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

Percentage Accurate: 49.8% → 80.6%

Time: 10.8s

Alternatives: 9

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: 49.8% 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: 80.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - t_1}{x \cdot x + t_1}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (cbrt
          (log1p (+ (* 0.5 (* (pow (/ x y) 2.0) (exp -1.0))) (expm1 -1.0)))))
        (t_1 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (* t_0 (* t_0 t_0))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_1) (+ (* x x) t_1))
       (+
        1.0
        (*
         (exp
          (*
           2.0
           (cast
            (!
             :precision
             binary32
             (cast (! :precision binary64 (log (/ y x))))))))
         -8.0))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = cbrt(log1p(((0.5 * (pow((x / y), 2.0) * exp(-1.0))) + expm1(-1.0))));
	double t_1 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = t_0 * (t_0 * t_0);
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_1) / ((x * x) + t_1);
	} else {
		double tmp_3 = log((y / x));
		double tmp_2 = (float) tmp_3;
		tmp = 1.0 + (exp((2.0 * ((double) tmp_2))) * -8.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = cbrt(log1p(Float64(Float64(0.5 * Float64((Float64(x / y) ^ 2.0) * exp(-1.0))) + expm1(-1.0))))
	t_1 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-70)
		tmp = Float64(t_0 * Float64(t_0 * t_0));
	elseif (Float64(x * x) <= 2e+220)
		tmp = Float64(Float64(Float64(x * x) - t_1) / Float64(Float64(x * x) + t_1));
	else
		tmp_3 = log(Float64(y / x))
		tmp_2 = Float32(tmp_3)
		tmp = Float64(1.0 + Float64(exp(Float64(2.0 * Float64(tmp_2))) * -8.0));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-70

   1. Initial program 57.7%

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

   2. Taylor expanded in x around 0 79.3%

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

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. times-frac 82.9%

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

      5. metadata-eval 82.9%

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

   4. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u_binary64 82.5%

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

   6. Applied rewrite-once 82.5%

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

   7. Taylor expanded in x around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. associate--l+ 79.6%

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

      3. associate-/l* 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

      6. associate-/r/ 79.6%

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

      7. times-frac 84.4%

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

      8. unpow2 84.4%

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

      9. expm1-def 84.4%

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

   9. Simplified 84.4%

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

       1. add-cube-cbrt_binary64 84.4%

          \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}} \]

   11. Applied rewrite-once 84.4%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}} \]

   if 4.9999999999999998e-70 < (*.f64 x x) < 2e220

   1. Initial program 74.6%

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

   if 2e220 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in y around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. unpow2 77.2%

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

   4. Simplified 77.2%

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

      1. frac-times 87.5%

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

      2. pow2 87.5%

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

      3. pow-to-exp 52.2%

         \[\leadsto 1 + \color{blue}{e^{\log \left(\frac{y}{x}\right) \cdot 2}} \cdot -8 \]

      4. *-commutative 52.2%

         \[\leadsto 1 + e^{\color{blue}{2 \cdot \log \left(\frac{y}{x}\right)}} \cdot -8 \]

   6. Applied egg-rr 52.2%

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

      1. rewrite-binary64/binary32-simplify 52.2%

         \[\leadsto \color{blue}{1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8} \]

   8. Applied rewrite-once 52.2%

      \[\leadsto 1 + e^{2 \cdot \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\log \left(\frac{y}{x}\right)} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}}} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)} \cdot \left(\sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8\\ \end{array} \]

Alternative 2: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-1\right) + \frac{e^{-1} \cdot \left(0.5 \cdot \frac{x}{y}\right)}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (log1p (+ (expm1 -1.0) (/ (* (exp -1.0) (* 0.5 (/ x y))) (/ y x))))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+
        1.0
        (*
         (exp
          (*
           2.0
           (cast
            (!
             :precision
             binary32
             (cast (! :precision binary64 (log (/ y x))))))))
         -8.0))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = log1p((expm1(-1.0) + ((exp(-1.0) * (0.5 * (x / y))) / (y / x))));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		double tmp_3 = log((y / x));
		double tmp_2 = (float) tmp_3;
		tmp = 1.0 + (exp((2.0 * ((double) tmp_2))) * -8.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-70)
		tmp = log1p(Float64(expm1(-1.0) + Float64(Float64(exp(-1.0) * Float64(0.5 * Float64(x / y))) / Float64(y / x))));
	elseif (Float64(x * x) <= 2e+220)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp_3 = log(Float64(y / x))
		tmp_2 = Float32(tmp_3)
		tmp = Float64(1.0 + Float64(exp(Float64(2.0 * Float64(tmp_2))) * -8.0));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-70

