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

Percentage Accurate: 77.9% → 99.4%

Time: 12.2s

Alternatives: 13

Speedup: 7.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -1.05) t_0 (if (<= x 0.21) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 0.21) {
		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 <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 0.21d0) 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 <= -1.05) {
		tmp = t_0;
	} else if (x <= 0.21) {
		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 <= -1.05:
		tmp = t_0
	elif x <= 0.21:
		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 <= -1.05)
		tmp = t_0;
	elseif (x <= 0.21)
		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 <= -1.05)
		tmp = t_0;
	elseif (x <= 0.21)
		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, -1.05], t$95$0, If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.209999999999999992 < x

   1. Initial program 70.8%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 -1.05000000000000004 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

Alternative 2: 87.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) \cdot \left(y \cdot y\right)}{x}}{x \cdot 2} + \frac{1 - y}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5 + \mathsf{fma}\left(y, \frac{0.3333333333333333}{x \cdot x} + \left(0.16666666666666666 + \frac{-0.5}{x}\right), \frac{-0.5}{x}\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (+ (/ (/ (* (+ x 1.0) (* y y)) x) (* x 2.0)) (/ (- 1.0 y) x))
   (if (<= x 0.21)
     (/ 1.0 x)
     (/
      1.0
      (*
       x
       (fma
        y
        (fma
         y
         (+
          0.5
          (fma
           y
           (+
            (/ 0.3333333333333333 (* x x))
            (+ 0.16666666666666666 (/ -0.5 x)))
           (/ -0.5 x)))
         1.0)
        1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = ((((x + 1.0) * (y * y)) / x) / (x * 2.0)) + ((1.0 - y) / x);
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * fma(y, fma(y, (0.5 + fma(y, ((0.3333333333333333 / (x * x)) + (0.16666666666666666 + (-0.5 / x))), (-0.5 / x))), 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x + 1.0) * Float64(y * y)) / x) / Float64(x * 2.0)) + Float64(Float64(1.0 - y) / x));
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * fma(y, fma(y, Float64(0.5 + fma(y, Float64(Float64(0.3333333333333333 / Float64(x * x)) + Float64(0.16666666666666666 + Float64(-0.5 / x))), Float64(-0.5 / x))), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(y * N[(y * N[(0.5 + N[(y * N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.6%

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

   if -1 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 76.9%

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

Alternative 3: 85.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.63:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, y, x\right) \cdot \mathsf{fma}\left(x, -y, x\right)}{\mathsf{fma}\left(x, y, x\right)}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5 + \frac{-0.5}{x}, x \cdot y, x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+94)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x -0.63)
     (/ (/ (* (fma x y x) (fma x (- y) x)) (fma x y x)) (* x x))
     (if (<= x 0.21)
       (/ 1.0 x)
       (/ 1.0 (fma y (fma (+ 0.5 (/ -0.5 x)) (* x y) x) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+94) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= -0.63) {
		tmp = ((fma(x, y, x) * fma(x, -y, x)) / fma(x, y, x)) / (x * x);
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma((0.5 + (-0.5 / x)), (x * y), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+94)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= -0.63)
		tmp = Float64(Float64(Float64(fma(x, y, x) * fma(x, Float64(-y), x)) / fma(x, y, x)) / Float64(x * x));
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(0.5 + Float64(-0.5 / x)), Float64(x * y), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+94], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.63], N[(N[(N[(N[(x * y + x), $MachinePrecision] * N[(x * (-y) + x), $MachinePrecision]), $MachinePrecision] / N[(x * y + x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.79999999999999998e94

   1. Initial program 59.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

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

   7. Applied rewrites 72.0%

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

   if -2.79999999999999998e94 < x < -0.630000000000000004

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 89.3%

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

   if -0.630000000000000004 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 75.4%

