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

Percentage Accurate: 49.8% → 80.6%

Time: 9.7s

Alternatives: 6

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

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (log1p (+ (* 0.5 (* (pow (/ x y) 2.0) (exp -1.0))) (expm1 -1.0)))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))

C

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

Java

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(((0.5 * (Math.pow((x / y), 2.0) * Math.exp(-1.0))) + Math.expm1(-1.0)));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-70:
		tmp = math.log1p(((0.5 * (math.pow((x / y), 2.0) * math.exp(-1.0))) + math.expm1(-1.0)))
	elif (x * x) <= 2e+220:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-70)
		tmp = log1p(Float64(Float64(0.5 * Float64((Float64(x / y) ^ 2.0) * exp(-1.0))) + expm1(-1.0)));
	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(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

Wolfram

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[(0.5 * N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[-1.0] - 1), $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[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $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. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. log1p-expm1-u 82.5%

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

      2. *-commutative 82.5%

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

      3. fma-def 82.5%

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

   8. Applied egg-rr 82.5%

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

   9. Taylor expanded in x around 0 79.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{-1} + 0.5 \cdot \frac{{x}^{2} \cdot e^{-1}}{{y}^{2}}\right) - 1}\right) \]
   10. 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. associate-/r/ 79.6%

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

       5. unpow2 79.6%

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

       6. unpow2 79.6%

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

       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%

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

   11. Simplified 84.4%

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

   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 x around inf 77.2%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 77.2%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. unpow2 77.2%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      3. associate-*r/ 77.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      4. *-commutative 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. unpow2 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-*r* 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      7. unpow2 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}}\right) \]

      8. associate-*r/ 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \color{blue}{\frac{4 \cdot {y}^{2}}{x \cdot x}}\right) \]

      9. *-commutative 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{{y}^{2} \cdot 4}}{x \cdot x}\right) \]

      10. unpow2 77.2%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{\left(y \cdot y\right)} \cdot 4}{x \cdot x}\right) \]

      11. associate-*r* 77.2%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x}\right) \]

   4. Simplified 77.2%

      \[\leadsto \color{blue}{1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right)} \]
   5. Step-by-step derivation

      1. frac-2neg 77.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x}} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right) \]

      2. frac-2neg 77.2%

         \[\leadsto 1 + \left(\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x} - \color{blue}{\frac{-y \cdot \left(y \cdot 4\right)}{-x \cdot x}}\right) \]

      3. sub-div 77.2%

         \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]

   6. Applied egg-rr 77.2%

      \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]
   7. Step-by-step derivation

      1. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \color{blue}{-1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      2. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right)\right)} - -1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x} \]

      3. distribute-lft-out-- 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      4. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-\left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      5. associate-*r* 77.2%

         \[\leadsto 1 + \frac{-\left(y \cdot \left(y \cdot -4\right) - \color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{-x \cdot x} \]

      6. associate-*r* 77.2%

         \[\leadsto 1 + \frac{-\left(\color{blue}{\left(y \cdot y\right) \cdot -4} - \left(y \cdot y\right) \cdot 4\right)}{-x \cdot x} \]

      7. distribute-lft-out-- 77.2%

         \[\leadsto 1 + \frac{-\color{blue}{\left(y \cdot y\right) \cdot \left(-4 - 4\right)}}{-x \cdot x} \]

      8. metadata-eval 77.2%

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

      9. distribute-rgt-neg-in 77.2%

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

      10. metadata-eval 77.2%

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

      11. distribute-rgt-neg-in 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in y around 0 77.2%

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

       1. unpow2 77.2%

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

       2. unpow2 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in y around 0 77.2%

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

       1. unpow2 77.2%

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

       2. unpow2 77.2%

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

       3. times-frac 87.5%

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

   14. Simplified 87.5%

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

4. Final simplification 83.0%

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

Alternative 2: 79.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-70)
     (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))
     (if (<= (* x x) 2e+220)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))

C

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) * (x / y)));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 5d-70) then
        tmp = (-1.0d0) + (0.5d0 * ((x / y) * (x / y)))
    else if ((x * x) <= 2d+220) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

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) * (x / y)));
	} else if ((x * x) <= 2e+220) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

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

Julia

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(x / y))));
	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(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

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) * (x / y)));
	elseif ((x * x) <= 2e+220)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

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[(x / y), $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[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $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. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. unpow2 82.9%

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

   8. Applied egg-rr 82.9%

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

   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 x around inf 77.2%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 77.2%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. unpow2 77.2%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      3. associate-*r/ 77.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      4. *-commutative 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. unpow2 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-*r* 77.2%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      7. unpow2 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}}\right) \]

      8. associate-*r/ 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \color{blue}{\frac{4 \cdot {y}^{2}}{x \cdot x}}\right) \]

