Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Percentage Accurate: 77.9% → 98.3%

Time: 8.4s

Alternatives: 10

Speedup: 6.1×

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Initial Program: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\mathsf{fma}\left(\frac{\frac{y}{x}}{x}, -0.3333333333333333, \frac{0.5}{x}\right) \cdot y - 1\right) \cdot y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+29)
   (/ (exp (- y)) x)
   (if (<= x 2.4e-17)
     (/ 1.0 x)
     (/
      (exp
       (* (- (* (fma (/ (/ y x) x) -0.3333333333333333 (/ 0.5 x)) y) 1.0) y))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+29) {
		tmp = exp(-y) / x;
	} else if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else {
		tmp = exp((((fma(((y / x) / x), -0.3333333333333333, (0.5 / x)) * y) - 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+29)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(exp(Float64(Float64(Float64(fma(Float64(Float64(y / x) / x), -0.3333333333333333, Float64(0.5 / x)) * y) - 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+29], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], N[(N[Exp[N[(N[(N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] * -0.3333333333333333 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4e29

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      2. lower-neg.f64 100.0

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

   if 2.39999999999999986e-17 < x

   1. Initial program 75.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \frac{e^{\left(\color{blue}{\mathsf{fma}\left(\frac{y}{{x}^{2}}, \frac{-1}{3}, \frac{1}{2} \cdot \frac{1}{x}\right)} \cdot y - 1\right) \cdot y}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, \frac{-1}{3}, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1\right) \cdot y}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{y}{x}}{x}}, \frac{-1}{3}, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1\right) \cdot y}}{x} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{y}{x}}{x}}, \frac{-1}{3}, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1\right) \cdot y}}{x} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\frac{\color{blue}{\frac{y}{x}}}{x}, \frac{-1}{3}, \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y - 1\right) \cdot y}}{x} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\frac{\frac{y}{x}}{x}, \frac{-1}{3}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{x}}\right) \cdot y - 1\right) \cdot y}}{x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\frac{\frac{y}{x}}{x}, \frac{-1}{3}, \frac{\color{blue}{\frac{1}{2}}}{x}\right) \cdot y - 1\right) \cdot y}}{x} \]

      14. lower-/.f64 100.0

          \[\leadsto \frac{e^{\left(\mathsf{fma}\left(\frac{\frac{y}{x}}{x}, -0.3333333333333333, \color{blue}{\frac{0.5}{x}}\right) \cdot y - 1\right) \cdot y}}{x} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{e^{\color{blue}{\left(\mathsf{fma}\left(\frac{\frac{y}{x}}{x}, -0.3333333333333333, \frac{0.5}{x}\right) \cdot y - 1\right) \cdot y}}}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{x \cdot x} + 0.16666666666666666\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+29)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 2.4e-17)
     (/ 1.0 x)
     (if (<= x 1.65e+195)
       (pow
        (fma
         (fma
          (*
           x
           (fma
            (-
             (+ (/ 0.3333333333333333 (* x x)) 0.16666666666666666)
             (/ 0.5 x))
            y
            (- 0.5 (/ 0.5 x))))
          y
          x)
         y
         x)
        -1.0)
       (/ (/ (fma (fma (- (* 0.5 y) 1.0) y 1.0) x (* (* y y) 0.5)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+29) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else if (x <= 1.65e+195) {
		tmp = pow(fma(fma((x * fma((((0.3333333333333333 / (x * x)) + 0.16666666666666666) - (0.5 / x)), y, (0.5 - (0.5 / x)))), y, x), y, x), -1.0);
	} else {
		tmp = (fma(fma(((0.5 * y) - 1.0), y, 1.0), x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+29)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.65e+195)
		tmp = fma(fma(Float64(x * fma(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 0.16666666666666666) - Float64(0.5 / x)), y, Float64(0.5 - Float64(0.5 / x)))), y, x), y, x) ^ -1.0;
	else
		tmp = Float64(Float64(fma(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0), x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+29], N[(N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.65e+195], N[Power[N[(N[(N[(x * N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] * y + x), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.4e29

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) + 1}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot y} + 1}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1, y, 1\right)}}{x} \]

   5. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 - \frac{-0.3333333333333333}{x \cdot x}\right) - \frac{-0.5}{x}, -y, \frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

   if 2.39999999999999986e-17 < x < 1.65e195

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\color{blue}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) + \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x} \]

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}{x} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\color{blue}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}}{x} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)} \cdot x}} \]

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{x + y \cdot \left(x + y \cdot \left(x \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot \left(x + y \cdot \left(x \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x + y \cdot \left(x \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x + y \cdot \left(x \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) + x \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)\right), y, x\right)}} \]

