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

Percentage Accurate: 78.1% → 99.5%

Time: 9.8s

Alternatives: 10

Speedup: 7.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% 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: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -0.64:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -0.64) t_0 (if (<= x 7e-7) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -0.64) {
		tmp = t_0;
	} else if (x <= 7e-7) {
		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 <= (-0.64d0)) then
        tmp = t_0
    else if (x <= 7d-7) 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 <= -0.64) {
		tmp = t_0;
	} else if (x <= 7e-7) {
		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 <= -0.64:
		tmp = t_0
	elif x <= 7e-7:
		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 <= -0.64)
		tmp = t_0;
	elseif (x <= 7e-7)
		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 <= -0.64)
		tmp = t_0;
	elseif (x <= 7e-7)
		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, -0.64], t$95$0, If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.640000000000000013 or 6.99999999999999968e-7 < x

   1. Initial program 77.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

   if -0.640000000000000013 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

Alternative 2: 86.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{0.5}{x}\\ t_1 := \frac{0.3333333333333333}{x \cdot x}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + t\_1\right) - \frac{0.5}{x}, y, t\_0\right) \cdot \left(-x\right), y, -x\right), y, -x\right)}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + t\_1, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ 0.5 x))) (t_1 (/ 0.3333333333333333 (* x x))))
   (if (<= x -7.4e+168)
     (/
      -1.0
      (fma
       (fma
        (* (fma (- (+ 0.16666666666666666 t_1) (/ 0.5 x)) y t_0) (- x))
        y
        (- x))
       y
       (- x)))
     (if (<= x -1.85)
       (*
        (fma
         (fma
          (fma (+ (+ (/ 0.5 x) 0.16666666666666666) t_1) y (- -0.5 (/ 0.5 x)))
          y
          1.0)
         y
         -1.0)
        (/ -1.0 x))
       (if (<= x 7e-7)
         (/ 1.0 x)
         (/ -1.0 (* (fma (fma t_0 y 1.0) y 1.0) (- x))))))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 / x);
	double t_1 = 0.3333333333333333 / (x * x);
	double tmp;
	if (x <= -7.4e+168) {
		tmp = -1.0 / fma(fma((fma(((0.16666666666666666 + t_1) - (0.5 / x)), y, t_0) * -x), y, -x), y, -x);
	} else if (x <= -1.85) {
		tmp = fma(fma(fma((((0.5 / x) + 0.16666666666666666) + t_1), y, (-0.5 - (0.5 / x))), y, 1.0), y, -1.0) * (-1.0 / x);
	} else if (x <= 7e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (fma(fma(t_0, y, 1.0), y, 1.0) * -x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 / x))
	t_1 = Float64(0.3333333333333333 / Float64(x * x))
	tmp = 0.0
	if (x <= -7.4e+168)
		tmp = Float64(-1.0 / fma(fma(Float64(fma(Float64(Float64(0.16666666666666666 + t_1) - Float64(0.5 / x)), y, t_0) * Float64(-x)), y, Float64(-x)), y, Float64(-x)));
	elseif (x <= -1.85)
		tmp = Float64(fma(fma(fma(Float64(Float64(Float64(0.5 / x) + 0.16666666666666666) + t_1), y, Float64(-0.5 - Float64(0.5 / x))), y, 1.0), y, -1.0) * Float64(-1.0 / x));
	elseif (x <= 7e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(fma(fma(t_0, y, 1.0), y, 1.0) * Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.4e+168], N[(-1.0 / N[(N[(N[(N[(N[(N[(0.16666666666666666 + t$95$1), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + t$95$0), $MachinePrecision] * (-x)), $MachinePrecision] * y + (-x)), $MachinePrecision] * y + (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85], N[(N[(N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + t$95$1), $MachinePrecision] * y + N[(-0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(N[(N[(t$95$0 * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.40000000000000018e168

   1. Initial program 55.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 55.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 74.7%

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

   if -7.40000000000000018e168 < x < -1.8500000000000001

   1. Initial program 91.5%

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

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 91.5

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 91.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 85.1%

