm1(-1.0)));
	elseif (x_m <= 5.1e+115)
		tmp = log1p(expm1(Float64(fma(y, Float64(y * -4.0), (x_m ^ 2.0)) / fma(4.0, (y ^ 2.0), (x_m ^ 2.0)))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x_m) * Float64(y / x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 9e-156], N[Log[1 + N[(N[(N[Exp[-1.0], $MachinePrecision] * N[(N[(N[(x$95$m / y), $MachinePrecision] * 0.5), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[-1.0] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$95$m, 5.1e+115], N[Log[1 + N[(Exp[N[(N[(y * N[(y * -4.0), $MachinePrecision] + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Power[y, 2.0], $MachinePrecision] + N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x$95$m), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.99999999999999971e-156

   1. Initial program 45.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 45.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right)} \]

      2. sub-neg 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      3. +-commutative 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\left(-\left(y \cdot 4\right) \cdot y\right) + x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      4. *-commutative 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right) + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      5. distribute-rgt-neg-in 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      6. fma-define 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      7. distribute-rgt-neg-in 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      8. metadata-eval 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      9. pow2 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, \color{blue}{{x}^{2}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      10. +-commutative 45.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}}\right)\right) \]

      11. *-commutative 45.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(4 \cdot y\right)} \cdot y + x \cdot x}\right)\right) \]

      12. associate-*l* 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{4 \cdot \left(y \cdot y\right)} + x \cdot x}\right)\right) \]

      13. fma-define 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}}\right)\right) \]

      14. pow2 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, \color{blue}{{y}^{2}}, x \cdot x\right)}\right)\right) \]

      15. pow2 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, \color{blue}{{x}^{2}}\right)}\right)\right) \]

   4. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)\right)} \]

   5. Taylor expanded in y around inf 50.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{-1} + \frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}}\right) - 1}\right) \]
   6. Step-by-step derivation

      1. +-commutative 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}} + e^{-1}\right)} - 1\right) \]

      2. associate--l+ 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}} + \left(e^{-1} - 1\right)}\right) \]

      3. distribute-rgt-out-- 50.7%

         \[\leadsto \mathsf{log1p}\left(\frac{e^{-1} \cdot \color{blue}{\left({x}^{2} \cdot \left(0.25 - -0.25\right)\right)}}{{y}^{2}} + \left(e^{-1} - 1\right)\right) \]

      4. metadata-eval 50.7%

         \[\leadsto \mathsf{log1p}\left(\frac{e^{-1} \cdot \left({x}^{2} \cdot \color{blue}{0.5}\right)}{{y}^{2}} + \left(e^{-1} - 1\right)\right) \]

      5. associate-/l* 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1} \cdot \frac{{x}^{2} \cdot 0.5}{{y}^{2}}} + \left(e^{-1} - 1\right)\right) \]

      6. associate-*l/ 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} \cdot 0.5\right)} + \left(e^{-1} - 1\right)\right) \]

      7. unpow2 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      8. unpow2 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      9. times-frac 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      10. unpow2 63.7%

          \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      11. expm1-define 63.7%

          \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right) + \color{blue}{\mathsf{expm1}\left(-1\right)}\right) \]

   7. Simplified 63.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1} \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right) + \mathsf{expm1}\left(-1\right)}\right) \]
   8. Step-by-step derivation

      1. pow2 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5\right) + \mathsf{expm1}\left(-1\right)\right) \]

      2. *-commutative 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\right)} + \mathsf{expm1}\left(-1\right)\right) \]

      3. associate-*r* 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x}{y}\right) \cdot \frac{x}{y}\right)} + \mathsf{expm1}\left(-1\right)\right) \]

      4. clear-num 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\left(0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) + \mathsf{expm1}\left(-1\right)\right) \]

      5. un-div-inv 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\frac{0.5 \cdot \frac{x}{y}}{\frac{y}{x}}} + \mathsf{expm1}\left(-1\right)\right) \]

      6. *-commutative 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \frac{\color{blue}{\frac{x}{y} \cdot 0.5}}{\frac{y}{x}} + \mathsf{expm1}\left(-1\right)\right) \]