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

Alternative 4: 87.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) \cdot \left(y \cdot y\right)}{x}}{x \cdot 2} + \frac{1 - y}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5 + \frac{-0.5}{x}, x \cdot y, x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (+ (/ (/ (* (+ x 1.0) (* y y)) x) (* x 2.0)) (/ (- 1.0 y) x))
   (if (<= x 0.21)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (+ 0.5 (/ -0.5 x)) (* x y) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = ((((x + 1.0) * (y * y)) / x) / (x * 2.0)) + ((1.0 - y) / x);
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma((0.5 + (-0.5 / x)), (x * y), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x + 1.0) * Float64(y * y)) / x) / Float64(x * 2.0)) + Float64(Float64(1.0 - y) / x));
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(0.5 + Float64(-0.5 / x)), Float64(x * y), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.6%

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

   if -1 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 75.4%

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

Alternative 5: 85.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.63:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right) \cdot \mathsf{fma}\left(x, -y, x\right)}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, y, x\right)}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5 + \frac{-0.5}{x}, x \cdot y, x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+61)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x -0.63)
     (/ (* (fma x y x) (fma x (- y) x)) (* (* x x) (fma x y x)))
     (if (<= x 0.21)
       (/ 1.0 x)
       (/ 1.0 (fma y (fma (+ 0.5 (/ -0.5 x)) (* x y) x) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+61) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= -0.63) {
		tmp = (fma(x, y, x) * fma(x, -y, x)) / ((x * x) * fma(x, y, x));
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma((0.5 + (-0.5 / x)), (x * y), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+61)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= -0.63)
		tmp = Float64(Float64(fma(x, y, x) * fma(x, Float64(-y), x)) / Float64(Float64(x * x) * fma(x, y, x)));
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(0.5 + Float64(-0.5 / x)), Float64(x * y), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e+61], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.63], N[(N[(N[(x * y + x), $MachinePrecision] * N[(x * (-y) + x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.7000000000000002e61

   1. Initial program 62.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 70.6%

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

   if -2.7000000000000002e61 < x < -0.630000000000000004

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 44.0

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 91.5%

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

   if -0.630000000000000004 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 75.4%

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

Alternative 6: 84.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5 + \frac{-0.5}{x}, x \cdot y, x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.39)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 0.21)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (+ 0.5 (/ -0.5 x)) (* x y) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma((0.5 + (-0.5 / x)), (x * y), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(0.5 + Float64(-0.5 / x)), Float64(x * y), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.39], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.39000000000000001

   1. Initial program 66.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 53.2

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

   5. Applied rewrites 53.2%

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

   7. Applied rewrites 67.1%

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

   if -0.39000000000000001 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 75.4%

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

Alternative 7: 85.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(1 + \mathsf{fma}\left(y, y, y\right)\right)}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.63:\\ \;\;\;\;\frac{\left(1 - y\right) - \left(y \cdot y\right) \cdot -0.5}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x (+ 1.0 (fma y y y))))))
   (if (<= x -2.5e+209)
     t_0
     (if (<= x -0.63)
       (/ (- (- 1.0 y) (* (* y y) -0.5)) x)
       (if (<= x 0.21) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + fma(y, y, y)));
	double tmp;
	if (x <= -2.5e+209) {
		tmp = t_0;
	} else if (x <= -0.63) {
		tmp = ((1.0 - y) - ((y * y) * -0.5)) / x;
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(1.0 + fma(y, y, y))))
	tmp = 0.0
	if (x <= -2.5e+209)
		tmp = t_0;
	elseif (x <= -0.63)
		tmp = Float64(Float64(Float64(1.0 - y) - Float64(Float64(y * y) * -0.5)) / x);
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e+209], t$95$0, If[LessEqual[x, -0.63], N[(N[(N[(1.0 - y), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.49999999999999982e209 or 0.209999999999999992 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 22.3%