      9. *-commutative 77.2%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{{y}^{2} \cdot 4}}{x \cdot x}\right) \]

      10. unpow2 77.2%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{\left(y \cdot y\right)} \cdot 4}{x \cdot x}\right) \]

      11. associate-*r* 77.2%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x}\right) \]

   4. Simplified 77.2%

      \[\leadsto \color{blue}{1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right)} \]
   5. Step-by-step derivation

      1. frac-2neg 77.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x}} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right) \]

      2. frac-2neg 77.2%

         \[\leadsto 1 + \left(\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x} - \color{blue}{\frac{-y \cdot \left(y \cdot 4\right)}{-x \cdot x}}\right) \]

      3. sub-div 77.2%

         \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]

   6. Applied egg-rr 77.2%

      \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]
   7. Step-by-step derivation

      1. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \color{blue}{-1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      2. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right)\right)} - -1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x} \]

      3. distribute-lft-out-- 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      4. neg-mul-1 77.2%

         \[\leadsto 1 + \frac{\color{blue}{-\left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      5. associate-*r* 77.2%

         \[\leadsto 1 + \frac{-\left(y \cdot \left(y \cdot -4\right) - \color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{-x \cdot x} \]

      6. associate-*r* 77.2%

         \[\leadsto 1 + \frac{-\left(\color{blue}{\left(y \cdot y\right) \cdot -4} - \left(y \cdot y\right) \cdot 4\right)}{-x \cdot x} \]

      7. distribute-lft-out-- 77.2%

         \[\leadsto 1 + \frac{-\color{blue}{\left(y \cdot y\right) \cdot \left(-4 - 4\right)}}{-x \cdot x} \]

      8. metadata-eval 77.2%

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

      9. distribute-rgt-neg-in 77.2%

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

      10. metadata-eval 77.2%

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

      11. distribute-rgt-neg-in 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in y around 0 77.2%

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

       1. unpow2 77.2%

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

       2. unpow2 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in y around 0 77.2%

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

       1. unpow2 77.2%

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

       2. unpow2 77.2%

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

       3. times-frac 87.5%

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

   14. Simplified 87.5%

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

4. Final simplification 82.3%

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

Alternative 3: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-66} \lor \neg \left(y \leq 7\right) \land y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 1.4e-66) (and (not (<= y 7.0)) (<= y 3.5e+35)))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
   -1.0))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 1.4d-66) .or. (.not. (y <= 7.0d0)) .and. (y <= 3.5d+35)) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= 1.4e-66) or (not (y <= 7.0) and (y <= 3.5e+35)):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 1.4e-66], And[N[Not[LessEqual[y, 7.0]], $MachinePrecision], LessEqual[y, 3.5e+35]]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

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

   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 51.0%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 51.1%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. unpow2 51.1%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      3. associate-*r/ 51.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      4. *-commutative 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. unpow2 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-*r* 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      7. unpow2 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}}\right) \]

      8. associate-*r/ 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \color{blue}{\frac{4 \cdot {y}^{2}}{x \cdot x}}\right) \]

      9. *-commutative 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{{y}^{2} \cdot 4}}{x \cdot x}\right) \]

      10. unpow2 51.1%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{\left(y \cdot y\right)} \cdot 4}{x \cdot x}\right) \]

      11. associate-*r* 51.1%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x}\right) \]

   4. Simplified 51.1%

      \[\leadsto \color{blue}{1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right)} \]
   5. Step-by-step derivation

      1. frac-2neg 51.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x}} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right) \]

      2. frac-2neg 51.1%

         \[\leadsto 1 + \left(\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x} - \color{blue}{\frac{-y \cdot \left(y \cdot 4\right)}{-x \cdot x}}\right) \]

      3. sub-div 51.1%

         \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]

   6. Applied egg-rr 51.1%

      \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]
   7. Step-by-step derivation

      1. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \color{blue}{-1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      2. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right)\right)} - -1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x} \]

      3. distribute-lft-out-- 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      4. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-\left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      5. associate-*r* 51.1%

         \[\leadsto 1 + \frac{-\left(y \cdot \left(y \cdot -4\right) - \color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{-x \cdot x} \]

      6. associate-*r* 51.1%

         \[\leadsto 1 + \frac{-\left(\color{blue}{\left(y \cdot y\right) \cdot -4} - \left(y \cdot y\right) \cdot 4\right)}{-x \cdot x} \]

      7. distribute-lft-out-- 51.1%

         \[\leadsto 1 + \frac{-\color{blue}{\left(y \cdot y\right) \cdot \left(-4 - 4\right)}}{-x \cdot x} \]

