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

Percentage Accurate: 77.6% → 98.7%

Time: 9.9s

Alternatives: 12

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: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -1.32e+22) t_0 (if (<= x 0.00045) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -1.32e+22) {
		tmp = t_0;
	} else if (x <= 0.00045) {
		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 <= (-1.32d+22)) then
        tmp = t_0
    else if (x <= 0.00045d0) 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 <= -1.32e+22) {
		tmp = t_0;
	} else if (x <= 0.00045) {
		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 <= -1.32e+22:
		tmp = t_0
	elif x <= 0.00045:
		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 <= -1.32e+22)
		tmp = t_0;
	elseif (x <= 0.00045)
		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 <= -1.32e+22)
		tmp = t_0;
	elseif (x <= 0.00045)
		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, -1.32e+22], t$95$0, If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e22 or 4.4999999999999999e-4 < x

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot y}}}{x} \]
   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 -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

Alternative 2: 86.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}}{x}, y, \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, 1\right), y, 1\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (fma (/ (/ (* (fma 0.5 y -1.0) x) x) x) y (/ 1.0 x))
   (if (<= x 0.00045)
     (/ 1.0 x)
     (/
      (- -1.0)
      (*
       (fma
        (fma
         (fma
          (- (+ 0.16666666666666666 (/ 0.3333333333333333 (* x x))) (/ 0.5 x))
          y
          (- 0.5 (/ 0.5 x)))
         y
         1.0)
        y
        1.0)
       x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = fma((((fma(0.5, y, -1.0) * x) / x) / x), y, (1.0 / x));
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = -(-1.0) / (fma(fma(fma(((0.16666666666666666 + (0.3333333333333333 / (x * x))) - (0.5 / x)), y, (0.5 - (0.5 / x))), y, 1.0), y, 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = fma(Float64(Float64(Float64(fma(0.5, y, -1.0) * x) / x) / x), y, Float64(1.0 / x));
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(fma(fma(Float64(Float64(0.16666666666666666 + Float64(0.3333333333333333 / Float64(x * x))) - Float64(0.5 / x)), y, Float64(0.5 - Float64(0.5 / x))), y, 1.0), y, 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * y + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], N[((--1.0) / N[(N[(N[(N[(N[(N[(0.16666666666666666 + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22

   1. Initial program 70.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 76.5%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x

   1. Initial program 80.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 80.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 82.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}}{x}, y, \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, 1\right), y, 1\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(t\_0, y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(1 - y\right) \cdot \left(x \cdot x\right)}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10^{+164}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-y, t\_0 \cdot x, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(0.5 \cdot y\right) \cdot y\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- 0.5 (/ 0.5 x)) y 1.0)))
   (if (<= x -3.2e+151)
     (/ -1.0 (* (fma t_0 y 1.0) (- x)))
     (if (<= x -1.15e+48)
       (/ (/ (* (- 1.0 y) (* x x)) x) (* x x))
       (if (<= x 0.00045)
         (/ 1.0 x)
         (if (<= x 1e+164)
           (/ -1.0 (fma (- y) (* t_0 x) (- x)))
           (/
            (/ (fma (fma (fma 0.5 y -1.0) y 1.0) x (* (* 0.5 y) y)) x)
            x)))))))

C

double code(double x, double y) {
	double t_0 = fma((0.5 - (0.5 / x)), y, 1.0);
	double tmp;
	if (x <= -3.2e+151) {
		tmp = -1.0 / (fma(t_0, y, 1.0) * -x);
	} else if (x <= -1.15e+48) {
		tmp = (((1.0 - y) * (x * x)) / x) / (x * x);
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else if (x <= 1e+164) {
		tmp = -1.0 / fma(-y, (t_0 * x), -x);
	} else {
		tmp = (fma(fma(fma(0.5, y, -1.0), y, 1.0), x, ((0.5 * y) * y)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(0.5 - Float64(0.5 / x)), y, 1.0)
	tmp = 0.0
	if (x <= -3.2e+151)
		tmp = Float64(-1.0 / Float64(fma(t_0, y, 1.0) * Float64(-x)));
	elseif (x <= -1.15e+48)
		tmp = Float64(Float64(Float64(Float64(1.0 - y) * Float64(x * x)) / x) / Float64(x * x));
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	elseif (x <= 1e+164)
		tmp = Float64(-1.0 / fma(Float64(-y), Float64(t_0 * x), Float64(-x)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), y, 1.0), x, Float64(Float64(0.5 * y) * y)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]}, If[LessEqual[x, -3.2e+151], N[(-1.0 / N[(N[(t$95$0 * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e+48], N[(N[(N[(N[(1.0 - y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1e+164], N[(-1.0 / N[((-y) * N[(t$95$0 * x), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -3.19999999999999994e151

