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

Percentage Accurate: 78.0% → 99.4%

Time: 11.5s

Alternatives: 9

Speedup: 6.4×

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: 78.0% 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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -0.96) t_0 (if (<= x 0.96) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -0.96) {
		tmp = t_0;
	} else if (x <= 0.96) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-y) / x
    if (x <= (-0.96d0)) then
        tmp = t_0
    else if (x <= 0.96d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -0.96) {
		tmp = t_0;
	} else if (x <= 0.96) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -0.96:
		tmp = t_0
	elif x <= 0.96:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = t_0;
	elseif (x <= 0.96)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -0.96)
		tmp = t_0;
	elseif (x <= 0.96)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -0.96], t$95$0, If[LessEqual[x, 0.96], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.95999999999999996 or 0.95999999999999996 < x

   1. Initial program 73.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -0.95999999999999996 < x < 0.95999999999999996

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

Alternative 2: 87.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)\\ \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, y \cdot \left(y \cdot t\_0\right), -1\right)}{\mathsf{fma}\left(y, -1, -1\right)}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(y, t\_0, -1\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (fma y -0.16666666666666666 0.5) -1.0)))
   (if (<= x -0.8)
     (/ (/ (fma t_0 (* y (* y t_0)) -1.0) (fma y -1.0 -1.0)) x)
     (if (<= x 0.75) (/ 1.0 x) (/ (/ -1.0 (fma y t_0 -1.0)) x)))))

C

double code(double x, double y) {
	double t_0 = fma(y, fma(y, -0.16666666666666666, 0.5), -1.0);
	double tmp;
	if (x <= -0.8) {
		tmp = (fma(t_0, (y * (y * t_0)), -1.0) / fma(y, -1.0, -1.0)) / x;
	} else if (x <= 0.75) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / fma(y, t_0, -1.0)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, fma(y, -0.16666666666666666, 0.5), -1.0)
	tmp = 0.0
	if (x <= -0.8)
		tmp = Float64(Float64(fma(t_0, Float64(y * Float64(y * t_0)), -1.0) / fma(y, -1.0, -1.0)) / x);
	elseif (x <= 0.75)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / fma(y, t_0, -1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.8], N[(N[(N[(t$95$0 * N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(y * -1.0 + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.75], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / N[(y * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.80000000000000004

   1. Initial program 68.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.4

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

   8. Simplified 68.4%

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

      1. flip-+ N/A

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

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

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

   10. Applied egg-rr 51.8%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 73.5%

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

   if -0.80000000000000004 < x < 0.75

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

   if 0.75 < x

   1. Initial program 78.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.6

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

   8. Simplified 68.6%

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

      1. flip-+ N/A

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

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

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

   10. Applied egg-rr 64.7%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 83.9%

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

Alternative 3: 86.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right)\\ \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, t\_0, 1\right)}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\mathsf{fma}\left(y, t\_0, -1\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (fma y -0.16666666666666666 0.5) -1.0)))
   (if (<= x -0.72)
     (/ (fma y t_0 1.0) x)
     (if (<= x 0.86) (/ 1.0 x) (/ (/ -1.0 (fma y t_0 -1.0)) x)))))

C

double code(double x, double y) {
	double t_0 = fma(y, fma(y, -0.16666666666666666, 0.5), -1.0);
	double tmp;
	if (x <= -0.72) {
		tmp = fma(y, t_0, 1.0) / x;
	} else if (x <= 0.86) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / fma(y, t_0, -1.0)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, fma(y, -0.16666666666666666, 0.5), -1.0)
	tmp = 0.0
	if (x <= -0.72)
		tmp = Float64(fma(y, t_0, 1.0) / x);
	elseif (x <= 0.86)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(-1.0 / fma(y, t_0, -1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.72], N[(N[(y * t$95$0 + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.86], N[(1.0 / x), $MachinePrecision], N[(N[(-1.0 / N[(y * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.71999999999999997

   1. Initial program 68.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.4

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

   8. Simplified 68.4%

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

   if -0.71999999999999997 < x < 0.859999999999999987

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

   if 0.859999999999999987 < x

   1. Initial program 78.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.6

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

   8. Simplified 68.6%

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

      1. flip-+ N/A

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

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

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

   10. Applied egg-rr 64.7%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 83.9%

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

Alternative 4: 80.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 92:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.9)
   (/ (fma y (fma y (fma y -0.16666666666666666 0.5) -1.0) 1.0) x)
   (if (<= x 92.0) (/ 1.0 x) (/ (fma y (fma y 0.5 -1.0) 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.9) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x;
	} else if (x <= 92.0) {
		tmp = 1.0 / x;
	} else {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.9)
		tmp = Float64(fma(y, fma(y, fma(y, -0.16666666666666666, 0.5), -1.0), 1.0) / x);
	elseif (x <= 92.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.9], N[(N[(y * N[(y * N[(y * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 92.0], N[(1.0 / x), $MachinePrecision], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.900000000000000022

   1. Initial program 68.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.4

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

   8. Simplified 68.4%

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

   if -0.900000000000000022 < x < 92

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

   if 92 < x

   1. Initial program 78.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

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

      12. associate--r- N/A

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

      13. div-sub N/A

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

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 68.6%

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

Alternative 5: 80.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 12:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.15)
   (/ (fma y (fma y (* y -0.16666666666666666) -1.0) 1.0) x)
   (if (<= x 12.0) (/ 1.0 x) (/ (fma y (fma y 0.5 -1.0) 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.15) {
		tmp = fma(y, fma(y, (y * -0.16666666666666666), -1.0), 1.0) / x;
	} else if (x <= 12.0) {
		tmp = 1.0 / x;
	} else {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.15)
		tmp = Float64(fma(y, fma(y, Float64(y * -0.16666666666666666), -1.0), 1.0) / x);
	elseif (x <= 12.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.15], N[(N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 12.0], N[(1.0 / x), $MachinePrecision], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.149999999999999994