   7. Applied rewrites 82.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{x \cdot x} + 0.16666666666666666\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, x\right), y, x\right)}} \]

   if 1.65e195 < x

   1. Initial program 64.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{x \cdot x} + 0.16666666666666666\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.5 - \frac{0.5}{x}\right) \cdot y, x, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+29)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 2.4e-17)
     (/ 1.0 x)
     (if (<= x 1.55e+195)
       (pow (fma (fma (* (- 0.5 (/ 0.5 x)) y) x x) y x) -1.0)
       (/ (/ (fma (fma (- (* 0.5 y) 1.0) y 1.0) x (* (* y y) 0.5)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+29) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else if (x <= 1.55e+195) {
		tmp = pow(fma(fma(((0.5 - (0.5 / x)) * y), x, x), y, x), -1.0);
	} else {
		tmp = (fma(fma(((0.5 * y) - 1.0), y, 1.0), x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+29)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.55e+195)
		tmp = fma(fma(Float64(Float64(0.5 - Float64(0.5 / x)) * y), x, x), y, x) ^ -1.0;
	else
		tmp = Float64(Float64(fma(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0), x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+29], N[(N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.55e+195], N[Power[N[(N[(N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision] * y + x), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.4e29

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) + 1}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot y} + 1}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1, y, 1\right)}}{x} \]

   5. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 - \frac{-0.3333333333333333}{x \cdot x}\right) - \frac{-0.5}{x}, -y, \frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

   if 2.39999999999999986e-17 < x < 1.5500000000000001e195

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\color{blue}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) + \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x} \]

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}{x} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\color{blue}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}}{x} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)} \cdot x}} \]

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if 1.5500000000000001e195 < x

   1. Initial program 64.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{\color{blue}{x}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.5 - \frac{0.5}{x}\right) \cdot y, x, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.5 - \frac{0.5}{x}\right) \cdot y, x, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{x} - 1, y, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+29)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 2.4e-17)
     (/ 1.0 x)
     (if (<= x 2.8e+195)
       (pow (fma (fma (* (- 0.5 (/ 0.5 x)) y) x x) y x) -1.0)
       (/ (fma (- (/ (* (* 0.5 y) x) x) 1.0) y 1.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+29) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else if (x <= 2.8e+195) {
		tmp = pow(fma(fma(((0.5 - (0.5 / x)) * y), x, x), y, x), -1.0);
	} else {
		tmp = fma(((((0.5 * y) * x) / x) - 1.0), y, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+29)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	elseif (x <= 2.8e+195)
		tmp = fma(fma(Float64(Float64(0.5 - Float64(0.5 / x)) * y), x, x), y, x) ^ -1.0;
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * y) * x) / x) - 1.0), y, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+29], N[(N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 2.8e+195], N[Power[N[(N[(N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x + x), $MachinePrecision] * y + x), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * y), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.4e29

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) + 1}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot y} + 1}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1, y, 1\right)}}{x} \]

   5. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 - \frac{-0.3333333333333333}{x \cdot x}\right) - \frac{-0.5}{x}, -y, \frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

   if 2.39999999999999986e-17 < x < 2.7999999999999998e195

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\color{blue}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) + \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x} \]

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}{x} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\color{blue}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}}{x} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)} \cdot x}} \]

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if 2.7999999999999998e195 < x

   1. Initial program 64.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot y + \frac{1}{2} \cdot \left(x \cdot y\right)}{x} - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{x} - 1, y, 1\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.6%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{x} - 1, y, 1\right)}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.5 - \frac{0.5}{x}\right) \cdot y, x, x\right), y, x\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{x} - 1, y, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.4e+29) (not (<= x 2.4e-17))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+29) || !(x <= 2.4e-17)) {
		tmp = exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5.4d+29)) .or. (.not. (x <= 2.4d-17))) then
        tmp = exp(-y) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+29) || !(x <= 2.4e-17)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.4e+29) or not (x <= 2.4e-17):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.4e+29) || !(x <= 2.4e-17))
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.4e+29) || ~((x <= 2.4e-17)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.4e+29], N[Not[LessEqual[x, 2.4e-17]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e29 or 2.39999999999999986e-17 < x

   1. Initial program 76.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(y\right)}}}{x} \]

      2. lower-neg.f64 99.7

         \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{e^{\color{blue}{-y}}}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y, x, x\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e-17) (/ 1.0 x) (pow (fma y x x) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else {
		tmp = pow(fma(y, x, x), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	else
		tmp = fma(y, x, x) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], N[Power[N[(y * x + x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999986e-17

   1. Initial program 83.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 88.1%

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

   if 2.39999999999999986e-17 < x

   1. Initial program 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{\color{blue}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) + \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x} \]

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}{x} \]

      5. sinh---cosh-rev N/A

         \[\leadsto \frac{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{\color{blue}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}}}}{x} \]

      6. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \cosh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) - \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right) \cdot \sinh \left(x \cdot \log \left(\frac{x}{x + y}\right)\right)}{e^{\mathsf{neg}\left(x \cdot \log \left(\frac{x}{x + y}\right)\right)} \cdot x}} \]

