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

Percentage Accurate: 77.2% → 98.5%

Time: 9.0s

Alternatives: 10

Speedup: 8.9×

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: 77.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -1.1e+28) t_0 (if (<= x 0.84) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -1.1e+28) {
		tmp = t_0;
	} else if (x <= 0.84) {
		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 <= (-1.1d+28)) then
        tmp = t_0
    else if (x <= 0.84d0) 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 <= -1.1e+28) {
		tmp = t_0;
	} else if (x <= 0.84) {
		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 <= -1.1e+28:
		tmp = t_0
	elif x <= 0.84:
		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 <= -1.1e+28)
		tmp = t_0;
	elseif (x <= 0.84)
		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 <= -1.1e+28)
		tmp = t_0;
	elseif (x <= 0.84)
		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, -1.1e+28], t$95$0, If[LessEqual[x, 0.84], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999993e28 or 0.839999999999999969 < x

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.09999999999999993e28 < x < 0.839999999999999969

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

Alternative 2: 84.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-y, \mathsf{fma}\left(-0.5, y, -0.5\right), \mathsf{fma}\left(-y, \mathsf{fma}\left(0.16666666666666666, y, 0.5\right), -1\right) \cdot x\right), x, -0.3333333333333333 \cdot \left(y \cdot y\right)\right)}{x}, y, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+28)
   (/ (fma (/ (* (fma 0.5 y -1.0) x) x) y 1.0) x)
   (if (<= x 0.84)
     (/ 1.0 x)
     (if (<= x 7.5e+156)
       (/
        -1.0
        (fma
         (/
          (fma
           (fma
            (- y)
            (fma -0.5 y -0.5)
            (* (fma (- y) (fma 0.16666666666666666 y 0.5) -1.0) x))
           x
           (* -0.3333333333333333 (* y y)))
          x)
         y
         (- x)))
       (/ (/ (fma (fma (fma 0.5 y -1.0) y 1.0) x (* (* y y) 0.5)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+28) {
		tmp = fma(((fma(0.5, y, -1.0) * x) / x), y, 1.0) / x;
	} else if (x <= 0.84) {
		tmp = 1.0 / x;
	} else if (x <= 7.5e+156) {
		tmp = -1.0 / fma((fma(fma(-y, fma(-0.5, y, -0.5), (fma(-y, fma(0.16666666666666666, y, 0.5), -1.0) * x)), x, (-0.3333333333333333 * (y * y))) / x), y, -x);
	} else {
		tmp = (fma(fma(fma(0.5, y, -1.0), y, 1.0), x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = Float64(fma(Float64(Float64(fma(0.5, y, -1.0) * x) / x), y, 1.0) / x);
	elseif (x <= 0.84)
		tmp = Float64(1.0 / x);
	elseif (x <= 7.5e+156)
		tmp = Float64(-1.0 / fma(Float64(fma(fma(Float64(-y), fma(-0.5, y, -0.5), Float64(fma(Float64(-y), fma(0.16666666666666666, y, 0.5), -1.0) * x)), x, Float64(-0.3333333333333333 * Float64(y * y))) / x), y, Float64(-x)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), y, 1.0), x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+28], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.84], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 7.5e+156], N[(-1.0 / N[(N[(N[(N[((-y) * N[(-0.5 * y + -0.5), $MachinePrecision] + N[(N[((-y) * N[(0.16666666666666666 * y + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + N[(-0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y + (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.09999999999999993e28

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 75.7%

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

   if -1.09999999999999993e28 < x < 0.839999999999999969

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.839999999999999969 < x < 7.50000000000000026e156

