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

Percentage Accurate: 77.6% → 98.3%

Time: 8.8s

Alternatives: 9

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: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41} \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.25e+41) (not (<= x 0.56))) (/ (exp (- y)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e+41) || !(x <= 0.56)) {
		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 <= (-1.25d+41)) .or. (.not. (x <= 0.56d0))) 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 <= -1.25e+41) || !(x <= 0.56)) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.25e+41) or not (x <= 0.56):
		tmp = math.exp(-y) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.25e+41) || !(x <= 0.56))
		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 <= -1.25e+41) || ~((x <= 0.56)))
		tmp = exp(-y) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.25e+41], N[Not[LessEqual[x, 0.56]], $MachinePrecision]], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000006e41 or 0.56000000000000005 < x

   1. Initial program 73.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.25000000000000006e41 < x < 0.56000000000000005

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41} \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), x, 0.5 \cdot y\right), y, x\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} - 0.16666666666666666\right) - \frac{0.3333333333333333}{x \cdot x}, y, \frac{0.5}{x} - 0.5\right), y, -1\right), y, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (/ (fma (fma (fma 0.5 y -1.0) x (* 0.5 y)) y x) x) x)
   (if (<= x 0.56)
     (/ 1.0 x)
     (/
      (/ -1.0 x)
      (fma
       (fma
        (fma
         (- (- (/ 0.5 x) 0.16666666666666666) (/ 0.3333333333333333 (* x x)))
         y
         (- (/ 0.5 x) 0.5))
        y
        -1.0)
       y
       -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = (fma(fma(fma(0.5, y, -1.0), x, (0.5 * y)), y, x) / x) / x;
	} else if (x <= 0.56) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / x) / fma(fma(fma((((0.5 / x) - 0.16666666666666666) - (0.3333333333333333 / (x * x))), y, ((0.5 / x) - 0.5)), y, -1.0), y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), x, Float64(0.5 * y)), y, x) / x) / x);
	elseif (x <= 0.56)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / x) / fma(fma(fma(Float64(Float64(Float64(0.5 / x) - 0.16666666666666666) - Float64(0.3333333333333333 / Float64(x * x))), y, Float64(Float64(0.5 / x) - 0.5)), y, -1.0), y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.56], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / N[(N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.5 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.6%

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

   10. Applied rewrites 77.9%

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

   if -1.25000000000000006e41 < x < 0.56000000000000005

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.56000000000000005 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 79.9%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} - 0.16666666666666666\right) - \frac{0.3333333333333333}{x \cdot x}, y, \frac{0.5}{x} - 0.5\right), y, -1\right), y, -1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), x, 0.5 \cdot y\right), y, x\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{x} - 0.5, y, -1\right), y, -1\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (/ (fma (fma (fma 0.5 y -1.0) x (* 0.5 y)) y x) x) x)
   (if (<= x 0.56)
     (/ 1.0 x)
     (/ -1.0 (* (fma (fma (- (/ 0.5 x) 0.5) y -1.0) y -1.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = (fma(fma(fma(0.5, y, -1.0), x, (0.5 * y)), y, x) / x) / x;
	} else if (x <= 0.56) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (fma(fma(((0.5 / x) - 0.5), y, -1.0), y, -1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), x, Float64(0.5 * y)), y, x) / x) / x);
	elseif (x <= 0.56)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(fma(fma(Float64(Float64(0.5 / x) - 0.5), y, -1.0), y, -1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.56], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] - 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.6%

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

   10. Applied rewrites 77.9%

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

   if -1.25000000000000006e41 < x < 0.56000000000000005

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.56000000000000005 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 78.5

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

   9. Applied rewrites 78.5%

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

Alternative 4: 85.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{x} - 0.5, y, -1\right), y, -1\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (/ (- x (* y x)) x) x)
   (if (<= x 0.56)
     (/ 1.0 x)
     (/ -1.0 (* (fma (fma (- (/ 0.5 x) 0.5) y -1.0) y -1.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = ((x - (y * x)) / x) / x;
	} else if (x <= 0.56) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (fma(fma(((0.5 / x) - 0.5), y, -1.0), y, -1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	elseif (x <= 0.56)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(fma(fma(Float64(Float64(0.5 / x) - 0.5), y, -1.0), y, -1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.56], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] - 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   7. Applied rewrites 72.7%

