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

Percentage Accurate: 78.1% → 99.2%

Time: 11.7s

Alternatives: 10

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: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{0 - y}}{x}\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- 0.0 y)) x)))
   (if (<= x -2.0) t_0 (if (<= x 3.3e-23) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp((0.0 - y)) / x;
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= 3.3e-23) {
		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((0.0d0 - y)) / x
    if (x <= (-2.0d0)) then
        tmp = t_0
    else if (x <= 3.3d-23) 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((0.0 - y)) / x;
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= 3.3e-23) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((0.0 - y)) / x
	tmp = 0
	if x <= -2.0:
		tmp = t_0
	elif x <= 3.3e-23:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(0.0 - y)) / x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= 3.3e-23)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((0.0 - y)) / x;
	tmp = 0.0;
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= 3.3e-23)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - y), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.0], t$95$0, If[LessEqual[x, 3.3e-23], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 3.30000000000000021e-23 < x

   1. Initial program 75.1%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 99.9

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -2 < x < 3.30000000000000021e-23

   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. Simplified 98.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{0 - y}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{1 + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{x} \cdot \left(y + 1\right), \frac{1}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.0)
   (/ (+ 1.0 (* y (fma y (fma y -0.16666666666666666 0.5) -1.0))) x)
   (if (<= x 2.9e-19)
     (/ 1.0 x)
     (if (<= x 1.2e+110)
       (/
        (/ 1.0 (* x (* x x)))
        (fma (/ y x) (* (/ 1.0 x) (+ y 1.0)) (/ 1.0 (* x x))))
       (/ (/ (- x (* x y)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.0) {
		tmp = (1.0 + (y * fma(y, fma(y, -0.16666666666666666, 0.5), -1.0))) / x;
	} else if (x <= 2.9e-19) {
		tmp = 1.0 / x;
	} else if (x <= 1.2e+110) {
		tmp = (1.0 / (x * (x * x))) / fma((y / x), ((1.0 / x) * (y + 1.0)), (1.0 / (x * x)));
	} else {
		tmp = ((x - (x * y)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(1.0 + Float64(y * fma(y, fma(y, -0.16666666666666666, 0.5), -1.0))) / x);
	elseif (x <= 2.9e-19)
		tmp = Float64(1.0 / x);
	elseif (x <= 1.2e+110)
		tmp = Float64(Float64(1.0 / Float64(x * Float64(x * x))) / fma(Float64(y / x), Float64(Float64(1.0 / x) * Float64(y + 1.0)), Float64(1.0 / Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.0], N[(N[(1.0 + N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.9e-19], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 1.2e+110], N[(N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y / x), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 100.0

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 77.7

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

   10. Applied egg-rr 77.7%

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

   if -2 < x < 2.9e-19

   1. Initial program 81.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.3%

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

   if 2.9e-19 < x < 1.20000000000000006e110

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 57.2

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

   5. Simplified 57.2%

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

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{x} \cdot \left(1 + y\right), \frac{1}{x \cdot x}\right)}} \]

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 86.4

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

   10. Simplified 86.4%

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

   if 1.20000000000000006e110 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 60.2

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

   5. Simplified 60.2%

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

      1. frac-sub N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. --lowering--.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 72.2

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

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{x - y \cdot x}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{1 + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)}{x}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{x} \cdot \left(y + 1\right), \frac{1}{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{1 + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.0)
   (/ (+ 1.0 (* y (fma y (fma y -0.16666666666666666 0.5) -1.0))) x)
   (if (<= x 3.3e-23) (/ 1.0 x) (/ (/ (- x (* x y)) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.0) {
		tmp = (1.0 + (y * fma(y, fma(y, -0.16666666666666666, 0.5), -1.0))) / x;
	} else if (x <= 3.3e-23) {
		tmp = 1.0 / x;
	} else {
		tmp = ((x - (x * y)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(1.0 + Float64(y * fma(y, fma(y, -0.16666666666666666, 0.5), -1.0))) / x);
	elseif (x <= 3.3e-23)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.0], N[(N[(1.0 + N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.3e-23], N[(1.0 / x), $MachinePrecision], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 100.0

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{x} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 77.7

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

   10. Applied egg-rr 77.7%

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

   if -2 < x < 3.30000000000000021e-23

   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. Simplified 98.3%

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

   if 3.30000000000000021e-23 < x

   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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 59.6

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

   5. Simplified 59.6%

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

      1. frac-sub N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. --lowering--.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.7

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

   7. Applied egg-rr 67.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{1 + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.0)
   (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)
   (if (<= x 3.3e-23) (/ 1.0 x) (/ (/ (- x (* x y)) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.0) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	} else if (x <= 3.3e-23) {
		tmp = 1.0 / x;
	} else {
		tmp = ((x - (x * y)) / x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x);
	elseif (x <= 3.3e-23)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.0], N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.3e-23], N[(1.0 / x), $MachinePrecision], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 100.0

