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

Percentage Accurate: 50.2% → 93.8%

Time: 8.7s

Alternatives: 8

Speedup: 19.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 93.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\mathsf{fma}\left(0.25, \frac{x \cdot x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 0.0)
   (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)
   (if (<= (* x x) 2e+306)
     (-
      (/ (* x x) (pow (hypot x (* y 2.0)) 2.0))
      (/ y (fma 0.25 (/ (* x x) y) y)))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 0.0) {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	} else if ((x * x) <= 2e+306) {
		tmp = ((x * x) / pow(hypot(x, (y * 2.0)), 2.0)) - (y / fma(0.25, ((x * x) / y), y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	elseif (Float64(x * x) <= 2e+306)
		tmp = Float64(Float64(Float64(x * x) / (hypot(x, Float64(y * 2.0)) ^ 2.0)) - Float64(y / fma(0.25, Float64(Float64(x * x) / y), y)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.0], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+306], N[(N[(N[(x * x), $MachinePrecision] / N[Power[N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(y / N[(0.25 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 0.0

   1. Initial program 50.0%

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

   3. Applied egg-rr 50.4%

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

   4. Taylor expanded in x around 0 70.0%

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

      1. *-commutative 70.0%

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

      2. fma-neg 70.0%

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

      3. unpow2 70.0%

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

      4. unpow2 70.0%

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

      5. times-frac 86.8%

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

      6. metadata-eval 86.8%

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

   6. Simplified 86.8%

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

      1. fma-udef 86.8%

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

      2. pow2 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. unpow2 86.8%

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

   10. Applied egg-rr 86.8%

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

   if 0.0 < (*.f64 x x) < 2.00000000000000003e306

   1. Initial program 73.7%

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

   3. Applied egg-rr 74.5%

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

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\color{blue}{0.25 \cdot \frac{{x}^{2}}{y} + y}} \]
   5. Step-by-step derivation

      1. fma-def 99.9%

         \[\leadsto \frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\color{blue}{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{y}, y\right)}} \]

      2. unpow2 99.9%

         \[\leadsto \frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\mathsf{fma}\left(0.25, \frac{\color{blue}{x \cdot x}}{y}, y\right)} \]

   6. Simplified 99.9%

      \[\leadsto \frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\color{blue}{\mathsf{fma}\left(0.25, \frac{x \cdot x}{y}, y\right)}} \]

   if 2.00000000000000003e306 < (*.f64 x x)

   1. Initial program 1.5%

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

   2. Taylor expanded in x around inf 87.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} - \frac{y}{\mathsf{fma}\left(0.25, \frac{x \cdot x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-205)
     1.0
     (if (<= t_0 2e+247)
       (/ (fma x x (* y (* y -4.0))) (fma y (* y 4.0) (* x x)))
       (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-205) {
		tmp = 1.0;
	} else if (t_0 <= 2e+247) {
		tmp = fma(x, x, (y * (y * -4.0))) / fma(y, (y * 4.0), (x * x));
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-205)
		tmp = 1.0;
	elseif (t_0 <= 2e+247)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / fma(y, Float64(y * 4.0), Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-205], 1.0, If[LessEqual[t$95$0, 2e+247], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-205

   1. Initial program 48.3%

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

   2. Taylor expanded in x around inf 87.3%

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

   if 2e-205 < (*.f64 (*.f64 y 4) y) < 1.9999999999999999e247

   1. Initial program 83.6%

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

   2. Taylor expanded in x around 0 83.6%

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

      1. unpow2 83.6%

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

      2. +-commutative 83.6%

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

      3. fma-def 83.6%

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

      4. unpow2 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

      1. expm1-log1p-u 83.6%

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

      2. expm1-udef 83.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)} - 1} \]

      3. +-commutative 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}}\right)} - 1 \]

      4. *-commutative 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{y \cdot \left(y \cdot 4\right)} + x \cdot x}\right)} - 1 \]