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

   if -1.25000000000000006e41 < x < 0.56000000000000005

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.56000000000000005 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 78.5

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

   9. Applied rewrites 78.5%

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

Alternative 5: 85.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.56:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right), y, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (/ (- x (* y x)) x) x)
   (if (<= x 0.56) (/ 1.0 x) (/ (/ -1.0 x) (fma (fma -0.5 y -1.0) y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = ((x - (y * x)) / x) / x;
	} else if (x <= 0.56) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / x) / fma(fma(-0.5, y, -1.0), y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	elseif (x <= 0.56)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / x) / fma(fma(-0.5, y, -1.0), y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.56], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] / N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   7. Applied rewrites 72.7%

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

   if -1.25000000000000006e41 < x < 0.56000000000000005

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.56000000000000005 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 77.1

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 77.1%

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

Alternative 6: 82.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (/ (- x (* y x)) x) x)
   (if (<= x 0.5) (/ 1.0 x) (/ 1.0 (* (+ 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = ((x - (y * x)) / x) / x;
	} else if (x <= 0.5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / ((1.0 + y) * 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 <= (-1.25d+41)) then
        tmp = ((x - (y * x)) / x) / x
    else if (x <= 0.5d0) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / ((1.0d0 + y) * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = ((x - (y * x)) / x) / x;
	} else if (x <= 0.5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / ((1.0 + y) * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e+41:
		tmp = ((x - (y * x)) / x) / x
	elif x <= 0.5:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / ((1.0 + y) * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	elseif (x <= 0.5)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + y) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e+41)
		tmp = ((x - (y * x)) / x) / x;
	elseif (x <= 0.5)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / ((1.0 + y) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.5], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(N[(1.0 + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

   7. Applied rewrites 72.7%

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

   if -1.25000000000000006e41 < x < 0.5

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.5 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 72.3

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

   7. Applied rewrites 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+166} \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.5e+166) (not (<= x 0.5))) (/ 1.0 (* (+ 1.0 y) x)) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e+166) || !(x <= 0.5)) {
		tmp = 1.0 / ((1.0 + 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 <= (-1.5d+166)) .or. (.not. (x <= 0.5d0))) then
        tmp = 1.0d0 / ((1.0d0 + 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 <= -1.5e+166) || !(x <= 0.5)) {
		tmp = 1.0 / ((1.0 + y) * x);
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.5e+166) or not (x <= 0.5):
		tmp = 1.0 / ((1.0 + y) * x)
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.5e+166) || !(x <= 0.5))
		tmp = Float64(1.0 / Float64(Float64(1.0 + y) * x));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.5e+166) || ~((x <= 0.5)))
		tmp = 1.0 / ((1.0 + y) * x);
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.5e+166], N[Not[LessEqual[x, 0.5]], $MachinePrecision]], N[(1.0 / N[(N[(1.0 + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999999e166 or 0.5 < x

   1. Initial program 69.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 68.4

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

   7. Applied rewrites 68.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 68.4

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

   9. Applied rewrites 68.4%

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

   if -1.49999999999999999e166 < x < 0.5

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+166} \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+41)
   (/ (fma (fma 0.5 y -1.0) y 1.0) x)
   (if (<= x 0.5) (/ 1.0 x) (/ 1.0 (* (+ 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+41) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else if (x <= 0.5) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / ((1.0 + y) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+41)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	elseif (x <= 0.5)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+41], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.5], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(N[(1.0 + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e41

   1. Initial program 72.4%

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

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.8%

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

   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 70.4%

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

   if -1.25000000000000006e41 < x < 0.5

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 0.5 < x

   1. Initial program 74.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-log.f64 N/A

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

      16. exp-to-pow N/A

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

      17. pow-flip N/A

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

      18. neg-mul-1 N/A

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

      19. pow-unpow N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 72.3

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

   7. Applied rewrites 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + y\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 72.1%

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

Developer Target 1: 77.4% 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 2024296 
(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))