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

   if -1.8500000000000001 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

   if 6.99999999999999968e-7 < x

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 77.9%

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

   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 80.9

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

   7. Applied rewrites 80.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right) \cdot \left(-x\right), y, -x\right), y, -x\right)}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{x \cdot x}\\ t_1 := 0.5 - \frac{0.5}{x}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + t\_0\right) - \frac{0.5}{x}, y, t\_1\right), y, 1\right), y, 1\right) \cdot x}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + t\_0, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* x x))) (t_1 (- 0.5 (/ 0.5 x))))
   (if (<= x -7.4e+168)
     (/
      (- -1.0)
      (*
       (fma
        (fma (fma (- (+ 0.16666666666666666 t_0) (/ 0.5 x)) y t_1) y 1.0)
        y
        1.0)
       x))
     (if (<= x -1.85)
       (*
        (fma
         (fma
          (fma (+ (+ (/ 0.5 x) 0.16666666666666666) t_0) y (- -0.5 (/ 0.5 x)))
          y
          1.0)
         y
         -1.0)
        (/ -1.0 x))
       (if (<= x 7e-7)
         (/ 1.0 x)
         (/ -1.0 (* (fma (fma t_1 y 1.0) y 1.0) (- x))))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 / (x * x);
	double t_1 = 0.5 - (0.5 / x);
	double tmp;
	if (x <= -7.4e+168) {
		tmp = -(-1.0) / (fma(fma(fma(((0.16666666666666666 + t_0) - (0.5 / x)), y, t_1), y, 1.0), y, 1.0) * x);
	} else if (x <= -1.85) {
		tmp = fma(fma(fma((((0.5 / x) + 0.16666666666666666) + t_0), y, (-0.5 - (0.5 / x))), y, 1.0), y, -1.0) * (-1.0 / x);
	} else if (x <= 7e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = -1.0 / (fma(fma(t_1, y, 1.0), y, 1.0) * -x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 / Float64(x * x))
	t_1 = Float64(0.5 - Float64(0.5 / x))
	tmp = 0.0
	if (x <= -7.4e+168)
		tmp = Float64(Float64(-(-1.0)) / Float64(fma(fma(fma(Float64(Float64(0.16666666666666666 + t_0) - Float64(0.5 / x)), y, t_1), y, 1.0), y, 1.0) * x));
	elseif (x <= -1.85)
		tmp = Float64(fma(fma(fma(Float64(Float64(Float64(0.5 / x) + 0.16666666666666666) + t_0), y, Float64(-0.5 - Float64(0.5 / x))), y, 1.0), y, -1.0) * Float64(-1.0 / x));
	elseif (x <= 7e-7)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-1.0 / Float64(fma(fma(t_1, y, 1.0), y, 1.0) * Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.4e+168], N[((--1.0) / N[(N[(N[(N[(N[(N[(0.16666666666666666 + t$95$0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + t$95$1), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85], N[(N[(N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + t$95$0), $MachinePrecision] * y + N[(-0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], N[(-1.0 / N[(N[(N[(t$95$1 * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.40000000000000018e168

   1. Initial program 55.4%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 55.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{-1}{\left(-x\right) \cdot \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\left(\frac{1}{2} + y \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)}} \]
   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 74.7%

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

   if -7.40000000000000018e168 < x < -1.8500000000000001

   1. Initial program 91.5%

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

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 91.5

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 91.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 85.1%