   1. Initial program 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 94.0%

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

   if 7.50000000000000026e156 < x

   1. Initial program 66.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-y, \mathsf{fma}\left(-0.5, y, -0.5\right), \mathsf{fma}\left(-y, \mathsf{fma}\left(0.16666666666666666, y, 0.5\right), -1\right) \cdot x\right), x, -0.3333333333333333 \cdot \left(y \cdot y\right)\right)}{x}, y, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+178}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right) \cdot \left(-x\right), y, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+28)
   (/ (fma (/ (* (fma 0.5 y -1.0) x) x) y 1.0) x)
   (if (<= x 0.84)
     (/ 1.0 x)
     (if (<= x 9e+178)
       (/
        -1.0
        (fma (* (fma (fma 0.16666666666666666 y 0.5) y 1.0) (- x)) y (- x)))
       (/ (/ (fma (fma (fma 0.5 y -1.0) y 1.0) x (* (* y y) 0.5)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+28) {
		tmp = fma(((fma(0.5, y, -1.0) * x) / x), y, 1.0) / x;
	} else if (x <= 0.84) {
		tmp = 1.0 / x;
	} else if (x <= 9e+178) {
		tmp = -1.0 / fma((fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * -x), y, -x);
	} else {
		tmp = (fma(fma(fma(0.5, y, -1.0), y, 1.0), x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = Float64(fma(Float64(Float64(fma(0.5, y, -1.0) * x) / x), y, 1.0) / x);
	elseif (x <= 0.84)
		tmp = Float64(1.0 / x);
	elseif (x <= 9e+178)
		tmp = Float64(-1.0 / fma(Float64(fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * Float64(-x)), y, Float64(-x)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), y, 1.0), x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+28], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.84], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9e+178], N[(-1.0 / N[(N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision] * y + (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.09999999999999993e28

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 75.7%

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

   if -1.09999999999999993e28 < x < 0.839999999999999969

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.839999999999999969 < x < 8.9999999999999994e178

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 87.3%

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

   if 8.9999999999999994e178 < x

   1. Initial program 65.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.5%

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

Alternative 4: 84.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right) \cdot \left(-x\right), y, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+28)
   (/ (fma (/ (* (fma 0.5 y -1.0) x) x) y 1.0) x)
   (if (<= x 0.84)
     (/ 1.0 x)
     (if (<= x 9.2e+178)
       (/
        -1.0
        (fma (* (fma (fma 0.16666666666666666 y 0.5) y 1.0) (- x)) y (- x)))
       (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+28) {
		tmp = fma(((fma(0.5, y, -1.0) * x) / x), y, 1.0) / x;
	} else if (x <= 0.84) {
		tmp = 1.0 / x;
	} else if (x <= 9.2e+178) {
		tmp = -1.0 / fma((fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * -x), y, -x);
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = Float64(fma(Float64(Float64(fma(0.5, y, -1.0) * x) / x), y, 1.0) / x);
	elseif (x <= 0.84)
		tmp = Float64(1.0 / x);
	elseif (x <= 9.2e+178)
		tmp = Float64(-1.0 / fma(Float64(fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * Float64(-x)), y, Float64(-x)));
	else
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+28], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.84], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9.2e+178], N[(-1.0 / N[(N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision] * y + (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.09999999999999993e28

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 75.7%

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

   if -1.09999999999999993e28 < x < 0.839999999999999969

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.839999999999999969 < x < 9.2000000000000003e178

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 87.3%

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

   if 9.2000000000000003e178 < x

   1. Initial program 65.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 82.6%

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

Alternative 5: 84.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+178}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right) \cdot \left(-x\right), y, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)))
   (if (<= x -1.1e+28)
     t_0
     (if (<= x 0.84)
       (/ 1.0 x)
       (if (<= x 9.2e+178)
         (/
          -1.0
          (fma (* (fma (fma 0.16666666666666666 y 0.5) y 1.0) (- x)) y (- x)))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	double tmp;
	if (x <= -1.1e+28) {
		tmp = t_0;
	} else if (x <= 0.84) {
		tmp = 1.0 / x;
	} else if (x <= 9.2e+178) {
		tmp = -1.0 / fma((fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * -x), y, -x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = t_0;
	elseif (x <= 0.84)
		tmp = Float64(1.0 / x);
	elseif (x <= 9.2e+178)
		tmp = Float64(-1.0 / fma(Float64(fma(fma(0.16666666666666666, y, 0.5), y, 1.0) * Float64(-x)), y, Float64(-x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.1e+28], t$95$0, If[LessEqual[x, 0.84], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9.2e+178], N[(-1.0 / N[(N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision] * y + (-x)), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999993e28 or 9.2000000000000003e178 < x

