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

Percentage Accurate: 77.6% → 99.3%

Time: 8.4s

Alternatives: 8

Speedup: 7.2×

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: 77.6% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{{\left(e^{y}\right)}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -600.0) (not (<= x 1.75e-6)))
   (/ (pow (exp y) -1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -600.0) || !(x <= 1.75e-6)) {
		tmp = pow(exp(y), -1.0) / 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 <= (-600.0d0)) .or. (.not. (x <= 1.75d-6))) then
        tmp = (exp(y) ** (-1.0d0)) / 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 <= -600.0) || !(x <= 1.75e-6)) {
		tmp = Math.pow(Math.exp(y), -1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -600.0) or not (x <= 1.75e-6):
		tmp = math.pow(math.exp(y), -1.0) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -600.0) || !(x <= 1.75e-6))
		tmp = Float64((exp(y) ^ -1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -600.0) || ~((x <= 1.75e-6)))
		tmp = (exp(y) ^ -1.0) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -600.0], N[Not[LessEqual[x, 1.75e-6]], $MachinePrecision]], N[(N[Power[N[Exp[y], $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -600 or 1.74999999999999997e-6 < x

   1. Initial program 76.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}}}{x} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\frac{1}{e^{\color{blue}{y}}}}{x} \]

      3. lower-exp.f64 99.9

         \[\leadsto \frac{\frac{1}{\color{blue}{e^{y}}}}{x} \]

   10. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{e^{y}}}}{x} \]

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{{\left(e^{y}\right)}^{-1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\right)}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -600.0)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 1.75e-6)
     (/ 1.0 x)
     (/
      (pow (fma (fma (fma 0.16666666666666666 y 0.5) y 1.0) y 1.0) -1.0)
      x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -600.0) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 1.75e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0), -1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 1.75e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64((fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0) ^ -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -600.0], 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, 1.75e-6], N[(1.0 / x), $MachinePrecision], N[(N[Power[N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -600

   1. Initial program 76.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 79.7%

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

   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 79.7%

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

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 1.74999999999999997e-6 < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}}}{x} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\frac{1}{e^{\color{blue}{y}}}}{x} \]

      3. lower-exp.f64 100.0

         \[\leadsto \frac{\frac{1}{\color{blue}{e^{y}}}}{x} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{e^{y}}}}{x} \]

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 80.9%

       \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), \color{blue}{y}, 1\right)}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\right)}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, 1\right)\right)}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -600.0)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 1.75e-6) (/ 1.0 x) (/ (pow (fma (fma 0.5 y 1.0) y 1.0) -1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -600.0) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 1.75e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = pow(fma(fma(0.5, y, 1.0), y, 1.0), -1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 1.75e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64((fma(fma(0.5, y, 1.0), y, 1.0) ^ -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -600.0], 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, 1.75e-6], N[(1.0 / x), $MachinePrecision], N[(N[Power[N[(N[(0.5 * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -600

   1. Initial program 76.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 79.7%

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

   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 79.7%

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

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 1.74999999999999997e-6 < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.8

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

   10. Applied rewrites 76.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 76.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, 1\right)\right)}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + y\right)}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -600.0)
   (/ (fma (- (* (fma -0.16666666666666666 y 0.5) y) 1.0) y 1.0) x)
   (if (<= x 1.75e-6) (/ 1.0 x) (/ (pow (+ 1.0 y) -1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -600.0) {
		tmp = fma(((fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x;
	} else if (x <= 1.75e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = pow((1.0 + y), -1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, y, 0.5) * y) - 1.0), y, 1.0) / x);
	elseif (x <= 1.75e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64((Float64(1.0 + y) ^ -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -600.0], 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, 1.75e-6], N[(1.0 / x), $MachinePrecision], N[(N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -600

   1. Initial program 76.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 79.7%

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

   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 79.7%

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

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 1.74999999999999997e-6 < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 71.3

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

   10. Applied rewrites 71.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right) \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + y\right)}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + y\right)}^{-1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -600.0)
   (/ (fma (- (* 0.5 y) 1.0) y 1.0) x)
   (if (<= x 1.75e-6) (/ 1.0 x) (/ (pow (+ 1.0 y) -1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -600.0) {
		tmp = fma(((0.5 * y) - 1.0), y, 1.0) / x;
	} else if (x <= 1.75e-6) {
		tmp = 1.0 / x;
	} else {
		tmp = pow((1.0 + y), -1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(fma(Float64(Float64(0.5 * y) - 1.0), y, 1.0) / x);
	elseif (x <= 1.75e-6)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64((Float64(1.0 + y) ^ -1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -600.0], N[(N[(N[(N[(0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.75e-6], N[(1.0 / x), $MachinePrecision], N[(N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -600

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 74.8%

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

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 1.74999999999999997e-6 < 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 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 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 71.3

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

   10. Applied rewrites 71.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + y\right)}^{-1}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -600.0) (not (<= x 1.75e-6))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -600.0) || !(x <= 1.75e-6)) {
		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 <= (-600.0d0)) .or. (.not. (x <= 1.75d-6))) 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 <= -600.0) || !(x <= 1.75e-6)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -600.0) or not (x <= 1.75e-6):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -600.0) || !(x <= 1.75e-6))
		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 <= -600.0) || ~((x <= 1.75e-6)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -600.0], N[Not[LessEqual[x, 1.75e-6]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -600 or 1.74999999999999997e-6 < x

   1. Initial program 76.3%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if -600 < x < 1.74999999999999997e-6

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 1.75 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot y - 1, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -600.0) (/ (fma (- (* 0.5 y) 1.0) y 1.0) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -600.0) {
		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 <= -600.0)
		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[LessEqual[x, -600.0], 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 < -600

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 74.8%

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

   if -600 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.4%

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

Alternative 8: 74.9% 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 79.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 74.9%

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

Developer Target 1: 78.4% 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 2024339 
(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))