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 77.7

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

   8. Simplified 77.7%

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

   if -2 < x < 3.30000000000000021e-23

   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. Simplified 98.3%

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

   if 3.30000000000000021e-23 < x

   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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 59.6

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

   5. Simplified 59.6%

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

      1. frac-sub N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. --lowering--.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 67.7

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

   7. Applied egg-rr 67.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.0) t_0 (if (<= x 1.1e+161) (/ 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 <= -2.0) {
		tmp = t_0;
	} else if (x <= 1.1e+161) {
		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 <= -2.0)
		tmp = t_0;
	elseif (x <= 1.1e+161)
		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, -2.0], t$95$0, If[LessEqual[x, 1.1e+161], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 1.1e161 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 100.0

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 75.6

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

   8. Simplified 75.6%

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

   if -2 < x < 1.1e161

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 86.3%

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

Alternative 6: 81.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{\mathsf{fma}\left(y, y \cdot y, -1\right)}{x}\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ (fma y (* y y) -1.0) x))))
   (if (<= x -2.0) t_0 (if (<= x 1.45e+165) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (fma(y, (y * y), -1.0) / x);
	double tmp;
	if (x <= -2.0) {
		tmp = t_0;
	} else if (x <= 1.45e+165) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(fma(y, Float64(y * y), -1.0) / x))
	tmp = 0.0
	if (x <= -2.0)
		tmp = t_0;
	elseif (x <= 1.45e+165)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(N[(y * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.0], t$95$0, If[LessEqual[x, 1.45e+165], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 1.45000000000000003e165 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 59.2

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

   5. Simplified 59.2%

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

   7. Applied egg-rr 10.9%

      \[\leadsto \color{blue}{\frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{x} \cdot \left(1 + y\right), \frac{1}{x \cdot x}\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\frac{1}{\color{blue}{x \cdot x}}} \]

      3. *-lowering-*.f64 10.1

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\frac{1}{\color{blue}{x \cdot x}}} \]

   10. Simplified 10.1%

       \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{x \cdot x}}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{{y}^{3}}{x} + \frac{1}{x}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{x} + -1 \cdot \frac{{y}^{3}}{x}} \]

       2. mul-1-neg N/A

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

       3. sub-neg N/A

          \[\leadsto \color{blue}{\frac{1}{x} - \frac{{y}^{3}}{x}} \]

       4. div-sub N/A

          \[\leadsto \color{blue}{\frac{1 - {y}^{3}}{x}} \]

       5. sub-neg N/A

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

       6. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{-1 \cdot {y}^{3}}}{x} \]

       7. +-commutative N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot {y}^{3} + 1}}{x} \]

       8. mul-1-neg N/A

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

       9. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\left(0 - {y}^{3}\right)} + 1}{x} \]

       10. associate-+l- N/A

           \[\leadsto \frac{\color{blue}{0 - \left({y}^{3} - 1\right)}}{x} \]

       11. div-sub N/A

           \[\leadsto \color{blue}{\frac{0}{x} - \frac{{y}^{3} - 1}{x}} \]

       12. div0 N/A

           \[\leadsto \color{blue}{0} - \frac{{y}^{3} - 1}{x} \]

       13. sub-neg N/A

           \[\leadsto 0 - \frac{\color{blue}{{y}^{3} + \left(\mathsf{neg}\left(1\right)\right)}}{x} \]

       14. *-rgt-identity N/A

           \[\leadsto 0 - \frac{\color{blue}{{y}^{3} \cdot 1} + \left(\mathsf{neg}\left(1\right)\right)}{x} \]

       15. rgt-mult-inverse N/A

           \[\leadsto 0 - \frac{{y}^{3} \cdot 1 + \left(\mathsf{neg}\left(\color{blue}{{y}^{3} \cdot \frac{1}{{y}^{3}}}\right)\right)}{x} \]

       16. distribute-rgt-neg-out N/A

           \[\leadsto 0 - \frac{{y}^{3} \cdot 1 + \color{blue}{{y}^{3} \cdot \left(\mathsf{neg}\left(\frac{1}{{y}^{3}}\right)\right)}}{x} \]

       17. distribute-lft-in N/A

           \[\leadsto 0 - \frac{\color{blue}{{y}^{3} \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{{y}^{3}}\right)\right)\right)}}{x} \]

       18. sub-neg N/A

           \[\leadsto 0 - \frac{{y}^{3} \cdot \color{blue}{\left(1 - \frac{1}{{y}^{3}}\right)}}{x} \]

       19. --lowering--.f64 N/A

           \[\leadsto \color{blue}{0 - \frac{{y}^{3} \cdot \left(1 - \frac{1}{{y}^{3}}\right)}{x}} \]

       20. /-lowering-/.f64 N/A

           \[\leadsto 0 - \color{blue}{\frac{{y}^{3} \cdot \left(1 - \frac{1}{{y}^{3}}\right)}{x}} \]