      5. fma-def 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}}\right)} - 1 \]

   6. Applied egg-rr 83.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

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

      2. expm1-log1p 83.6%

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

   8. Simplified 83.6%

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

   if 1.9999999999999999e247 < (*.f64 (*.f64 y 4) y)

   1. Initial program 12.3%

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

   3. Applied egg-rr 14.3%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. fma-neg 76.8%

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

      3. unpow2 76.8%

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

      4. unpow2 76.8%

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

      5. times-frac 88.3%

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

      6. metadata-eval 88.3%

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

   6. Simplified 88.3%

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

      1. fma-udef 88.3%

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

      2. pow2 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow2 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \]

Alternative 3: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-205)
     1.0
     (if (<= t_0 2e+247)
       (/ (+ (* x x) (* y (* y -4.0))) (fma y (* y 4.0) (* x x)))
       (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-205) {
		tmp = 1.0;
	} else if (t_0 <= 2e+247) {
		tmp = ((x * x) + (y * (y * -4.0))) / fma(y, (y * 4.0), (x * x));
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-205)
		tmp = 1.0;
	elseif (t_0 <= 2e+247)
		tmp = Float64(Float64(Float64(x * x) + Float64(y * Float64(y * -4.0))) / fma(y, Float64(y * 4.0), Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-205], 1.0, If[LessEqual[t$95$0, 2e+247], N[(N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-205

   1. Initial program 48.3%

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

   2. Taylor expanded in x around inf 87.3%

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

   if 2e-205 < (*.f64 (*.f64 y 4) y) < 1.9999999999999999e247

   1. Initial program 83.6%

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

   2. Taylor expanded in x around 0 83.6%

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

      1. unpow2 83.6%

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

      2. +-commutative 83.6%

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

      3. fma-def 83.6%

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

      4. unpow2 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

      1. expm1-log1p-u 83.6%

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

      2. expm1-udef 83.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)} - 1} \]

      3. +-commutative 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}}\right)} - 1 \]

      4. *-commutative 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{y \cdot \left(y \cdot 4\right)} + x \cdot x}\right)} - 1 \]

      5. fma-def 83.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}}\right)} - 1 \]

   6. Applied egg-rr 83.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 83.6%

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

      2. expm1-log1p 83.6%

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

   8. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}} \]
   9. Step-by-step derivation

      1. fma-udef 83.6%

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

   10. Applied egg-rr 83.6%

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

   if 1.9999999999999999e247 < (*.f64 (*.f64 y 4) y)

   1. Initial program 12.3%

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

   3. Applied egg-rr 14.3%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. fma-neg 76.8%

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

      3. unpow2 76.8%

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

      4. unpow2 76.8%

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

      5. times-frac 88.3%

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

      6. metadata-eval 88.3%

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

   6. Simplified 88.3%

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

      1. fma-udef 88.3%

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

      2. pow2 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow2 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x + y \cdot \left(y \cdot -4\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \]

Alternative 4: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-205)
     1.0
     (if (<= t_0 2e+247)
       (/ (fma x x (* y (* y -4.0))) (+ (* x x) t_0))
       (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-205) {
		tmp = 1.0;
	} else if (t_0 <= 2e+247) {
		tmp = fma(x, x, (y * (y * -4.0))) / ((x * x) + t_0);
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-205)
		tmp = 1.0;
	elseif (t_0 <= 2e+247)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-205], 1.0, If[LessEqual[t$95$0, 2e+247], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-205

   1. Initial program 48.3%

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

   2. Taylor expanded in x around inf 87.3%

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

   if 2e-205 < (*.f64 (*.f64 y 4) y) < 1.9999999999999999e247

   1. Initial program 83.6%

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

   2. Taylor expanded in x around 0 83.6%

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

      1. unpow2 83.6%

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

      2. +-commutative 83.6%

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

      3. fma-def 83.6%

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

      4. unpow2 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.6%

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

   4. Simplified 83.6%

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

   if 1.9999999999999999e247 < (*.f64 (*.f64 y 4) y)