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

   if -1.8500000000000001 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

   if 6.99999999999999968e-7 < x

   1. Initial program 77.9%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 77.9%

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

   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 80.9

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

   7. Applied rewrites 80.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{--1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{x \cdot x}\right) - \frac{0.5}{x}, y, 0.5 - \frac{0.5}{x}\right), y, 1\right), y, 1\right) \cdot x}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -7.8e+168)
     t_0
     (if (<= x -1.85)
       (*
        (fma
         (fma
          (fma
           (+ (+ (/ 0.5 x) 0.16666666666666666) (/ 0.3333333333333333 (* x x)))
           y
           (- -0.5 (/ 0.5 x)))
          y
          1.0)
         y
         -1.0)
        (/ -1.0 x))
       (if (<= x 7e-7) (/ 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 <= -7.8e+168) {
		tmp = t_0;
	} else if (x <= -1.85) {
		tmp = fma(fma(fma((((0.5 / x) + 0.16666666666666666) + (0.3333333333333333 / (x * x))), y, (-0.5 - (0.5 / x))), y, 1.0), y, -1.0) * (-1.0 / x);
	} else if (x <= 7e-7) {
		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 <= -7.8e+168)
		tmp = t_0;
	elseif (x <= -1.85)
		tmp = Float64(fma(fma(fma(Float64(Float64(Float64(0.5 / x) + 0.16666666666666666) + Float64(0.3333333333333333 / Float64(x * x))), y, Float64(-0.5 - Float64(0.5 / x))), y, 1.0), y, -1.0) * Float64(-1.0 / x));
	elseif (x <= 7e-7)
		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, -7.8e+168], t$95$0, If[LessEqual[x, -1.85], N[(N[(N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999999e168 or 6.99999999999999968e-7 < x

   1. Initial program 71.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 71.5%

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

   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 79.0

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

   7. Applied rewrites 79.0%

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

   if -7.79999999999999999e168 < x < -1.8500000000000001

   1. Initial program 91.5%

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

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 91.5

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 91.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 85.1%

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

   if -1.8500000000000001 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, y, -0.5 - \frac{0.5}{x}\right), y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, -y, \frac{0.5}{x} + 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -7.8e+168)
     t_0
     (if (<= x -1.85)
       (/
        (fma
         (fma
          (fma
           (+ (+ (/ 0.5 x) 0.16666666666666666) (/ 0.3333333333333333 (* x x)))
           (- y)
           (+ (/ 0.5 x) 0.5))
          y
          -1.0)
         y
         1.0)
        x)
       (if (<= x 7e-7) (/ 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 <= -7.8e+168) {
		tmp = t_0;
	} else if (x <= -1.85) {
		tmp = fma(fma(fma((((0.5 / x) + 0.16666666666666666) + (0.3333333333333333 / (x * x))), -y, ((0.5 / x) + 0.5)), y, -1.0), y, 1.0) / x;
	} else if (x <= 7e-7) {
		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 <= -7.8e+168)
		tmp = t_0;
	elseif (x <= -1.85)
		tmp = Float64(fma(fma(fma(Float64(Float64(Float64(0.5 / x) + 0.16666666666666666) + Float64(0.3333333333333333 / Float64(x * x))), Float64(-y), Float64(Float64(0.5 / x) + 0.5)), y, -1.0), y, 1.0) / x);
	elseif (x <= 7e-7)
		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, -7.8e+168], t$95$0, If[LessEqual[x, -1.85], N[(N[(N[(N[(N[(N[(N[(0.5 / x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-y) + N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999999e168 or 6.99999999999999968e-7 < x

   1. Initial program 71.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 71.5%

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

   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 79.0

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

   7. Applied rewrites 79.0%

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

   if -7.79999999999999999e168 < x < -1.8500000000000001

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   if -1.8500000000000001 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -1.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.5}{x} + 0.16666666666666666\right) + \frac{0.3333333333333333}{x \cdot x}, -y, \frac{0.5}{x} + 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.64:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 - \frac{0.5}{x}, y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -7.8e+168)
     t_0
     (if (<= x -0.64)
       (* (fma (fma (- -0.5 (/ 0.5 x)) y 1.0) y -1.0) (/ -1.0 x))
       (if (<= x 7e-7) (/ 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 <= -7.8e+168) {
		tmp = t_0;
	} else if (x <= -0.64) {
		tmp = fma(fma((-0.5 - (0.5 / x)), y, 1.0), y, -1.0) * (-1.0 / x);
	} else if (x <= 7e-7) {
		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 <= -7.8e+168)
		tmp = t_0;
	elseif (x <= -0.64)
		tmp = Float64(fma(fma(Float64(-0.5 - Float64(0.5 / x)), y, 1.0), y, -1.0) * Float64(-1.0 / x));
	elseif (x <= 7e-7)
		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, -7.8e+168], t$95$0, If[LessEqual[x, -0.64], N[(N[(N[(N[(-0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999999e168 or 6.99999999999999968e-7 < x