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

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

   if -1.09999999999999993e28 < x < 0.839999999999999969

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.839999999999999969 < x < 9.2000000000000003e178

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 87.3%

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

Alternative 6: 82.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+178}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)))
   (if (<= x -1.1e+28)
     t_0
     (if (<= x 0.49)
       (/ 1.0 x)
       (if (<= x 9e+178) (/ -1.0 (- (fma y x x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	double tmp;
	if (x <= -1.1e+28) {
		tmp = t_0;
	} else if (x <= 0.49) {
		tmp = 1.0 / x;
	} else if (x <= 9e+178) {
		tmp = -1.0 / -fma(y, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = t_0;
	elseif (x <= 0.49)
		tmp = Float64(1.0 / x);
	elseif (x <= 9e+178)
		tmp = Float64(-1.0 / Float64(-fma(y, x, x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.1e+28], t$95$0, If[LessEqual[x, 0.49], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9e+178], N[(-1.0 / (-N[(y * x + x), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999993e28 or 8.9999999999999994e178 < x

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

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

   if -1.09999999999999993e28 < x < 0.48999999999999999

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.48999999999999999 < x < 8.9999999999999994e178

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 7: 82.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, -1\right), y, 1\right)}{x}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+178}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (fma (fma (* -0.16666666666666666 y) y -1.0) y 1.0) x)))
   (if (<= x -1.1e+28)
     t_0
     (if (<= x 0.49)
       (/ 1.0 x)
       (if (<= x 9e+178) (/ -1.0 (- (fma y x x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma((-0.16666666666666666 * y), y, -1.0), y, 1.0) / x;
	double tmp;
	if (x <= -1.1e+28) {
		tmp = t_0;
	} else if (x <= 0.49) {
		tmp = 1.0 / x;
	} else if (x <= 9e+178) {
		tmp = -1.0 / -fma(y, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(Float64(-0.16666666666666666 * y), y, -1.0), y, 1.0) / x)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = t_0;
	elseif (x <= 0.49)
		tmp = Float64(1.0 / x);
	elseif (x <= 9e+178)
		tmp = Float64(-1.0 / Float64(-fma(y, x, x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.1e+28], t$95$0, If[LessEqual[x, 0.49], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 9e+178], N[(-1.0 / (-N[(y * x + x), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999993e28 or 8.9999999999999994e178 < x

   1. Initial program 71.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 77.3%

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

   if -1.09999999999999993e28 < x < 0.48999999999999999

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.48999999999999999 < x < 8.9999999999999994e178

   1. Initial program 89.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 8: 82.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+28)
   (/ (fma (fma 0.5 y -1.0) y 1.0) x)
   (if (<= x 0.49) (/ 1.0 x) (/ -1.0 (- (fma y x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+28) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else if (x <= 0.49) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / -fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+28)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	elseif (x <= 0.49)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(-fma(y, x, x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+28], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.49], N[(1.0 / x), $MachinePrecision], N[(-1.0 / (-N[(y * x + x), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999993e28

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.0%

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

   if -1.09999999999999993e28 < x < 0.48999999999999999

   1. Initial program 82.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.2%

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

   if 0.48999999999999999 < x

   1. Initial program 77.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.3

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

   7. Applied rewrites 73.3%

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

Alternative 9: 78.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.49:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.49) (/ 1.0 x) (/ -1.0 (- (fma y x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.49) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / -fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.49)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(-fma(y, x, x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.49], N[(1.0 / x), $MachinePrecision], N[(-1.0 / (-N[(y * x + x), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.48999999999999999

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.8%

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

   if 0.48999999999999999 < x

   1. Initial program 77.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-out N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.3

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

   7. Applied rewrites 73.3%

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

Alternative 10: 74.5% 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 78.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 76.5%

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

Developer Target 1: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-1.0d0) / y)) / x
    t_1 = ((x / (y + x)) ** x) / x
    if (y < (-3.7311844206647956d+94)) then
        tmp = t_0
    else if (y < 2.817959242728288d+37) then
        tmp = t_1
    else if (y < 2.347387415166998d+178) then
        tmp = log(exp(t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(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))