   13. Simplified 74.5%

       \[\leadsto \color{blue}{0 - \frac{\mathsf{fma}\left(y, y \cdot y, -1\right)}{x}} \]

   if -2 < x < 1.45000000000000003e165

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 86.3%

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

Alternative 7: 80.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot y\right)}{0 - x}\\ \mathbf{elif}\;y \leq 95:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.4e+102)
   (/ (* y (* y y)) (- 0.0 x))
   (if (<= y 95.0) (/ 1.0 x) (/ x (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.4e+102) {
		tmp = (y * (y * y)) / (0.0 - x);
	} else if (y <= 95.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.4d+102)) then
        tmp = (y * (y * y)) / (0.0d0 - x)
    else if (y <= 95.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.4e+102) {
		tmp = (y * (y * y)) / (0.0 - x);
	} else if (y <= 95.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.4e+102:
		tmp = (y * (y * y)) / (0.0 - x)
	elif y <= 95.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.4e+102)
		tmp = Float64(Float64(y * Float64(y * y)) / Float64(0.0 - x));
	elseif (y <= 95.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.4e+102)
		tmp = (y * (y * y)) / (0.0 - x);
	elseif (y <= 95.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.4e+102], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(0.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 95.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.40000000000000045e102

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 4.2

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

   5. Simplified 4.2%

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

   7. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{x} \cdot \left(1 + y\right), \frac{1}{x \cdot x}\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\frac{1}{\color{blue}{x \cdot x}}} \]

      3. *-lowering-*.f64 13.1

         \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\frac{1}{\color{blue}{x \cdot x}}} \]

   10. Simplified 13.1%

       \[\leadsto \frac{\frac{1 - y \cdot \mathsf{fma}\left(y, y, 0\right)}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{x \cdot x}}} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{{y}^{3}}{x}} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{y}^{3}}{x}\right)} \]

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{0 - \frac{{y}^{3}}{x}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \color{blue}{0 - \frac{{y}^{3}}{x}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto 0 - \color{blue}{\frac{{y}^{3}}{x}} \]

       5. cube-mult N/A

          \[\leadsto 0 - \frac{\color{blue}{y \cdot \left(y \cdot y\right)}}{x} \]

       6. unpow2 N/A

          \[\leadsto 0 - \frac{y \cdot \color{blue}{{y}^{2}}}{x} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto 0 - \frac{\color{blue}{y \cdot {y}^{2}}}{x} \]

       8. unpow2 N/A

          \[\leadsto 0 - \frac{y \cdot \color{blue}{\left(y \cdot y\right)}}{x} \]

       9. *-lowering-*.f64 61.5

          \[\leadsto 0 - \frac{y \cdot \color{blue}{\left(y \cdot y\right)}}{x} \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{0 - \frac{y \cdot \left(y \cdot y\right)}{x}} \]

   if -7.40000000000000045e102 < y < 95

   1. Initial program 94.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 94.1%

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

   if 95 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 2.5

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

   5. Simplified 2.5%

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

      1. frac-sub N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 13.5

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

   7. Applied egg-rr 13.5%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 51.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot y\right)}{0 - x}\\ \mathbf{elif}\;y \leq 95:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;y \leq 95:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+154)
   (/ (fma y (fma y 0.5 -1.0) 1.0) x)
   (if (<= y 95.0) (/ 1.0 x) (/ x (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+154) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (y <= 95.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+154)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (y <= 95.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+154], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 95.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000003e154

   1. Initial program 45.2%

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

   3. Taylor expanded in x around inf

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

      1. exp-lowering-exp.f64 N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

      4. --lowering--.f64 64.0

         \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

   5. Simplified 64.0%

      \[\leadsto \frac{\color{blue}{e^{0 - y}}}{x} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 64.0

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

   8. Simplified 64.0%

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

   if -4.8000000000000003e154 < y < 95

   1. Initial program 91.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 91.2%

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

   if 95 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 2.5

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

   5. Simplified 2.5%

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

      1. frac-sub N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 13.5

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

   7. Applied egg-rr 13.5%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 51.4%

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

Alternative 9: 78.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 105:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 105.0) (/ 1.0 x) (/ x (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 105.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 105.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = x / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 105.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 105.0:
		tmp = 1.0 / x
	else:
		tmp = x / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 105.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 105.0)
		tmp = 1.0 / x;
	else
		tmp = x / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 105.0], N[(1.0 / x), $MachinePrecision], N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 105

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 85.7%

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

   if 105 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 2.5

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

   5. Simplified 2.5%

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

      1. frac-sub N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lft-identity N/A

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

      4. --lowering--.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 13.5

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

   7. Applied egg-rr 13.5%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 51.4%

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

Alternative 10: 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 77.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 74.0%

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

Developer Target 1: 77.6% 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 2024195 
(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))