   1. Initial program 57.7%

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

   2. Taylor expanded in x around 0 79.3%

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

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. times-frac 82.9%

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

      5. metadata-eval 82.9%

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

   4. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u_binary64 82.5%

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

   6. Applied rewrite-once 82.5%

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

   7. Taylor expanded in x around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. associate--l+ 79.6%

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

      3. associate-/l* 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

      6. associate-/r/ 79.6%

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

      7. times-frac 84.4%

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

      8. unpow2 84.4%

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

      9. expm1-def 84.4%

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

   9. Simplified 84.4%

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

       1. associate-*r* 84.4%

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

       2. unpow2 84.4%

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

       3. clear-num 84.4%

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

       4. div-inv 84.4%

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

       5. *-commutative 84.4%

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

       6. associate-*l/ 84.4%

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

       7. associate-*l/ 84.4%

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

       8. *-commutative 84.4%

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

       9. *-commutative 84.4%

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

   11. Applied egg-rr 84.4%

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

   if 4.9999999999999998e-70 < (*.f64 x x) < 2e220

   1. Initial program 74.6%

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

   if 2e220 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in y around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. unpow2 77.2%

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

   4. Simplified 77.2%

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

      1. frac-times 87.5%

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

      2. pow2 87.5%

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

      3. pow-to-exp 52.2%

         \[\leadsto 1 + \color{blue}{e^{\log \left(\frac{y}{x}\right) \cdot 2}} \cdot -8 \]

      4. *-commutative 52.2%

         \[\leadsto 1 + e^{\color{blue}{2 \cdot \log \left(\frac{y}{x}\right)}} \cdot -8 \]

   6. Applied egg-rr 52.2%

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

      1. rewrite-binary64/binary32-simplify 52.2%

         \[\leadsto \color{blue}{1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8} \]

   8. Applied rewrite-once 52.2%

      \[\leadsto 1 + e^{2 \cdot \color{blue}{\langle \color{blue}{\left( \color{blue}{\langle \color{blue}{\left( \color{blue}{\log \left(\frac{y}{x}\right)} \right)_{\text{binary64}}} \rangle_{\text{binary32}}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}}} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-1\right) + \frac{e^{-1} \cdot \left(0.5 \cdot \frac{x}{y}\right)}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{2 \cdot \langle \left( \langle \left( \log \left(\frac{y}{x}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \right)_{\text{binary32}} \rangle_{\text{binary64}}} \cdot -8\\ \end{array} \]

Alternative 3: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-1\right) + \frac{e^{-1} \cdot \left(0.5 \cdot \frac{x}{y}\right)}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (log1p (+ (expm1 -1.0) (/ (* (exp -1.0) (* 0.5 (/ x y))) (/ y x))))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (/ (/ (* y -8.0) x) (/ x y)))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = log1p((expm1(-1.0) + ((exp(-1.0) * (0.5 * (x / y))) / (y / x))));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	}
	return tmp;
}

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = Math.log1p((Math.expm1(-1.0) + ((Math.exp(-1.0) * (0.5 * (x / y))) / (y / x))));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-70:
		tmp = math.log1p((math.expm1(-1.0) + ((math.exp(-1.0) * (0.5 * (x / y))) / (y / x))))
	elif (x * x) <= 2e+220:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (((y * -8.0) / x) / (x / y))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-70)
		tmp = log1p(Float64(expm1(-1.0) + Float64(Float64(exp(-1.0) * Float64(0.5 * Float64(x / y))) / Float64(y / x))));
	elseif (Float64(x * x) <= 2e+220)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y * -8.0) / x) / Float64(x / y)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-70], N[Log[1 + N[(N[(Exp[-1.0] - 1), $MachinePrecision] + N[(N[(N[Exp[-1.0], $MachinePrecision] * N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+220], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-70