   1. Initial program 57.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 57.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if -3.19999999999999994e151 < x < -1.15e48

   1. Initial program 83.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 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 70.5%

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

   9. Applied rewrites 87.1%

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

   if -1.15e48 < x < 4.4999999999999999e-4

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.4%

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

   if 4.4999999999999999e-4 < x < 1e164

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 91.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if 1e164 < x

   1. Initial program 65.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.4%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(0.5 \cdot y\right) \cdot y\right)}{x}}{\color{blue}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(1 - y\right) \cdot \left(x \cdot x\right)}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10^{+164}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-y, \mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right) \cdot x, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(0.5 \cdot y\right) \cdot y\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(0.5, y, -1\right) \cdot x}{x}}{x}, y, \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10^{+164}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-y, \mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right) \cdot x, -x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(0.5 \cdot y\right) \cdot y\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (fma (/ (/ (* (fma 0.5 y -1.0) x) x) x) y (/ 1.0 x))
   (if (<= x 0.00045)
     (/ 1.0 x)
     (if (<= x 1e+164)
       (/ -1.0 (fma (- y) (* (fma (- 0.5 (/ 0.5 x)) y 1.0) x) (- x)))
       (/ (/ (fma (fma (fma 0.5 y -1.0) y 1.0) x (* (* 0.5 y) y)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = fma((((fma(0.5, y, -1.0) * x) / x) / x), y, (1.0 / x));
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else if (x <= 1e+164) {
		tmp = -1.0 / fma(-y, (fma((0.5 - (0.5 / x)), y, 1.0) * x), -x);
	} else {
		tmp = (fma(fma(fma(0.5, y, -1.0), y, 1.0), x, ((0.5 * y) * y)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = fma(Float64(Float64(Float64(fma(0.5, y, -1.0) * x) / x) / x), y, Float64(1.0 / x));
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	elseif (x <= 1e+164)
		tmp = Float64(-1.0 / fma(Float64(-y), Float64(fma(Float64(0.5 - Float64(0.5 / x)), y, 1.0) * x), Float64(-x)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), y, 1.0), x, Float64(Float64(0.5 * y) * y)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * y + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1e+164], N[(-1.0 / N[((-y) * N[(N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.32e22

   1. Initial program 70.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 76.5%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x < 1e164

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 91.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if 1e164 < x

   1. Initial program 65.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 82.4%

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

Alternative 5: 85.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(1 - y\right) \cdot \left(x \cdot x\right)}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (* (fma (fma (- 0.5 (/ 0.5 x)) y 1.0) y 1.0) (- x)))))
   (if (<= x -3.2e+151)
     t_0
     (if (<= x -1.15e+48)
       (/ (/ (* (- 1.0 y) (* x x)) x) (* x x))
       (if (<= x 0.00045) (/ 1.0 x) t_0)))))

C

double code(double x, double y) {
	double t_0 = -1.0 / (fma(fma((0.5 - (0.5 / x)), y, 1.0), y, 1.0) * -x);
	double tmp;
	if (x <= -3.2e+151) {
		tmp = t_0;
	} else if (x <= -1.15e+48) {
		tmp = (((1.0 - y) * (x * x)) / x) / (x * x);
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(fma(fma(Float64(0.5 - Float64(0.5 / x)), y, 1.0), y, 1.0) * Float64(-x)))
	tmp = 0.0
	if (x <= -3.2e+151)
		tmp = t_0;
	elseif (x <= -1.15e+48)
		tmp = Float64(Float64(Float64(Float64(1.0 - y) * Float64(x * x)) / x) / Float64(x * x));
	elseif (x <= 0.00045)
		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[(N[(N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+151], t$95$0, If[LessEqual[x, -1.15e+48], N[(N[(N[(N[(1.0 - y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999994e151 or 4.4999999999999999e-4 < x

   1. Initial program 73.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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. 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 73.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -3.19999999999999994e151 < x < -1.15e48

   1. Initial program 83.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 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 70.5%