   9. Applied egg-rr 63.7%

      \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\frac{\frac{x}{y} \cdot 0.5}{\frac{y}{x}}} + \mathsf{expm1}\left(-1\right)\right) \]

   if 8.99999999999999971e-156 < x < 5.0999999999999996e115

   1. Initial program 80.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 80.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right)} \]

      2. sub-neg 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      3. +-commutative 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\left(-\left(y \cdot 4\right) \cdot y\right) + x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      4. *-commutative 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right) + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      5. distribute-rgt-neg-in 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      6. fma-define 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      7. distribute-rgt-neg-in 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      8. metadata-eval 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      9. pow2 80.8%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, \color{blue}{{x}^{2}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      10. +-commutative 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}}\right)\right) \]

      11. *-commutative 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(4 \cdot y\right)} \cdot y + x \cdot x}\right)\right) \]

      12. associate-*l* 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{4 \cdot \left(y \cdot y\right)} + x \cdot x}\right)\right) \]

      13. fma-define 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}}\right)\right) \]

      14. pow2 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, \color{blue}{{y}^{2}}, x \cdot x\right)}\right)\right) \]

      15. pow2 80.8%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, \color{blue}{{x}^{2}}\right)}\right)\right) \]

   4. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)\right)} \]

   if 5.0999999999999996e115 < x

   1. Initial program 14.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.1%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 92.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   5. Applied egg-rr 92.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{-1} \cdot \frac{\frac{x}{y} \cdot 0.5}{\frac{y}{x}} + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x\_m \leq 3 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{-1} \cdot \frac{\frac{x\_m}{y} \cdot 0.5}{\frac{y}{x\_m}} + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x\_m \leq 6.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x\_m} \cdot \frac{y}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= x_m 3e-156)
     (log1p (+ (* (exp -1.0) (/ (* (/ x_m y) 0.5) (/ y x_m))) (expm1 -1.0)))
     (if (<= x_m 6.3e+115)
       (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0))
       (+ 1.0 (* -8.0 (* (/ y x_m) (/ y x_m))))))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (x_m <= 3e-156) {
		tmp = log1p(((exp(-1.0) * (((x_m / y) * 0.5) / (y / x_m))) + expm1(-1.0)));
	} else if (x_m <= 6.3e+115) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (x_m <= 3e-156) {
		tmp = Math.log1p(((Math.exp(-1.0) * (((x_m / y) * 0.5) / (y / x_m))) + Math.expm1(-1.0)));
	} else if (x_m <= 6.3e+115) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if x_m <= 3e-156:
		tmp = math.log1p(((math.exp(-1.0) * (((x_m / y) * 0.5) / (y / x_m))) + math.expm1(-1.0)))
	elif x_m <= 6.3e+115:
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)
	else:
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (x_m <= 3e-156)
		tmp = log1p(Float64(Float64(exp(-1.0) * Float64(Float64(Float64(x_m / y) * 0.5) / Float64(y / x_m))) + expm1(-1.0)));
	elseif (x_m <= 6.3e+115)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x_m) * Float64(y / x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3e-156], N[Log[1 + N[(N[(N[Exp[-1.0], $MachinePrecision] * N[(N[(N[(x$95$m / y), $MachinePrecision] * 0.5), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[-1.0] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$95$m, 6.3e+115], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x$95$m), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3e-156

   1. Initial program 45.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 45.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right)} \]

      2. sub-neg 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      3. +-commutative 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\left(-\left(y \cdot 4\right) \cdot y\right) + x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      4. *-commutative 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right) + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      5. distribute-rgt-neg-in 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      6. fma-define 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      7. distribute-rgt-neg-in 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      8. metadata-eval 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      9. pow2 45.3%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, \color{blue}{{x}^{2}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right) \]

      10. +-commutative 45.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}}\right)\right) \]

      11. *-commutative 45.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\left(4 \cdot y\right)} \cdot y + x \cdot x}\right)\right) \]

      12. associate-*l* 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{4 \cdot \left(y \cdot y\right)} + x \cdot x}\right)\right) \]

      13. fma-define 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\color{blue}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}}\right)\right) \]