   1. Initial program 57.7%

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

   2. Taylor expanded in x around 0 79.3%

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

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. times-frac 82.9%

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

      5. metadata-eval 82.9%

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

   4. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u_binary64 82.5%

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

   6. Applied rewrite-once 82.5%

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

   7. Taylor expanded in x around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. associate--l+ 79.6%

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

      3. associate-/l* 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

      6. associate-/r/ 79.6%

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

      7. times-frac 84.4%

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

      8. unpow2 84.4%

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

      9. expm1-def 84.4%

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

   9. Simplified 84.4%

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

       1. associate-*r* 84.4%

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

       2. unpow2 84.4%

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

       3. clear-num 84.4%

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

       4. div-inv 84.4%

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

       5. *-commutative 84.4%

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

       6. associate-*l/ 84.4%

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

       7. associate-*l/ 84.4%

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

       8. *-commutative 84.4%

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

       9. *-commutative 84.4%

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

   11. Applied egg-rr 84.4%

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

   if 4.9999999999999998e-70 < (*.f64 x x) < 2e220

   1. Initial program 74.6%

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

   if 2e220 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in y around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. unpow2 77.2%

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

   4. Simplified 77.2%

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

      1. frac-times 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

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

      4. associate-*l/ 87.5%

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

      5. associate-*l/ 87.5%

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

   6. Applied egg-rr 87.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-1\right) + \frac{e^{-1} \cdot \left(0.5 \cdot \frac{x}{y}\right)}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 4: 79.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (+ -1.0 (* 0.5 (/ (/ x y) (/ y x))))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (/ (/ (* y -8.0) x) (/ x y)))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 5d-70) then
        tmp = (-1.0d0) + (0.5d0 * ((x / y) / (y / x)))
    else if ((x * x) <= 2d+220) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + (((y * (-8.0d0)) / x) / (x / y))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-70) {
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-70:
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)))
	elif (x * x) <= 2e+220:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (((y * -8.0) / x) / (x / y))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-70)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))));
	elseif (Float64(x * x) <= 2e+220)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y * -8.0) / x) / Float64(x / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 5e-70)
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	elseif ((x * x) <= 2e+220)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-70], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+220], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-70

   1. Initial program 57.7%

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

   2. Taylor expanded in x around 0 79.3%

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

      1. fma-neg 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. times-frac 82.9%

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

      5. metadata-eval 82.9%

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

   4. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 82.9%

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

      2. *-commutative 82.9%

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

      3. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   8. Applied egg-rr 82.9%

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

   if 4.9999999999999998e-70 < (*.f64 x x) < 2e220

   1. Initial program 74.6%

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

   if 2e220 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in y around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. unpow2 77.2%

