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

Percentage Accurate: 78.4% → 98.5%

Time: 9.8s

Alternatives: 8

Speedup: 7.7×

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: 78.4% 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: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -22000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -22000.0) t_0 (if (<= x 7e-38) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -22000.0) {
		tmp = t_0;
	} else if (x <= 7e-38) {
		tmp = 1.0 / x;
	} 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) :: tmp
    t_0 = exp(-y) / x
    if (x <= (-22000.0d0)) then
        tmp = t_0
    else if (x <= 7d-38) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -22000.0) {
		tmp = t_0;
	} else if (x <= 7e-38) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -22000.0:
		tmp = t_0
	elif x <= 7e-38:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -22000.0)
		tmp = t_0;
	elseif (x <= 7e-38)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -22000.0)
		tmp = t_0;
	elseif (x <= 7e-38)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -22000.0], t$95$0, If[LessEqual[x, 7e-38], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -22000 or 7.0000000000000003e-38 < x

   1. Initial program 71.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -22000 < x < 7.0000000000000003e-38

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 76.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 1850:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, 0.5, -1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+154)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x -5.8e+36)
     (/ (+ x (* y (fma x (fma y 0.5 -1.0) (* y 0.5)))) (* x x))
     (if (<= x 1850.0)
       (/ 1.0 x)
       (if (<= x 1.65e+180)
         (/
          (/
           (fma (fma y 0.5 -1.0) (* y (* y (fma y 0.5 -1.0))) -1.0)
           (fma y (fma y 0.5 -1.0) -1.0))
          x)
         (* x (/ 1.0 (* x (- x)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+154) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= -5.8e+36) {
		tmp = (x + (y * fma(x, fma(y, 0.5, -1.0), (y * 0.5)))) / (x * x);
	} else if (x <= 1850.0) {
		tmp = 1.0 / x;
	} else if (x <= 1.65e+180) {
		tmp = (fma(fma(y, 0.5, -1.0), (y * (y * fma(y, 0.5, -1.0))), -1.0) / fma(y, fma(y, 0.5, -1.0), -1.0)) / x;
	} else {
		tmp = x * (1.0 / (x * -x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+154)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= -5.8e+36)
		tmp = Float64(Float64(x + Float64(y * fma(x, fma(y, 0.5, -1.0), Float64(y * 0.5)))) / Float64(x * x));
	elseif (x <= 1850.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.65e+180)
		tmp = Float64(Float64(fma(fma(y, 0.5, -1.0), Float64(y * Float64(y * fma(y, 0.5, -1.0))), -1.0) / fma(y, fma(y, 0.5, -1.0), -1.0)) / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(x * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e+154], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e+36], N[(N[(x + N[(y * N[(x * N[(y * 0.5 + -1.0), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1850.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.65e+180], N[(N[(N[(N[(y * 0.5 + -1.0), $MachinePrecision] * N[(y * N[(y * N[(y * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(1.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.20000000000000007e154

   1. Initial program 53.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 62.5%

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

   if -1.20000000000000007e154 < x < -5.8e36

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 87.8%

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

   if -5.8e36 < x < 1850

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.5%

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

   if 1850 < x < 1.64999999999999995e180

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.2%

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

   10. Applied rewrites 71.1%

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

   if 1.64999999999999995e180 < x

   1. Initial program 52.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   7. Applied rewrites 13.3%

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 55.9%

       \[\leadsto \left(-x\right) \cdot \frac{1}{\color{blue}{x} \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 1850:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, 0.5, -1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5 + \left(\frac{0.5}{x} - y \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right)\right)\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1850:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, 0.5, -1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -22000.0)
   (/
    (fma
     y
     (fma
      y
      (+
       0.5
       (-
        (/ 0.5 x)
        (*
         y
         (+
          (/ 0.5 x)
          (+ 0.16666666666666666 (/ 0.3333333333333333 (* x x)))))))
      -1.0)
     1.0)
    x)
   (if (<= x 1850.0)
     (/ 1.0 x)
     (if (<= x 1.65e+180)
       (/
        (/
         (fma (fma y 0.5 -1.0) (* y (* y (fma y 0.5 -1.0))) -1.0)
         (fma y (fma y 0.5 -1.0) -1.0))
        x)
       (* x (/ 1.0 (* x (- x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -22000.0) {
		tmp = fma(y, fma(y, (0.5 + ((0.5 / x) - (y * ((0.5 / x) + (0.16666666666666666 + (0.3333333333333333 / (x * x))))))), -1.0), 1.0) / x;
	} else if (x <= 1850.0) {
		tmp = 1.0 / x;
	} else if (x <= 1.65e+180) {
		tmp = (fma(fma(y, 0.5, -1.0), (y * (y * fma(y, 0.5, -1.0))), -1.0) / fma(y, fma(y, 0.5, -1.0), -1.0)) / x;
	} else {
		tmp = x * (1.0 / (x * -x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -22000.0)
		tmp = Float64(fma(y, fma(y, Float64(0.5 + Float64(Float64(0.5 / x) - Float64(y * Float64(Float64(0.5 / x) + Float64(0.16666666666666666 + Float64(0.3333333333333333 / Float64(x * x))))))), -1.0), 1.0) / x);
	elseif (x <= 1850.0)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.65e+180)
		tmp = Float64(Float64(fma(fma(y, 0.5, -1.0), Float64(y * Float64(y * fma(y, 0.5, -1.0))), -1.0) / fma(y, fma(y, 0.5, -1.0), -1.0)) / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(x * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -22000.0], N[(N[(y * N[(y * N[(0.5 + N[(N[(0.5 / x), $MachinePrecision] - N[(y * N[(N[(0.5 / x), $MachinePrecision] + N[(0.16666666666666666 + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1850.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.65e+180], N[(N[(N[(N[(y * 0.5 + -1.0), $MachinePrecision] * N[(y * N[(y * N[(y * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(1.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -22000