      14. pow2 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, \color{blue}{{y}^{2}}, x \cdot x\right)}\right)\right) \]

      15. pow2 45.2%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, \color{blue}{{x}^{2}}\right)}\right)\right) \]

   4. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)\right)} \]

   5. Taylor expanded in y around inf 50.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{-1} + \frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}}\right) - 1}\right) \]
   6. Step-by-step derivation

      1. +-commutative 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}} + e^{-1}\right)} - 1\right) \]

      2. associate--l+ 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{e^{-1} \cdot \left(0.25 \cdot {x}^{2} - -0.25 \cdot {x}^{2}\right)}{{y}^{2}} + \left(e^{-1} - 1\right)}\right) \]

      3. distribute-rgt-out-- 50.7%

         \[\leadsto \mathsf{log1p}\left(\frac{e^{-1} \cdot \color{blue}{\left({x}^{2} \cdot \left(0.25 - -0.25\right)\right)}}{{y}^{2}} + \left(e^{-1} - 1\right)\right) \]

      4. metadata-eval 50.7%

         \[\leadsto \mathsf{log1p}\left(\frac{e^{-1} \cdot \left({x}^{2} \cdot \color{blue}{0.5}\right)}{{y}^{2}} + \left(e^{-1} - 1\right)\right) \]

      5. associate-/l* 50.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1} \cdot \frac{{x}^{2} \cdot 0.5}{{y}^{2}}} + \left(e^{-1} - 1\right)\right) \]

      6. associate-*l/ 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}} \cdot 0.5\right)} + \left(e^{-1} - 1\right)\right) \]

      7. unpow2 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      8. unpow2 50.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      9. times-frac 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      10. unpow2 63.7%

          \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.5\right) + \left(e^{-1} - 1\right)\right) \]

      11. expm1-define 63.7%

          \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right) + \color{blue}{\mathsf{expm1}\left(-1\right)}\right) \]

   7. Simplified 63.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1} \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right) + \mathsf{expm1}\left(-1\right)}\right) \]
   8. Step-by-step derivation

      1. pow2 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.5\right) + \mathsf{expm1}\left(-1\right)\right) \]

      2. *-commutative 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\right)} + \mathsf{expm1}\left(-1\right)\right) \]

      3. associate-*r* 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x}{y}\right) \cdot \frac{x}{y}\right)} + \mathsf{expm1}\left(-1\right)\right) \]

      4. clear-num 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \left(\left(0.5 \cdot \frac{x}{y}\right) \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) + \mathsf{expm1}\left(-1\right)\right) \]

      5. un-div-inv 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\frac{0.5 \cdot \frac{x}{y}}{\frac{y}{x}}} + \mathsf{expm1}\left(-1\right)\right) \]

      6. *-commutative 63.7%

         \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \frac{\color{blue}{\frac{x}{y} \cdot 0.5}}{\frac{y}{x}} + \mathsf{expm1}\left(-1\right)\right) \]

   9. Applied egg-rr 63.7%

      \[\leadsto \mathsf{log1p}\left(e^{-1} \cdot \color{blue}{\frac{\frac{x}{y} \cdot 0.5}{\frac{y}{x}}} + \mathsf{expm1}\left(-1\right)\right) \]