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

   4. Simplified 77.2%

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

      1. frac-times 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

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

      4. associate-*l/ 87.5%

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

      5. associate-*l/ 87.5%

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

   6. Applied egg-rr 87.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-70}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 5: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 6 \lor \neg \left(y \leq 5.8 \cdot 10^{+35}\right):\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 9.8e-67)
   (+ 1.0 (/ (/ (* y -8.0) x) (/ x y)))
   (if (or (<= y 6.0) (not (<= y 5.8e+35)))
     (+ -1.0 (* 0.5 (/ (/ x y) (/ y x))))
     (+ 1.0 (* -8.0 (/ (* y y) (* x x)))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-67) {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	} else if ((y <= 6.0) || !(y <= 5.8e+35)) {
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	} else {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.8d-67) then
        tmp = 1.0d0 + (((y * (-8.0d0)) / x) / (x / y))
    else if ((y <= 6.0d0) .or. (.not. (y <= 5.8d+35))) then
        tmp = (-1.0d0) + (0.5d0 * ((x / y) / (y / x)))
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y * y) / (x * x)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-67) {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	} else if ((y <= 6.0) || !(y <= 5.8e+35)) {
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	} else {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 9.8e-67:
		tmp = 1.0 + (((y * -8.0) / x) / (x / y))
	elif (y <= 6.0) or not (y <= 5.8e+35):
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)))
	else:
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 9.8e-67)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * -8.0) / x) / Float64(x / y)));
	elseif ((y <= 6.0) || !(y <= 5.8e+35))
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y * y) / Float64(x * x))));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.8e-67)
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	elseif ((y <= 6.0) || ~((y <= 5.8e+35)))
		tmp = -1.0 + (0.5 * ((x / y) / (y / x)));
	else
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 9.8e-67], N[(1.0 + N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 6.0], N[Not[LessEqual[y, 5.8e+35]], $MachinePrecision]], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.79999999999999987e-67

   1. Initial program 48.2%

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

   2. Taylor expanded in y around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

   4. Simplified 50.7%

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

      1. frac-times 59.4%

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

      2. clear-num 59.4%

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

      3. un-div-inv 59.4%

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

      4. associate-*l/ 59.4%

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

      5. associate-*l/ 59.4%

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

   6. Applied egg-rr 59.4%

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

   if 9.79999999999999987e-67 < y < 6 or 5.79999999999999989e35 < y

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 76.7%

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

      1. fma-neg 76.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 76.7%

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

      3. unpow2 76.7%

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

      4. times-frac 78.3%

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

      5. metadata-eval 78.3%

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

   4. Simplified 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 78.3%

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

      2. *-commutative 78.3%

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

      3. pow2 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. pow2 78.3%

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

      2. clear-num 78.3%

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

      3. un-div-inv 78.3%

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

   8. Applied egg-rr 78.3%

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

   if 6 < y < 5.79999999999999989e35

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. unpow2 61.4%

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

      3. unpow2 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 6 \lor \neg \left(y \leq 5.8 \cdot 10^{+35}\right):\\ \;\;\;\;-1 + 0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \end{array} \]

Alternative 6: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.2:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-66)
   1.0
   (if (<= y 7.2)
     -1.0
     (if (<= y 3.8e+35) (+ 1.0 (* -8.0 (/ (* y y) (* x x)))) -1.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-66) {
		tmp = 1.0;
	} else if (y <= 7.2) {
		tmp = -1.0;
	} else if (y <= 3.8e+35) {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d-66) then
        tmp = 1.0d0
    else if (y <= 7.2d0) then
        tmp = -1.0d0
    else if (y <= 3.8d+35) then
        tmp = 1.0d0 + ((-8.0d0) * ((y * y) / (x * x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-66) {
		tmp = 1.0;
	} else if (y <= 7.2) {
		tmp = -1.0;
	} else if (y <= 3.8e+35) {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.1e-66:
		tmp = 1.0
	elif y <= 7.2:
		tmp = -1.0
	elif y <= 3.8e+35:
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)))
	else:
		tmp = -1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-66)
		tmp = 1.0;
	elseif (y <= 7.2)
		tmp = -1.0;
	elseif (y <= 3.8e+35)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-66)
		tmp = 1.0;
	elseif (y <= 7.2)
		tmp = -1.0;
	elseif (y <= 3.8e+35)
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.1e-66], 1.0, If[LessEqual[y, 7.2], -1.0, If[LessEqual[y, 3.8e+35], N[(1.0 + N[(-8.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1000000000000001e-66