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

   9. Applied rewrites 53.3%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 74.2%

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

   if -2.49999999999999982e209 < x < -0.630000000000000004

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 72.6%

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

   if -0.630000000000000004 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+209}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \mathsf{fma}\left(y, y, y\right)\right)}\\ \mathbf{elif}\;x \leq -0.63:\\ \;\;\;\;\frac{\left(1 - y\right) - \left(y \cdot y\right) \cdot -0.5}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \mathsf{fma}\left(y, y, y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \left(1 + \mathsf{fma}\left(y, y, y\right)\right)}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.63:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x (+ 1.0 (fma y y y))))))
   (if (<= x -2.5e+209)
     t_0
     (if (<= x -0.63)
       (/ (fma y (fma y 0.5 -1.0) 1.0) x)
       (if (<= x 0.21) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * (1.0 + fma(y, y, y)));
	double tmp;
	if (x <= -2.5e+209) {
		tmp = t_0;
	} else if (x <= -0.63) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * Float64(1.0 + fma(y, y, y))))
	tmp = 0.0
	if (x <= -2.5e+209)
		tmp = t_0;
	elseif (x <= -0.63)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e+209], t$95$0, If[LessEqual[x, -0.63], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.49999999999999982e209 or 0.209999999999999992 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 51.0

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 22.3%

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

   9. Applied rewrites 53.3%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 74.2%

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

   if -2.49999999999999982e209 < x < -0.630000000000000004

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.8%

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

   if -0.630000000000000004 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

Alternative 9: 84.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \mathsf{fma}\left(y, y, y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.39)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 0.21) (/ 1.0 x) (/ 1.0 (* x (+ 1.0 (fma y y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 0.21) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * (1.0 + fma(y, y, y)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 0.21)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * Float64(1.0 + fma(y, y, y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.39], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.21], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.39000000000000001

   1. Initial program 66.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 53.2

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

   5. Applied rewrites 53.2%

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

   7. Applied rewrites 67.1%

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

   if -0.39000000000000001 < x < 0.209999999999999992

   1. Initial program 85.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 0.209999999999999992 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 25.0%

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

   9. Applied rewrites 56.0%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 75.3%

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

Alternative 10: 82.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.63:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 17000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.63)
   (/ (fma y (fma y 0.5 -1.0) 1.0) x)
   (if (<= x 17000000.0) (/ 1.0 x) (/ 1.0 (fma x y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.63) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (x <= 17000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.63)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (x <= 17000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(x, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.63], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 17000000.0], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.630000000000000004

   1. Initial program 66.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.7%

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

   if -0.630000000000000004 < x < 1.7e7

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.1%

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

   if 1.7e7 < x

   1. Initial program 74.4%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 11: 80.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 17000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma x y x))))
   (if (<= x -1.45e+129) t_0 (if (<= x 17000000.0) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(x, y, x);
	double tmp;
	if (x <= -1.45e+129) {
		tmp = t_0;
	} else if (x <= 17000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(x, y, x))
	tmp = 0.0
	if (x <= -1.45e+129)
		tmp = t_0;
	elseif (x <= 17000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+129], t$95$0, If[LessEqual[x, 17000000.0], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000001e129 or 1.7e7 < x

   1. Initial program 66.2%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if -1.45000000000000001e129 < x < 1.7e7

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.4%

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

Alternative 12: 74.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+191}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e+191) (/ 1.0 x) (/ 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+191) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d+191) then
        tmp = 1.0d0 / x
    else
        tmp = 1.0d0 / (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+191) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e+191:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e+191)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e+191)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.35e+191], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.34999999999999998e191

   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 76.9%

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

   if 1.34999999999999998e191 < y

   1. Initial program 58.6%

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

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. exp-to-pow N/A

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

      11. pow-flip N/A

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

      12. neg-mul-1 N/A

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

      13. pow-unpow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-/.f64 58.7

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

   4. Applied rewrites 58.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 69.8%

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

Alternative 13: 74.6% 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 77.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 74.0%

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

Developer Target 1: 77.5% 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 2024221 
(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))