   1. Initial program 71.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 71.5%

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

   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 79.0

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

   7. Applied rewrites 79.0%

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

   if -7.79999999999999999e168 < x < -0.640000000000000013

   1. Initial program 91.5%

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

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 91.5

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 91.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -0.640000000000000013 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+168}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq -0.64:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 - \frac{0.5}{x}, y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right) \cdot \left(-x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.64:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 - \frac{0.5}{x}, y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (- (fma y x x)))))
   (if (<= x -7.5e+208)
     t_0
     (if (<= x -0.64)
       (* (fma (fma (- -0.5 (/ 0.5 x)) y 1.0) y -1.0) (/ -1.0 x))
       (if (<= x 7e-7) (/ 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 <= -7.5e+208) {
		tmp = t_0;
	} else if (x <= -0.64) {
		tmp = fma(fma((-0.5 - (0.5 / x)), y, 1.0), y, -1.0) * (-1.0 / x);
	} else if (x <= 7e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(-fma(y, x, x)))
	tmp = 0.0
	if (x <= -7.5e+208)
		tmp = t_0;
	elseif (x <= -0.64)
		tmp = Float64(fma(fma(Float64(-0.5 - Float64(0.5 / x)), y, 1.0), y, -1.0) * Float64(-1.0 / x));
	elseif (x <= 7e-7)
		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, -7.5e+208], t$95$0, If[LessEqual[x, -0.64], N[(N[(N[(N[(-0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999964e208 or 6.99999999999999968e-7 < x

   1. Initial program 72.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 72.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if -7.49999999999999964e208 < x < -0.640000000000000013

   1. Initial program 86.4%

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

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 86.4

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 86.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -0.640000000000000013 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{elif}\;x \leq -0.64:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 - \frac{0.5}{x}, y, 1\right), y, -1\right) \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -0.64:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (- (fma y x x)))))
   (if (<= x -7.5e+208)
     t_0
     (if (<= x -0.64)
       (/ (fma (fma 0.5 y -1.0) y 1.0) x)
       (if (<= x 7e-7) (/ 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 <= -7.5e+208) {
		tmp = t_0;
	} else if (x <= -0.64) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else if (x <= 7e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(-fma(y, x, x)))
	tmp = 0.0
	if (x <= -7.5e+208)
		tmp = t_0;
	elseif (x <= -0.64)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	elseif (x <= 7e-7)
		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, -7.5e+208], t$95$0, If[LessEqual[x, -0.64], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999964e208 or 6.99999999999999968e-7 < x

   1. Initial program 72.5%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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 72.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if -7.49999999999999964e208 < x < -0.640000000000000013

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.4%

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

   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 81.3%

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

   if -0.640000000000000013 < x < 6.99999999999999968e-7

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.1%

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

Alternative 9: 80.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{-\mathsf{fma}\left(y, x, x\right)}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ -1.0 (- (fma y x x)))))
   (if (<= x -1.65e+51) t_0 (if (<= x 7e-7) (/ 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 <= -1.65e+51) {
		tmp = t_0;
	} else if (x <= 7e-7) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 / Float64(-fma(y, x, x)))
	tmp = 0.0
	if (x <= -1.65e+51)
		tmp = t_0;
	elseif (x <= 7e-7)
		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, -1.65e+51], t$95$0, If[LessEqual[x, 7e-7], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e51 or 6.99999999999999968e-7 < x

   1. Initial program 74.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. exp-to-pow N/A

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

      15. pow-flip N/A

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

      16. neg-mul-1 N/A

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

      17. pow-unpow N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if -1.6499999999999999e51 < x < 6.99999999999999968e-7

   1. Initial program 82.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.2%

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

Alternative 10: 74.3% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 73.0%

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

Developer Target 1: 77.2% 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 2024276 
(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))