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

Percentage Accurate: 78.3% → 99.0%

Time: 11.7s

Alternatives: 11

Speedup: 9.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -2800000000000.0) t_0 (if (<= x 7.1e-9) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = t_0;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-y) / x
    if (x <= (-2800000000000.0d0)) then
        tmp = t_0
    else if (x <= 7.1d-9) 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 <= -2800000000000.0) {
		tmp = t_0;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -2800000000000.0:
		tmp = t_0
	elif x <= 7.1e-9:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = t_0;
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -2800000000000.0)
		tmp = t_0;
	elseif (x <= 7.1e-9)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2800000000000.0], t$95$0, If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e12 or 7.09999999999999988e-9 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

Alternative 2: 87.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, \mathsf{fma}\left(y, -3, -3\right), -2\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+26)
   (/
    (*
     (fma (* y (* y y)) (fma y (fma y -3.0 -3.0) -2.0) 1.0)
     (fma (fma y y y) (+ (fma y y y) -1.0) 1.0))
    x)
   (if (<= x 7.1e-9)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (- y) (fma x (+ 0.5 (/ 0.5 x)) (- x)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+26) {
		tmp = (fma((y * (y * y)), fma(y, fma(y, -3.0, -3.0), -2.0), 1.0) * fma(fma(y, y, y), (fma(y, y, y) + -1.0), 1.0)) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma(-y, fma(x, (0.5 + (0.5 / x)), -x), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+26)
		tmp = Float64(Float64(fma(Float64(y * Float64(y * y)), fma(y, fma(y, -3.0, -3.0), -2.0), 1.0) * fma(fma(y, y, y), Float64(fma(y, y, y) + -1.0), 1.0)) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(-y), fma(x, Float64(0.5 + Float64(0.5 / x)), Float64(-x)), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+26], N[(N[(N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * -3.0 + -3.0), $MachinePrecision] + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y + y), $MachinePrecision] * N[(N[(y * y + y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[((-y) * N[(x * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.59999999999999999e26

   1. Initial program 76.6%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 86.5%

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

   if -5.59999999999999999e26 < x < 7.09999999999999988e-9

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.4%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-*r* N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-+.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 75.2

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

   13. Applied rewrites 75.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, \mathsf{fma}\left(y, -3, -3\right), -2\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot -2, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/
    (* (fma (fma y y y) (+ (fma y y y) -1.0) 1.0) (fma y (* (* y y) -2.0) 1.0))
    x)
   (if (<= x 7.1e-9)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (- y) (fma x (+ 0.5 (/ 0.5 x)) (- x)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = (fma(fma(y, y, y), (fma(y, y, y) + -1.0), 1.0) * fma(y, ((y * y) * -2.0), 1.0)) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma(-y, fma(x, (0.5 + (0.5 / x)), -x), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(fma(fma(y, y, y), Float64(fma(y, y, y) + -1.0), 1.0) * fma(y, Float64(Float64(y * y) * -2.0), 1.0)) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(-y), fma(x, Float64(0.5 + Float64(0.5 / x)), Float64(-x)), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(N[(y * y + y), $MachinePrecision] * N[(N[(y * y + y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[((-y) * N[(x * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 64.7%

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

   9. Applied rewrites 61.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 82.7%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-*r* N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-+.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 75.2

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

   13. Applied rewrites 75.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot -2, 1\right)}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\frac{x + y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.5, -1\right), y \cdot 0.5\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (/ (+ x (* y (fma x (fma y 0.5 -1.0) (* y 0.5)))) x) x)
   (if (<= x 7.1e-9)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (- y) (fma x (+ 0.5 (/ 0.5 x)) (- x)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = ((x + (y * fma(x, fma(y, 0.5, -1.0), (y * 0.5)))) / x) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma(-y, fma(x, (0.5 + (0.5 / x)), -x), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(Float64(x + Float64(y * fma(x, fma(y, 0.5, -1.0), Float64(y * 0.5)))) / x) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(-y), fma(x, Float64(0.5 + Float64(0.5 / x)), Float64(-x)), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(x + N[(y * N[(x * N[(y * 0.5 + -1.0), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[((-y) * N[(x * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-*r* N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-+.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 75.2