   if 3e-156 < x < 6.2999999999999997e115

   1. Initial program 80.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   if 6.2999999999999997e115 < x

   1. Initial program 14.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.1%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 92.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   5. Applied egg-rr 92.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{-1} \cdot \frac{\frac{x}{y} \cdot 0.5}{\frac{y}{x}} + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x\_m \leq 7.3 \cdot 10^{-156}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x\_m}{y} \cdot \frac{x\_m}{y}\right)\\ \mathbf{elif}\;x\_m \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x\_m} \cdot \frac{y}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= x_m 7.3e-156)
     (+ -1.0 (* 0.5 (* (/ x_m y) (/ x_m y))))
     (if (<= x_m 4e+115)
       (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0))
       (+ 1.0 (* -8.0 (* (/ y x_m) (/ y x_m))))))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (x_m <= 7.3e-156) {
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	} else if (x_m <= 4e+115) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (x_m <= 7.3d-156) then
        tmp = (-1.0d0) + (0.5d0 * ((x_m / y) * (x_m / y)))
    else if (x_m <= 4d+115) then
        tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x_m) * (y / x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (x_m <= 7.3e-156) {
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	} else if (x_m <= 4e+115) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if x_m <= 7.3e-156:
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)))
	elif x_m <= 4e+115:
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)
	else:
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (x_m <= 7.3e-156)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x_m / y) * Float64(x_m / y))));
	elseif (x_m <= 4e+115)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x_m) * Float64(y / x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (x_m <= 7.3e-156)
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	elseif (x_m <= 4e+115)
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	else
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 7.3e-156], N[(-1.0 + N[(0.5 * N[(N[(x$95$m / y), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4e+115], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x$95$m), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.3e-156

   1. Initial program 45.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. unpow2 50.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 50.2%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 61.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   5. Applied egg-rr 61.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   if 7.3e-156 < x < 4.0000000000000001e115

   1. Initial program 80.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   if 4.0000000000000001e115 < x

   1. Initial program 14.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.1%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 80.1%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 92.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   5. Applied egg-rr 92.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.3 \cdot 10^{-156}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.35 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x\_m} \cdot \frac{y}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 2.35e-21) -1.0 (+ 1.0 (* -8.0 (* (/ y x_m) (/ y x_m))))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.35e-21) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 2.35d-21) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x_m) * (y / x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.35e-21) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 2.35e-21:
		tmp = -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2.35e-21)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x_m) * Float64(y / x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 2.35e-21)
		tmp = -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2.35e-21], -1.0, N[(1.0 + N[(-8.0 * N[(N[(y / x$95$m), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.35000000000000015e-21

   1. Initial program 50.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.6%

      \[\leadsto \color{blue}{-1} \]

   if 2.35000000000000015e-21 < x

   1. Initial program 34.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 75.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 83.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   5. Applied egg-rr 83.9%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-21}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{-21}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x\_m}{y} \cdot \frac{x\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x\_m} \cdot \frac{y}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 2.4e-21)
   (+ -1.0 (* 0.5 (* (/ x_m y) (/ x_m y))))
   (+ 1.0 (* -8.0 (* (/ y x_m) (/ y x_m))))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.4e-21) {
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 2.4d-21) then
        tmp = (-1.0d0) + (0.5d0 * ((x_m / y) * (x_m / y)))
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x_m) * (y / x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.4e-21) {
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 2.4e-21:
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)))
	else:
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2.4e-21)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x_m / y) * Float64(x_m / y))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x_m) * Float64(y / x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 2.4e-21)
		tmp = -1.0 + (0.5 * ((x_m / y) * (x_m / y)));
	else
		tmp = 1.0 + (-8.0 * ((y / x_m) * (y / x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2.4e-21], N[(-1.0 + N[(0.5 * N[(N[(x$95$m / y), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x$95$m), $MachinePrecision] * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e-21

   1. Initial program 50.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   4. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 52.9%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 62.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   5. Applied egg-rr 62.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

   if 2.3999999999999999e-21 < x

   1. Initial program 34.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 75.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 83.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   5. Applied egg-rr 83.9%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-21}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (if (<= x_m 2.6e-14) -1.0 1.0))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.6e-14) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 2.6d-14) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.6e-14) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 2.6e-14:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2.6e-14)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 2.6e-14)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2.6e-14], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.59999999999999997e-14

   1. Initial program 50.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.3%

      \[\leadsto \color{blue}{-1} \]

   if 2.59999999999999997e-14 < x

   1. Initial program 33.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 82.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ -1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 -1.0)

C

x_m = fabs(x);
double code(double x_m, double y) {
	return -1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return -1.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return -1.0

Julia

x_m = abs(x)
function code(x_m, y)
	return -1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y)
	tmp = -1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := -1.0

Derivation

1. Initial program 45.7%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 49.5%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 49.5%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 51.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_3}{x \cdot x + t\_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t\_1} - t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t\_1}}\right)}^{2} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024079 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))