   1. Initial program 48.2%

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

   2. Taylor expanded in x around inf 58.3%

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

   if 1.1000000000000001e-66 < y < 7.20000000000000018 or 3.8e35 < y

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 77.3%

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

   if 7.20000000000000018 < y < 3.8e35

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. unpow2 61.4%

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

      3. unpow2 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.2:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-66}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 7:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-66)
   (+ 1.0 (/ (/ (* y -8.0) x) (/ x y)))
   (if (<= y 7.0)
     -1.0
     (if (<= y 3.5e+35) (+ 1.0 (* -8.0 (/ (* y y) (* x x)))) -1.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-66) {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	} else if (y <= 7.0) {
		tmp = -1.0;
	} else if (y <= 3.5e+35) {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.4d-66) then
        tmp = 1.0d0 + (((y * (-8.0d0)) / x) / (x / y))
    else if (y <= 7.0d0) then
        tmp = -1.0d0
    else if (y <= 3.5d+35) then
        tmp = 1.0d0 + ((-8.0d0) * ((y * y) / (x * x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-66) {
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	} else if (y <= 7.0) {
		tmp = -1.0;
	} else if (y <= 3.5e+35) {
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.4e-66:
		tmp = 1.0 + (((y * -8.0) / x) / (x / y))
	elif y <= 7.0:
		tmp = -1.0
	elif y <= 3.5e+35:
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)))
	else:
		tmp = -1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-66)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * -8.0) / x) / Float64(x / y)));
	elseif (y <= 7.0)
		tmp = -1.0;
	elseif (y <= 3.5e+35)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y * y) / Float64(x * x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-66)
		tmp = 1.0 + (((y * -8.0) / x) / (x / y));
	elseif (y <= 7.0)
		tmp = -1.0;
	elseif (y <= 3.5e+35)
		tmp = 1.0 + (-8.0 * ((y * y) / (x * x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.4e-66], N[(1.0 + N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.0], -1.0, If[LessEqual[y, 3.5e+35], N[(1.0 + N[(-8.0 * N[(N[(y * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4e-66

   1. Initial program 48.2%

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

   2. Taylor expanded in y around 0 50.7%

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

      1. *-commutative 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

   4. Simplified 50.7%

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

      1. frac-times 59.4%

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

      2. clear-num 59.4%

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

      3. un-div-inv 59.4%

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

      4. associate-*l/ 59.4%

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

      5. associate-*l/ 59.4%

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

   6. Applied egg-rr 59.4%

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

   if 1.4e-66 < y < 7 or 3.5000000000000001e35 < y

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 77.3%

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

   if 7 < y < 3.5000000000000001e35

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0 61.4%

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

      1. *-commutative 61.4%

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

      2. unpow2 61.4%

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

      3. unpow2 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-66}:\\ \;\;\;\;1 + \frac{\frac{y \cdot -8}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 7:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \frac{y \cdot y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 73.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.16:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-67) 1.0 (if (<= y 0.16) -1.0 (if (<= y 5e+35) 1.0 -1.0))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-67) {
		tmp = 1.0;
	} else if (y <= 0.16) {
		tmp = -1.0;
	} else if (y <= 5e+35) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.4d-67) then
        tmp = 1.0d0
    else if (y <= 0.16d0) then
        tmp = -1.0d0
    else if (y <= 5d+35) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-67) {
		tmp = 1.0;
	} else if (y <= 0.16) {
		tmp = -1.0;
	} else if (y <= 5e+35) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.4e-67:
		tmp = 1.0
	elif y <= 0.16:
		tmp = -1.0
	elif y <= 5e+35:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-67)
		tmp = 1.0;
	elseif (y <= 0.16)
		tmp = -1.0;
	elseif (y <= 5e+35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.4e-67)
		tmp = 1.0;
	elseif (y <= 0.16)
		tmp = -1.0;
	elseif (y <= 5e+35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.4e-67], 1.0, If[LessEqual[y, 0.16], -1.0, If[LessEqual[y, 5e+35], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.40000000000000032e-67 or 0.160000000000000003 < y < 5.00000000000000021e35

   1. Initial program 48.6%

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

   2. Taylor expanded in x around inf 58.3%

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

   if 5.40000000000000032e-67 < y < 0.160000000000000003 or 5.00000000000000021e35 < y

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 77.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.16:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 50.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ -1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

x = abs(x)
y = abs(y)
def code(x, y):
	return -1.0

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 49.6%

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

2. Taylor expanded in x around 0 53.5%

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

3. Final simplification 53.5%

   \[\leadsto -1 \]

Developer target: 50.2% 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 2023297 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (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))))