   1. Initial program 68.0%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   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 68.4

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

   8. Simplified 68.4%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 67.9

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

   11. Simplified 67.9%

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

   if -0.149999999999999994 < x < 12

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

   if 12 < x

   1. Initial program 78.1%

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

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

      12. associate--r- N/A

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

      13. div-sub N/A

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

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 68.6%

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

Alternative 6: 79.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 47:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (fma y (fma y 0.5 -1.0) 1.0) x)))
   (if (<= x -0.92) t_0 (if (<= x 47.0) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	double tmp;
	if (x <= -0.92) {
		tmp = t_0;
	} else if (x <= 47.0) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x)
	tmp = 0.0
	if (x <= -0.92)
		tmp = t_0;
	elseif (x <= 47.0)
		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 * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -0.92], t$95$0, If[LessEqual[x, 47.0], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 47 < x

   1. Initial program 73.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

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

      12. associate--r- N/A

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

      13. div-sub N/A

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

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 67.8%

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

   if -0.92000000000000004 < x < 47

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.3%

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

Alternative 7: 73.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;y \leq 3600000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+143)
   (/ (* y (* y 0.5)) x)
   (if (<= y 3600000000000.0) (/ 1.0 x) (- (* y (/ x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+143) {
		tmp = (y * (y * 0.5)) / x;
	} else if (y <= 3600000000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = -(y * (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 <= (-1.45d+143)) then
        tmp = (y * (y * 0.5d0)) / x
    else if (y <= 3600000000000.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = -(y * (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+143) {
		tmp = (y * (y * 0.5)) / x;
	} else if (y <= 3600000000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = -(y * (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+143:
		tmp = (y * (y * 0.5)) / x
	elif y <= 3600000000000.0:
		tmp = 1.0 / x
	else:
		tmp = -(y * (x / (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+143)
		tmp = Float64(Float64(y * Float64(y * 0.5)) / x);
	elseif (y <= 3600000000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(-Float64(y * Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+143)
		tmp = (y * (y * 0.5)) / x;
	elseif (y <= 3600000000000.0)
		tmp = 1.0 / x;
	else
		tmp = -(y * (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+143], N[(N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3600000000000.0], N[(1.0 / x), $MachinePrecision], (-N[(y * N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4499999999999999e143

   1. Initial program 48.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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-lowering-neg.f64 68.2

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

   5. Simplified 68.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

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

      12. associate--r- N/A

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

      13. div-sub N/A

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

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 64.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {y}^{2}}{x}} \]

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

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {y}^{2}}{x}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \frac{1}{2}}}{x} \]

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 64.9

          \[\leadsto \frac{y \cdot \color{blue}{\left(y \cdot 0.5\right)}}{x} \]

   11. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{y \cdot \left(y \cdot 0.5\right)}{x}} \]

   if -1.4499999999999999e143 < y < 3.6e12

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 90.5%

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

   if 3.6e12 < y

   1. Initial program 46.6%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 2.6

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

   5. Simplified 2.6%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. neg-mul-1 N/A

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

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

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

      4. neg-lowering-neg.f64 2.6

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

   8. Simplified 2.6%

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

      1. neg-sub0 N/A

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

      2. div-sub N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 N/A

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

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

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

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

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

      11. /-lowering-/.f64 21.3

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

   10. Applied egg-rr 21.3%

       \[\leadsto \color{blue}{\frac{0 \cdot \frac{x}{y} - x \cdot 1}{x \cdot \frac{x}{y}}} \]
   11. Step-by-step derivation

       1. mul0-lft N/A

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

       2. *-rgt-identity N/A

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

       3. neg-sub0 N/A

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

       4. associate-*r/ N/A

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

       5. associate-/r/ N/A

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

   12. Applied egg-rr 45.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot 0.5\right)}{x}\\ \mathbf{elif}\;y \leq 3600000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot 0.5\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e+143) (/ (* y (* y 0.5)) x) (/ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+143) {
		tmp = (y * (y * 0.5)) / x;
	} else {
		tmp = 1.0 / 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 <= (-1.45d+143)) then
        tmp = (y * (y * 0.5d0)) / x
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e+143) {
		tmp = (y * (y * 0.5)) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e+143:
		tmp = (y * (y * 0.5)) / x
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e+143)
		tmp = Float64(Float64(y * Float64(y * 0.5)) / x);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e+143)
		tmp = (y * (y * 0.5)) / x;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e+143], N[(N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e143

   1. Initial program 48.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]
   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-lowering-neg.f64 68.2

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

   5. Simplified 68.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-sub N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg N/A

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

      11. neg-sub0 N/A

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

      12. associate--r- N/A

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

      13. div-sub N/A

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

      14. neg-sub0 N/A

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

      15. distribute-neg-frac N/A

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

   8. Simplified 64.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {y}^{2}}{x}} \]

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

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot {y}^{2}}{x}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \frac{1}{2}}}{x} \]

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 64.9

          \[\leadsto \frac{y \cdot \color{blue}{\left(y \cdot 0.5\right)}}{x} \]

   11. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{y \cdot \left(y \cdot 0.5\right)}{x}} \]

   if -1.4499999999999999e143 < y

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.3%

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

Alternative 9: 74.4% 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 77.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 73.2%

   \[\leadsto \frac{\color{blue}{1}}{x} \]
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 2024198 
(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))