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 68.9

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

   7. Applied rewrites 68.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, x, x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y, x, x\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{x} - 1, y, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+29)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 2.4e-17)
     (/ 1.0 x)
     (/ (fma (- (/ (* (* 0.5 y) x) x) 1.0) y 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+29) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 2.4e-17) {
		tmp = 1.0 / x;
	} else {
		tmp = fma(((((0.5 * y) * x) / x) - 1.0), y, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+29)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 2.4e-17)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(0.5 * y) * x) / x) - 1.0), y, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+29], N[(N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.4e-17], N[(1.0 / x), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * y), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4e29

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) + 1}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot y} + 1}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1, y, 1\right)}}{x} \]

   5. Applied rewrites 85.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 - \frac{-0.3333333333333333}{x \cdot x}\right) - \frac{-0.5}{x}, -y, \frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

   if 2.39999999999999986e-17 < x

   1. Initial program 75.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot y + \frac{1}{2} \cdot \left(x \cdot y\right)}{x} - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{x} - 1, y, 1\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{x} - 1, y, 1\right)}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.4e+29) (not (<= x 2.4e-17)))
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+29) || !(x <= 2.4e-17)) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.4e+29) || !(x <= 2.4e-17))
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.4e+29], N[Not[LessEqual[x, 2.4e-17]], $MachinePrecision]], N[(N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e29 or 2.39999999999999986e-17 < x

   1. Initial program 76.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) + 1}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot y} + 1}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1, y, 1\right)}}{x} \]

   5. Applied rewrites 76.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 - \frac{-0.3333333333333333}{x \cdot x}\right) - \frac{-0.5}{x}, -y, \frac{0.5}{x} + 0.5\right) \cdot y - 1, y, 1\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot y\right) \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x} \]

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.4e+29) (not (<= x 2.4e-17)))
   (/ (fma (- (* 0.5 y) 1.0) y 1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.4e+29) || !(x <= 2.4e-17)) {
		tmp = fma(((0.5 * y) - 1.0), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.4e+29) || !(x <= 2.4e-17))
		tmp = Float64(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.4e+29], N[Not[LessEqual[x, 2.4e-17]], $MachinePrecision]], N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e29 or 2.39999999999999986e-17 < x

   1. Initial program 76.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} \cdot y - 1, y, 1\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

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

   if -5.4e29 < x < 2.39999999999999986e-17

   1. Initial program 85.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+29} \lor \neg \left(x \leq 2.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 81.1%

   \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{x} \]
4. Step-by-step derivation

5. Applied rewrites 80.5%

   \[\leadsto \frac{\color{blue}{1}}{x} \]
6. Add Preprocessing

Developer Target 1: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-1}{y}}}{x}\\ t_1 := \frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (/ -1.0 y)) x)) (t_1 (/ (pow (/ x (+ y x)) x) x)))
   (if (< y -3.7311844206647956e+94)
     t_0
     (if (< y 2.817959242728288e+37)
       t_1
       (if (< y 2.347387415166998e+178) (log (exp t_1)) t_0)))))

C

double code(double x, double y) {
	double t_0 = exp((-1.0 / y)) / x;
	double t_1 = pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = log(exp(t_1));
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-1.0 / y)) / x;
	double t_1 = Math.pow((x / (y + x)), x) / x;
	double tmp;
	if (y < -3.7311844206647956e+94) {
		tmp = t_0;
	} else if (y < 2.817959242728288e+37) {
		tmp = t_1;
	} else if (y < 2.347387415166998e+178) {
		tmp = Math.log(Math.exp(t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-1.0 / y)) / x
	t_1 = math.pow((x / (y + x)), x) / x
	tmp = 0
	if y < -3.7311844206647956e+94:
		tmp = t_0
	elif y < 2.817959242728288e+37:
		tmp = t_1
	elif y < 2.347387415166998e+178:
		tmp = math.log(math.exp(t_1))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-1.0 / y)) / x)
	t_1 = Float64((Float64(x / Float64(y + x)) ^ x) / x)
	tmp = 0.0
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-1.0 / y)) / x;
	t_1 = ((x / (y + x)) ^ x) / x;
	tmp = 0.0;
	if (y < -3.7311844206647956e+94)
		tmp = t_0;
	elseif (y < 2.817959242728288e+37)
		tmp = t_1;
	elseif (y < 2.347387415166998e+178)
		tmp = log(exp(t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]}, If[Less[y, -3.7311844206647956e+94], t$95$0, If[Less[y, 2.817959242728288e+37], t$95$1, If[Less[y, 2.347387415166998e+178], N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Reproduce

herbie shell --seed 2024337 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -37311844206647956000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (exp (/ -1 y)) x) (if (< y 28179592427282880000000000000000000000) (/ (pow (/ x (+ y x)) x) x) (if (< y 23473874151669980000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x)))))

  (/ (exp (* x (log (/ x (+ x y))))) x))