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

   9. Applied rewrites 87.1%

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

   if -1.15e48 < x < 4.4999999999999999e-4

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(1 - y\right) \cdot \left(x \cdot x\right)}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y, x, x\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- x (* y x)) x) x)))
   (if (<= x -1.32e+22)
     t_0
     (if (<= x 0.00045)
       (/ 1.0 x)
       (if (<= x 1.1e+157) (* (/ (/ 1.0 x) (fma y x x)) x) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - (y * x)) / x) / x;
	double tmp;
	if (x <= -1.32e+22) {
		tmp = t_0;
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else if (x <= 1.1e+157) {
		tmp = ((1.0 / x) / fma(y, x, x)) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - Float64(y * x)) / x) / x)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = t_0;
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.1e+157)
		tmp = Float64(Float64(Float64(1.0 / x) / fma(y, x, x)) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.32e+22], t$95$0, If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.1e+157], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(y * x + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22 or 1.1000000000000001e157 < x

   1. Initial program 68.6%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 74.8%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x < 1.1000000000000001e157

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.5

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

   5. Applied rewrites 63.5%

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 73.6%

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

   12. Applied rewrites 76.0%

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

Alternative 7: 83.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{x - y \cdot x}{x}}{x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (/ (/ (- x (* y x)) x) x)
   (if (<= x 0.00045) (/ 1.0 x) (/ 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = ((x - (y * x)) / x) / x;
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = Float64(Float64(Float64(x - Float64(y * x)) / x) / x);
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, x, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22

   1. Initial program 70.2%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 73.3%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.6%

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

   12. Applied rewrites 71.2%

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

Alternative 8: 82.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{-x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (* (/ -1.0 (- x)) (fma (fma 0.5 y -1.0) y 1.0))
   (if (<= x 0.00045) (/ 1.0 x) (/ 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = (-1.0 / -x) * fma(fma(0.5, y, -1.0), y, 1.0);
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = Float64(Float64(-1.0 / Float64(-x)) * fma(fma(0.5, y, -1.0), y, 1.0));
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, x, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(N[(-1.0 / (-x)), $MachinePrecision] * N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22

   1. Initial program 70.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.4%

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

   10. Applied rewrites 72.5%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.6%

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

   12. Applied rewrites 71.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{-x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+22)
   (/ (fma (fma 0.5 y -1.0) y 1.0) x)
   (if (<= x 0.00045) (/ 1.0 x) (/ 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+22) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+22)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	elseif (x <= 0.00045)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, x, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+22], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e22

   1. Initial program 70.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.4%

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

   if -1.32e22 < x < 4.4999999999999999e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 4.4999999999999999e-4 < x

   1. Initial program 80.2%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.6%

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

   12. Applied rewrites 71.2%

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

Alternative 10: 74.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(y \cdot y\right) \cdot 0.5}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e+132)
   (/ (* (* y y) 0.5) x)
   (if (<= y 7e+87) (/ 1.0 x) (/ 1.0 (fma y x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.95e+132) {
		tmp = ((y * y) * 0.5) / x;
	} else if (y <= 7e+87) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.95e+132)
		tmp = Float64(Float64(Float64(y * y) * 0.5) / x);
	elseif (y <= 7e+87)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, x, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.95e+132], N[(N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 7e+87], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * x + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.95000000000000001e132

   1. Initial program 47.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 24.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.6%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{1}{2} \cdot {y}^{2}}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.6%

       \[\leadsto \frac{\left(y \cdot y\right) \cdot 0.5}{x} \]

   if -1.95000000000000001e132 < y < 6.99999999999999972e87

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 86.8%

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

   if 6.99999999999999972e87 < y

   1. Initial program 62.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 58.0%

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

   12. Applied rewrites 54.7%

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

Alternative 11: 80.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fma y x x))))
   (if (<= x -1.8e+177) t_0 (if (<= x 0.00045) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 / fma(y, x, x);
	double tmp;
	if (x <= -1.8e+177) {
		tmp = t_0;
	} else if (x <= 0.00045) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 / fma(y, x, x))
	tmp = 0.0
	if (x <= -1.8e+177)
		tmp = t_0;
	elseif (x <= 0.00045)
		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[(y * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e+177], t$95$0, If[LessEqual[x, 0.00045], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000001e177 or 4.4999999999999999e-4 < x

   1. Initial program 72.7%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 43.8%

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

   12. Applied rewrites 70.5%

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

   if -1.80000000000000001e177 < x < 4.4999999999999999e-4

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.9%

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

Alternative 12: 74.7% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 74.2%

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

Developer Target 1: 77.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(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))