      8. metadata-eval 51.1%

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

      9. distribute-rgt-neg-in 51.1%

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

      10. metadata-eval 51.1%

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

      11. distribute-rgt-neg-in 51.1%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in y around 0 51.1%

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

       1. unpow2 51.1%

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

       2. unpow2 51.1%

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

   11. Simplified 51.1%

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

   12. Taylor expanded in y around 0 51.1%

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

       1. unpow2 51.1%

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

       2. unpow2 51.1%

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

       3. times-frac 59.4%

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

   14. Simplified 59.4%

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

   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%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-66} \lor \neg \left(y \leq 7\right) \land y \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 63.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 9.8e-67) (and (not (<= y 6.0)) (<= y 5.8e+35)))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
   (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 9.8e-67], And[N[Not[LessEqual[y, 6.0]], $MachinePrecision], LessEqual[y, 5.8e+35]]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 51.0%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 51.1%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. unpow2 51.1%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      3. associate-*r/ 51.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-4 \cdot {y}^{2}}{x \cdot x}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      4. *-commutative 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{{y}^{2} \cdot -4}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. unpow2 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{\left(y \cdot y\right)} \cdot -4}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-*r* 51.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{y \cdot \left(y \cdot -4\right)}}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      7. unpow2 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - 4 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}}\right) \]

      8. associate-*r/ 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \color{blue}{\frac{4 \cdot {y}^{2}}{x \cdot x}}\right) \]

      9. *-commutative 51.1%

         \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{{y}^{2} \cdot 4}}{x \cdot x}\right) \]

      10. unpow2 51.1%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{\left(y \cdot y\right)} \cdot 4}{x \cdot x}\right) \]

      11. associate-*r* 51.1%

          \[\leadsto 1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{\color{blue}{y \cdot \left(y \cdot 4\right)}}{x \cdot x}\right) \]

   4. Simplified 51.1%

      \[\leadsto \color{blue}{1 + \left(\frac{y \cdot \left(y \cdot -4\right)}{x \cdot x} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right)} \]
   5. Step-by-step derivation

      1. frac-2neg 51.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x}} - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}\right) \]

      2. frac-2neg 51.1%

         \[\leadsto 1 + \left(\frac{-y \cdot \left(y \cdot -4\right)}{-x \cdot x} - \color{blue}{\frac{-y \cdot \left(y \cdot 4\right)}{-x \cdot x}}\right) \]

      3. sub-div 51.1%

         \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]

   6. Applied egg-rr 51.1%

      \[\leadsto 1 + \color{blue}{\frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \left(-y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x}} \]
   7. Step-by-step derivation

      1. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\left(-y \cdot \left(y \cdot -4\right)\right) - \color{blue}{-1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      2. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right)\right)} - -1 \cdot \left(y \cdot \left(y \cdot 4\right)\right)}{-x \cdot x} \]

      3. distribute-lft-out-- 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      4. neg-mul-1 51.1%

         \[\leadsto 1 + \frac{\color{blue}{-\left(y \cdot \left(y \cdot -4\right) - y \cdot \left(y \cdot 4\right)\right)}}{-x \cdot x} \]

      5. associate-*r* 51.1%

         \[\leadsto 1 + \frac{-\left(y \cdot \left(y \cdot -4\right) - \color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{-x \cdot x} \]

      6. associate-*r* 51.1%

         \[\leadsto 1 + \frac{-\left(\color{blue}{\left(y \cdot y\right) \cdot -4} - \left(y \cdot y\right) \cdot 4\right)}{-x \cdot x} \]

      7. distribute-lft-out-- 51.1%

         \[\leadsto 1 + \frac{-\color{blue}{\left(y \cdot y\right) \cdot \left(-4 - 4\right)}}{-x \cdot x} \]

      8. metadata-eval 51.1%

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

      9. distribute-rgt-neg-in 51.1%

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

      10. metadata-eval 51.1%

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

      11. distribute-rgt-neg-in 51.1%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in y around 0 51.1%

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

       1. unpow2 51.1%

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

       2. unpow2 51.1%

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

   11. Simplified 51.1%

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

   12. Taylor expanded in y around 0 51.1%

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

       1. unpow2 51.1%

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

       2. unpow2 51.1%

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

       3. times-frac 59.4%

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

   14. Simplified 59.4%

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

   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. pow2 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. unpow2 78.3%

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

   8. Applied egg-rr 78.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

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

Alternative 5: 62.2% accurate, 2.6× speedup

Math

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

FPCore

(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 -1.0))))

C

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;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

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
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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
	else:
		tmp = -1.0
	return tmp

Julia

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 = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

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;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e-66], 1.0, If[LessEqual[y, 7.2], -1.0, If[LessEqual[y, 3.8e+35], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

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

   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 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%

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

4. Final simplification 64.3%

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

Alternative 6: 50.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

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))))