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

Percentage Accurate: 76.8% → 99.5%

Time: 9.2s

Alternatives: 7

Speedup: 7.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -0.05) t_0 (if (<= x 0.02) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -0.05) {
		tmp = t_0;
	} else if (x <= 0.02) {
		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 <= (-0.05d0)) then
        tmp = t_0
    else if (x <= 0.02d0) 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 <= -0.05) {
		tmp = t_0;
	} else if (x <= 0.02) {
		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 <= -0.05:
		tmp = t_0
	elif x <= 0.02:
		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 <= -0.05)
		tmp = t_0;
	elseif (x <= 0.02)
		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 <= -0.05)
		tmp = t_0;
	elseif (x <= 0.02)
		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, -0.05], t$95$0, If[LessEqual[x, 0.02], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 0.0200000000000000004 < x

   1. Initial program 74.4%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.050000000000000003 < x < 0.0200000000000000004

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

Alternative 2: 86.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{x \cdot x} + 0.16666666666666666\right) - \frac{0.5}{x}, y, \frac{-0.5}{x}\right) + 0.5, y, 1\right), y, 1\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.05)
   (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)
   (if (<= x 0.02)
     (/ 1.0 x)
     (/
      (/
       1.0
       (fma
        (fma
         (+
          (fma
           (- (+ (/ 0.3333333333333333 (* x x)) 0.16666666666666666) (/ 0.5 x))
           y
           (/ -0.5 x))
          0.5)
         y
         1.0)
        y
        1.0))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.05) {
		tmp = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	} else if (x <= 0.02) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / fma(fma((fma((((0.3333333333333333 / (x * x)) + 0.16666666666666666) - (0.5 / x)), y, (-0.5 / x)) + 0.5), y, 1.0), y, 1.0)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x);
	elseif (x <= 0.02)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / fma(fma(Float64(fma(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 0.16666666666666666) - Float64(0.5 / x)), y, Float64(-0.5 / x)) + 0.5), y, 1.0), y, 1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.05], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.02], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.6%

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

   if -0.050000000000000003 < x < 0.0200000000000000004

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

   if 0.0200000000000000004 < x

   1. Initial program 72.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. remove-double-neg N/A

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

      7. pow-flip N/A

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

      8. pow-flip N/A

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

      9. exp-to-pow N/A

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

      10. lift-log.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-log.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 81.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{x \cdot x} + 0.16666666666666666\right) - \frac{0.5}{x}, y, \frac{-0.5}{x}\right) + 0.5, y, 1\right), y, 1\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 - \frac{0.5}{x}, y, 1\right), y, 1\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.05)
   (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)
   (if (<= x 0.02)
     (/ 1.0 x)
     (/ (/ 1.0 (fma (fma (- 0.5 (/ 0.5 x)) y 1.0) y 1.0)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.05) {
		tmp = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	} else if (x <= 0.02) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / fma(fma((0.5 - (0.5 / x)), y, 1.0), y, 1.0)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x);
	elseif (x <= 0.02)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / fma(fma(Float64(0.5 - Float64(0.5 / x)), y, 1.0), y, 1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.05], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.02], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.6%

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

   if -0.050000000000000003 < x < 0.0200000000000000004

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

   if 0.0200000000000000004 < x

   1. Initial program 72.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. remove-double-neg N/A

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

      7. pow-flip N/A

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

      8. pow-flip N/A

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

      9. exp-to-pow N/A

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

      10. lift-log.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-log.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

Alternative 4: 83.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.05)
   (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)
   (if (<= x 0.02) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.05) {
		tmp = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	} else if (x <= 0.02) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x);
	elseif (x <= 0.02)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.05], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.02], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.6%

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

   if -0.050000000000000003 < x < 0.0200000000000000004

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

   if 0.0200000000000000004 < x

   1. Initial program 72.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. remove-double-neg N/A

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

      7. pow-flip N/A

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

      8. pow-flip N/A

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

      9. exp-to-pow N/A

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

      10. lift-log.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-log.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.2

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

   7. Applied rewrites 70.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.05)
   (/ (fma (fma 0.5 y -1.0) y 1.0) x)
   (if (<= x 0.02) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.05) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else if (x <= 0.02) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	elseif (x <= 0.02)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.05], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.02], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right) + \frac{1}{x}} \]
   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 66.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.0%

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

   if -0.050000000000000003 < x < 0.0200000000000000004

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

   if 0.0200000000000000004 < x

   1. Initial program 72.6%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. remove-double-neg N/A

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

      7. pow-flip N/A

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

      8. pow-flip N/A

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

      9. exp-to-pow N/A

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

      10. lift-log.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-log.f64 N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.2

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

   7. Applied rewrites 70.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.05) (/ (fma (fma 0.5 y -1.0) y 1.0) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.05) {
		tmp = fma(fma(0.5, y, -1.0), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(fma(fma(0.5, y, -1.0), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.05], N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right) + \frac{1}{x}} \]
   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 66.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.0%

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

   if -0.050000000000000003 < x

   1. Initial program 80.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 80.5%

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

Alternative 7: 74.5% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 73.4%

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

Developer Target 1: 77.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024249 
(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))