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

Percentage Accurate: 77.6% → 98.4%

Time: 11.4s

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: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.285:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -1.25e+34) t_0 (if (<= x 0.285) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -1.25e+34) {
		tmp = t_0;
	} else if (x <= 0.285) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-y) / x
    if (x <= (-1.25d+34)) then
        tmp = t_0
    else if (x <= 0.285d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -1.25e+34) {
		tmp = t_0;
	} else if (x <= 0.285) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -1.25e+34:
		tmp = t_0
	elif x <= 0.285:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = t_0;
	elseif (x <= 0.285)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -1.25e+34)
		tmp = t_0;
	elseif (x <= 0.285)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.25e+34], t$95$0, If[LessEqual[x, 0.285], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e34 or 0.284999999999999976 < x

   1. Initial program 77.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.25e34 < x < 0.284999999999999976

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.5%

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

Alternative 2: 86.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 0.285:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+34)
   (/ (/ (+ x (* y (fma x (fma y 0.5 -1.0) (* y 0.5)))) x) x)
   (if (<= x 0.285) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (fma y y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+34) {
		tmp = ((x + (y * fma(x, fma(y, 0.5, -1.0), (y * 0.5)))) / x) / x;
	} else if (x <= 0.285) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + fma(y, y, y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = Float64(Float64(Float64(x + Float64(y * fma(x, fma(y, 0.5, -1.0), Float64(y * 0.5)))) / x) / x);
	elseif (x <= 0.285)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + fma(y, y, y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+34], N[(N[(N[(x + N[(y * N[(x * N[(y * 0.5 + -1.0), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.285], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e34

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.3%

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

   if -1.25e34 < x < 0.284999999999999976

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.5%

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

   if 0.284999999999999976 < x

   1. Initial program 77.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 82.6%

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

Alternative 3: 85.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x}, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.285:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+34)
   (/ (fma y (/ (fma x (fma y 0.5 -1.0) (* y 0.5)) x) 1.0) x)
   (if (<= x 0.285) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (fma y y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+34) {
		tmp = fma(y, (fma(x, fma(y, 0.5, -1.0), (y * 0.5)) / x), 1.0) / x;
	} else if (x <= 0.285) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + fma(y, y, y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = Float64(fma(y, Float64(fma(x, fma(y, 0.5, -1.0), Float64(y * 0.5)) / x), 1.0) / x);
	elseif (x <= 0.285)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + fma(y, y, y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+34], N[(N[(y * N[(N[(x * N[(y * 0.5 + -1.0), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.285], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e34

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.8%

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

   if -1.25e34 < x < 0.284999999999999976

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.5%

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

   if 0.284999999999999976 < x

   1. Initial program 77.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 82.6%

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

Alternative 4: 85.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.285:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+34)
   (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)
   (if (<= x 0.285) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (fma y y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+34) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	} else if (x <= 0.285) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + fma(y, y, y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x);
	elseif (x <= 0.285)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + fma(y, y, y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+34], N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.285], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e34

   1. Initial program 78.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   7. 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. lower-fma.f64 N/A

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 80.8%

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

   if -1.25e34 < x < 0.284999999999999976

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.5%

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

   if 0.284999999999999976 < x

   1. Initial program 77.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 82.6%

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

Alternative 5: 82.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)))
   (if (<= x -1.25e+34) t_0 (if (<= x 5.5e+38) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	double tmp;
	if (x <= -1.25e+34) {
		tmp = t_0;
	} else if (x <= 5.5e+38) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = t_0;
	elseif (x <= 5.5e+38)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -1.25e+34], t$95$0, If[LessEqual[x, 5.5e+38], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e34 or 5.5000000000000003e38 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   7. 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. lower-fma.f64 N/A

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 74.8%

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

   if -1.25e34 < x < 5.5000000000000003e38

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.0%

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

Alternative 6: 78.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+34) (/ (fma y (fma y 0.5 -1.0) 1.0) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+34) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+34)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+34], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e34

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.4%

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

   if -1.25e34 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.2%

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

Alternative 7: 75.0% 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 x around 0

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

5. Applied rewrites 74.4%

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

Developer Target 1: 77.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))