   1. Initial program 70.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. lower-fma.f64 N/A

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

   5. Applied rewrites 71.9%

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

   if -22000 < x < 1850

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 1850 < x < 1.64999999999999995e180

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.2%

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

   10. Applied rewrites 71.1%

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

   if 1.64999999999999995e180 < x

   1. Initial program 52.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   7. Applied rewrites 13.3%

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 55.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5 + \left(\frac{0.5}{x} - y \cdot \left(\frac{0.5}{x} + \left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right)\right)\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1850:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, 0.5, -1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, 0.5, -1\right)\right), -1\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+154)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x -5.8e+36)
     (/ (+ x (* y (fma x (fma y 0.5 -1.0) (* y 0.5)))) (* x x))
     (if (<= x 1.65e+180) (/ 1.0 x) (* x (/ 1.0 (* x (- x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+154) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= -5.8e+36) {
		tmp = (x + (y * fma(x, fma(y, 0.5, -1.0), (y * 0.5)))) / (x * x);
	} else if (x <= 1.65e+180) {
		tmp = 1.0 / x;
	} else {
		tmp = x * (1.0 / (x * -x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+154)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= -5.8e+36)
		tmp = Float64(Float64(x + Float64(y * fma(x, fma(y, 0.5, -1.0), Float64(y * 0.5)))) / Float64(x * x));
	elseif (x <= 1.65e+180)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(x * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e+154], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e+36], N[(N[(x + N[(y * N[(x * N[(y * 0.5 + -1.0), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+180], N[(1.0 / x), $MachinePrecision], N[(x * N[(1.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.20000000000000007e154

   1. Initial program 53.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 62.5%

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

   if -1.20000000000000007e154 < x < -5.8e36

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 87.8%

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

   if -5.8e36 < x < 1.64999999999999995e180

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.1%

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

   if 1.64999999999999995e180 < x

   1. Initial program 52.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   7. Applied rewrites 13.3%

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 55.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right) \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -22000.0)
   (* (fma y (fma y 0.5 -1.0) 1.0) (/ 1.0 x))
   (if (<= x 1.65e+180) (/ 1.0 x) (* x (/ 1.0 (* x (- x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -22000.0) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) * (1.0 / x);
	} else if (x <= 1.65e+180) {
		tmp = 1.0 / x;
	} else {
		tmp = x * (1.0 / (x * -x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -22000.0)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) * Float64(1.0 / x));
	elseif (x <= 1.65e+180)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(x * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -22000.0], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+180], N[(1.0 / x), $MachinePrecision], N[(x * N[(1.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -22000

   1. Initial program 70.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 67.9%

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

   10. Applied rewrites 68.0%

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

   if -22000 < x < 1.64999999999999995e180

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   if 1.64999999999999995e180 < x

   1. Initial program 52.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   7. Applied rewrites 13.3%

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 55.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right) \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -22000.0)
   (/ (fma y (fma y 0.5 -1.0) 1.0) x)
   (if (<= x 1.65e+180) (/ 1.0 x) (* x (/ 1.0 (* x (- x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -22000.0) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (x <= 1.65e+180) {
		tmp = 1.0 / x;
	} else {
		tmp = x * (1.0 / (x * -x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -22000.0)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (x <= 1.65e+180)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(x * Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -22000.0], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.65e+180], N[(1.0 / x), $MachinePrecision], N[(x * N[(1.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -22000

   1. Initial program 70.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 67.9%

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

   if -22000 < x < 1.64999999999999995e180

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   if 1.64999999999999995e180 < x

   1. Initial program 52.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 29.5

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

   5. Applied rewrites 29.5%

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

   7. Applied rewrites 13.3%

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

   9. Applied rewrites 55.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 55.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -22000.0) (/ (fma y (fma y 0.5 -1.0) 1.0) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -22000.0) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -22000.0)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -22000.0], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -22000

   1. Initial program 70.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 67.9%

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

   if -22000 < x

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.0%

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

Alternative 8: 75.0% 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 78.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 74.1%

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

Developer Target 1: 78.0% 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 2024233 
(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))