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

   13. Applied rewrites 75.2%

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

Alternative 5: 87.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot 1}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (* (fma (fma y y y) (+ (fma y y y) -1.0) 1.0) 1.0) x)
   (if (<= x 7.1e-9)
     (/ 1.0 x)
     (/ 1.0 (fma y (fma (- y) (fma x (+ 0.5 (/ 0.5 x)) (- x)) x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = (fma(fma(y, y, y), (fma(y, y, y) + -1.0), 1.0) * 1.0) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(y, fma(-y, fma(x, (0.5 + (0.5 / x)), -x), x), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(fma(fma(y, y, y), Float64(fma(y, y, y) + -1.0), 1.0) * 1.0) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(y, fma(Float64(-y), fma(x, Float64(0.5 + Float64(0.5 / x)), Float64(-x)), x), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(N[(y * y + y), $MachinePrecision] * N[(N[(y * y + y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y * N[((-y) * N[(x * N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 64.7%

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

   9. Applied rewrites 61.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 78.0%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-*r* N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower-fma.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-+.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 75.2

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

   13. Applied rewrites 75.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot 1}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(-y, \mathsf{fma}\left(x, 0.5 + \frac{0.5}{x}, -x\right), x\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot 1}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (* (fma (fma y y y) (+ (fma y y y) -1.0) 1.0) 1.0) x)
   (if (<= x 7.1e-9) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (fma y y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = (fma(fma(y, y, y), (fma(y, y, y) + -1.0), 1.0) * 1.0) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + fma(y, y, y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(fma(fma(y, y, y), Float64(fma(y, y, y) + -1.0), 1.0) * 1.0) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + fma(y, y, y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(N[(y * y + y), $MachinePrecision] * N[(N[(y * y + y), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 64.7%

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

   9. Applied rewrites 61.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 78.0%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 75.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y, y\right), \mathsf{fma}\left(y, y, y\right) + -1, 1\right) \cdot 1}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + \mathsf{fma}\left(y, y, y\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 7.1e-9) (/ 1.0 x) (/ (/ 1.0 (+ 1.0 (fma y y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / (1.0 + fma(y, y, y))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + fma(y, y, y))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[(y * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 73.2%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 75.1%

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

Alternative 8: 82.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (/ (- x (* x y)) x) x)
   (if (<= x 7.1e-9) (/ 1.0 x) (/ 1.0 (fma x y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = ((x - (x * y)) / x) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(Float64(Float64(x - Float64(x * y)) / x) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(x, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 73.2%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 71.7

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

   13. Applied rewrites 71.7%

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

Alternative 9: 83.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.5, -1\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2800000000000.0)
   (/ (fma y (fma y 0.5 -1.0) 1.0) x)
   (if (<= x 7.1e-9) (/ 1.0 x) (/ 1.0 (fma x y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2800000000000.0) {
		tmp = fma(y, fma(y, 0.5, -1.0), 1.0) / x;
	} else if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2800000000000.0)
		tmp = Float64(fma(y, fma(y, 0.5, -1.0), 1.0) / x);
	elseif (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(x, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2800000000000.0], N[(N[(y * N[(y * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e12

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.6%

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

   if -2.8e12 < x < 7.09999999999999988e-9

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.1%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 71.7

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

   13. Applied rewrites 71.7%

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

Alternative 10: 78.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.1e-9) (/ 1.0 x) (/ 1.0 (fma x y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.1e-9) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / fma(x, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.1e-9)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / fma(x, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.1e-9], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(x * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.09999999999999988e-9

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.9%

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

   if 7.09999999999999988e-9 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 71.7

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

   13. Applied rewrites 71.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, y, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.3% 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 80.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 76.6%

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

Developer Target 1: 77.5% 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 2024222 
(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))