   1. Initial program 12.3%

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

   3. Applied egg-rr 14.3%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. fma-neg 76.8%

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

      3. unpow2 76.8%

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

      4. unpow2 76.8%

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

      5. times-frac 88.3%

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

      6. metadata-eval 88.3%

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

   6. Simplified 88.3%

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

      1. fma-udef 88.3%

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

      2. pow2 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow2 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \]

Alternative 5: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-205)
     1.0
     (if (<= t_0 2e+247)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-205) {
		tmp = 1.0;
	} else if (t_0 <= 2e+247) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	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 (t_0 <= 2d-205) then
        tmp = 1.0d0
    else if (t_0 <= 2d+247) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = (((x / y) * (x / y)) * 0.5d0) + (-1.0d0)
    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 (t_0 <= 2e-205) {
		tmp = 1.0;
	} else if (t_0 <= 2e+247) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 2e-205:
		tmp = 1.0
	elif t_0 <= 2e+247:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-205)
		tmp = 1.0;
	elseif (t_0 <= 2e+247)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 2e-205)
		tmp = 1.0;
	elseif (t_0 <= 2e+247)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-205], 1.0, If[LessEqual[t$95$0, 2e+247], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-205

   1. Initial program 48.3%

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

   2. Taylor expanded in x around inf 87.3%

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

   if 2e-205 < (*.f64 (*.f64 y 4) y) < 1.9999999999999999e247

   1. Initial program 83.6%

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

   if 1.9999999999999999e247 < (*.f64 (*.f64 y 4) y)

   1. Initial program 12.3%

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

   3. Applied egg-rr 14.3%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. fma-neg 76.8%

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

      3. unpow2 76.8%

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

      4. unpow2 76.8%

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

      5. times-frac 88.3%

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

      6. metadata-eval 88.3%

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

   6. Simplified 88.3%

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

      1. fma-udef 88.3%

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

      2. pow2 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow2 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \]

Alternative 6: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 2e-68) 1.0 (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 2e-68) {
		tmp = 1.0;
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -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 * (y * 4.0d0)) <= 2d-68) then
        tmp = 1.0d0
    else
        tmp = (((x / y) * (x / y)) * 0.5d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 2e-68) {
		tmp = 1.0;
	} else {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 2e-68:
		tmp = 1.0
	else:
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 2e-68)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 2e-68)
		tmp = 1.0;
	else
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 2e-68], 1.0, N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 2.00000000000000013e-68

   1. Initial program 54.9%

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

   2. Taylor expanded in x around inf 82.9%

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

   if 2.00000000000000013e-68 < (*.f64 (*.f64 y 4) y)

   1. Initial program 47.0%

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

   3. Applied egg-rr 48.1%

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

   4. Taylor expanded in x around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. fma-neg 69.0%

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

      3. unpow2 69.0%

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

      4. unpow2 69.0%

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

      5. times-frac 75.2%

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

      6. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

      1. fma-udef 75.2%

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

      2. pow2 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. unpow2 75.2%

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

   10. Applied egg-rr 75.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \end{array} \]

Alternative 7: 62.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5000000000000.0) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5000000000000.0) {
		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 <= 5000000000000.0d0) 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 <= 5000000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5000000000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5000000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5000000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5000000000000.0], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5e12

   1. Initial program 53.1%

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

   2. Taylor expanded in x around inf 63.3%

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

   if 5e12 < y

   1. Initial program 43.9%

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

   2. Taylor expanded in x around 0 76.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 50.6% 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 50.7%

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

2. Taylor expanded in x around 0 46.7%

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

3. Final simplification 46.7%

   \[\leadsto -1 \]

Developer target: 50.7% 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 2023258 
(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))))