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

Percentage Accurate: 78.3% → 98.5%

Time: 8.9s

Alternatives: 8

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: 78.3% 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: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -5e+33) t_0 (if (<= x 0.9) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -5e+33) {
		tmp = t_0;
	} else if (x <= 0.9) {
		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 <= (-5d+33)) then
        tmp = t_0
    else if (x <= 0.9d0) 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 <= -5e+33) {
		tmp = t_0;
	} else if (x <= 0.9) {
		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 <= -5e+33:
		tmp = t_0
	elif x <= 0.9:
		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 <= -5e+33)
		tmp = t_0;
	elseif (x <= 0.9)
		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 <= -5e+33)
		tmp = t_0;
	elseif (x <= 0.9)
		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, -5e+33], t$95$0, If[LessEqual[x, 0.9], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999973e33 or 0.900000000000000022 < x

   1. Initial program 73.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot y}}}{x} \]
   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 -4.99999999999999973e33 < x < 0.900000000000000022

   1. Initial program 81.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

Alternative 2: 82.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), x, 0.5 \cdot y\right)}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right), x, \left(y \cdot y\right) \cdot 0.5\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+35)
   (/ (fma (/ (fma (fma 0.5 y -1.0) x (* 0.5 y)) x) y 1.0) x)
   (if (<= x 0.9)
     (/ 1.0 x)
     (/ (/ (fma (fma (fma 0.5 y -1.0) y 1.0) x (* (* y y) 0.5)) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+35) {
		tmp = fma((fma(fma(0.5, y, -1.0), x, (0.5 * y)) / x), y, 1.0) / x;
	} else if (x <= 0.9) {
		tmp = 1.0 / x;
	} else {
		tmp = (fma(fma(fma(0.5, y, -1.0), y, 1.0), x, ((y * y) * 0.5)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+35)
		tmp = Float64(fma(Float64(fma(fma(0.5, y, -1.0), x, Float64(0.5 * y)) / x), y, 1.0) / x);
	elseif (x <= 0.9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(fma(fma(fma(0.5, y, -1.0), y, 1.0), x, Float64(Float64(y * y) * 0.5)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e+35], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.9], N[(1.0 / x), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] * x + N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000011e35

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.1%

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

   if -2.50000000000000011e35 < x < 0.900000000000000022

   1. Initial program 81.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

   if 0.900000000000000022 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 57.7

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

   5. Applied rewrites 57.7%

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

   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 61.9%

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

Alternative 3: 81.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), x, 0.5 \cdot y\right)}{x}, y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-y, x, x\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+35)
   (/ (fma (/ (fma (fma 0.5 y -1.0) x (* 0.5 y)) x) y 1.0) x)
   (if (<= x 7e+232) (/ 1.0 x) (/ (/ (fma (- y) x x) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+35) {
		tmp = fma((fma(fma(0.5, y, -1.0), x, (0.5 * y)) / x), y, 1.0) / x;
	} else if (x <= 7e+232) {
		tmp = 1.0 / x;
	} else {
		tmp = (fma(-y, x, x) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+35)
		tmp = Float64(fma(Float64(fma(fma(0.5, y, -1.0), x, Float64(0.5 * y)) / x), y, 1.0) / x);
	elseif (x <= 7e+232)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(fma(Float64(-y), x, x) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e+35], N[(N[(N[(N[(N[(0.5 * y + -1.0), $MachinePrecision] * x + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e+232], N[(1.0 / x), $MachinePrecision], N[(N[(N[((-y) * x + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000011e35

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.1%

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

   if -2.50000000000000011e35 < x < 7.00000000000000026e232

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.9%

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

   if 7.00000000000000026e232 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 65.5

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

   5. Applied rewrites 65.5%

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

   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 72.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 72.3%

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

Alternative 4: 81.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+232}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-y, x, x\right)}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+35)
   (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)
   (if (<= x 7e+232) (/ 1.0 x) (/ (/ (fma (- y) x x) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+35) {
		tmp = fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x;
	} else if (x <= 7e+232) {
		tmp = 1.0 / x;
	} else {
		tmp = (fma(-y, x, x) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+35)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 0.5), y, -1.0), y, 1.0) / x);
	elseif (x <= 7e+232)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(fma(Float64(-y), x, x) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e+35], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e+232], N[(1.0 / x), $MachinePrecision], N[(N[(N[((-y) * x + x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000011e35

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   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 85.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.0%

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

   if -2.50000000000000011e35 < x < 7.00000000000000026e232

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.9%

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

   if 7.00000000000000026e232 < x

   1. Initial program 55.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 65.5

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

   5. Applied rewrites 65.5%

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

   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 72.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 72.3%

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

Alternative 5: 80.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, 0.5\right), y, -1\right), y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+35)
   (/ (fma (fma (fma -0.16666666666666666 y 0.5) y -1.0) y 1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+35) {
		tmp = fma(fma(fma(-0.16666666666666666, y, 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 <= -2.5e+35)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, 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, -2.5e+35], N[(N[(N[(N[(-0.16666666666666666 * y + 0.5), $MachinePrecision] * 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 < -2.50000000000000011e35

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   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 85.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.0%

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

   if -2.50000000000000011e35 < 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

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

   5. Applied rewrites 79.0%

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

Alternative 6: 79.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot y\right) \cdot y, y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+70)
   (/ (fma (* (* -0.16666666666666666 y) y) y 1.0) x)
   (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+70) {
		tmp = fma(((-0.16666666666666666 * y) * y), y, 1.0) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e+70)
		tmp = Float64(fma(Float64(Float64(-0.16666666666666666 * y) * y), y, 1.0) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e+70], N[(N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999988e70

   1. Initial program 72.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(y \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{x}^{2}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - 1\right)}}{x} \]
   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 83.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 83.6%

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

   if -1.49999999999999988e70 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 79.5%

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

Alternative 7: 79.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, -1\right), y, 1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+35) (/ (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 <= -2.5e+35) {
		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 <= -2.5e+35)
		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, -2.5e+35], 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 < -2.50000000000000011e35

   1. Initial program 74.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{1 + y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - 1\right)}}{x} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.1%

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

   if -2.50000000000000011e35 < 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

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

   5. Applied rewrites 79.0%

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

Alternative 8: 75.5% 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 76.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 75.4%

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

Developer Target 1: 77.9% 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 2024270 
(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))