Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Percentage Accurate: 95.1% → 99.7%

Time: 10.8s

Alternatives: 10

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+
      x
      (/
       y
       (-
        (*
         1.1283791670955126
         (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))
        (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / ((1.1283791670955126 * fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0)) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(N[(1.1283791670955126 * N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\left(1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right) + 1\right)} - x \cdot y} \]

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

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right), 1\right)} - x \cdot y} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right) + 1}, 1\right) - x \cdot y} \]

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

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} + \frac{1}{6} \cdot z, 1\right)}, 1\right) - x \cdot y} \]

      5. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{1}{6} \cdot z + \frac{1}{2}}, 1\right), 1\right) - x \cdot y} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) - x \cdot y} \]

      7. accelerator-lowering-fma.f64 99.9

         \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) - x \cdot y} \]

   5. Simplified 99.9%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)} - x \cdot y} \]

   if 2 < (exp.f64 z)

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+
      x
      (/
       y
       (fma
        z
        (fma
         z
         (fma z 0.18806319451591877 0.5641895835477563)
         1.1283791670955126)
        (- 1.1283791670955126 (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), (1.1283791670955126 - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), Float64(1.1283791670955126 - Float64(x * y)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. associate--l+ N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}, \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{30000000000000000} \cdot z + \frac{5641895835477563}{10000000000000000}}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{30000000000000000}} + \frac{5641895835477563}{10000000000000000}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right)}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - x \cdot y\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}\right)} \]

      11. *-lowering-*.f64 99.8

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - \color{blue}{y \cdot x}\right)} \]

   5. Simplified 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - y \cdot x\right)}} \]

   if 2 < (exp.f64 z)

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+ x (/ y (fma y (- x) (fma z 1.1283791670955126 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / fma(y, -x, fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(y * (-x) + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \frac{5641895835477563}{5000000000000000}} + \frac{5641895835477563}{5000000000000000}\right)} \]

      11. accelerator-lowering-fma.f64 99.5

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)}\right)} \]

   5. Simplified 99.5%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}} \]

   if 2 < (exp.f64 z)

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0) (+ x (/ y (- 1.1283791670955126 (* x y)))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 2.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 2.0d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 2.0) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 2.0:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 2.0)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}} \]

      5. *-lowering-*.f64 99.2

         \[\leadsto x + \frac{y}{1.1283791670955126 - \color{blue}{y \cdot x}} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{x + \frac{y}{1.1283791670955126 - y \cdot x}} \]

   if 2 < (exp.f64 z)

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.95:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000000005:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.95)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.000000005) (+ x (/ y 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.95) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.000000005) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.95d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.000000005d0) then
        tmp = x + (y / 1.1283791670955126d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.95) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.000000005) {
		tmp = x + (y / 1.1283791670955126);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.95:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.000000005:
		tmp = x + (y / 1.1283791670955126)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.95)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.000000005)
		tmp = Float64(x + Float64(y / 1.1283791670955126));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.95)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.000000005)
		tmp = x + (y / 1.1283791670955126);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.95], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.000000005], N[(x + N[(y / 1.1283791670955126), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.94999999999999996

   1. Initial program 87.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.94999999999999996 < (exp.f64 z) < 1.000000005

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}} \]

      5. *-lowering-*.f64 99.6

         \[\leadsto x + \frac{y}{1.1283791670955126 - \color{blue}{y \cdot x}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x + \frac{y}{1.1283791670955126 - y \cdot x}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000}}} \]
   7. Step-by-step derivation

   8. Simplified 76.0%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126}} \]

   if 1.000000005 < (exp.f64 z)

   1. Initial program 89.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

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

Alternative 6: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.95:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.000000005:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.95)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.000000005) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.95) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.000000005) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.95)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.000000005)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.95], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.000000005], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.94999999999999996

   1. Initial program 87.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.94999999999999996 < (exp.f64 z) < 1.000000005

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}} \]

      5. *-lowering-*.f64 99.6

         \[\leadsto x + \frac{y}{1.1283791670955126 - \color{blue}{y \cdot x}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x + \frac{y}{1.1283791670955126 - y \cdot x}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} + x \]

      3. accelerator-lowering-fma.f64 75.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   if 1.000000005 < (exp.f64 z)

   1. Initial program 89.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

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

Alternative 7: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.95:\\ \;\;\;\;x\\ \mathbf{elif}\;e^{z} \leq 1.000000005:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.95)
   x
   (if (<= (exp z) 1.000000005) (fma y 0.8862269254527579 x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.95) {
		tmp = x;
	} else if (exp(z) <= 1.000000005) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.95)
		tmp = x;
	elseif (exp(z) <= 1.000000005)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.95], x, If[LessEqual[N[Exp[z], $MachinePrecision], 1.000000005], N[(y * 0.8862269254527579 + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.94999999999999996 or 1.000000005 < (exp.f64 z)

   1. Initial program 88.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 82.7%

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

   if 0.94999999999999996 < (exp.f64 z) < 1.000000005

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}} \]

      5. *-lowering-*.f64 99.6

         \[\leadsto x + \frac{y}{1.1283791670955126 - \color{blue}{y \cdot x}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x + \frac{y}{1.1283791670955126 - y \cdot x}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} + x \]

      3. accelerator-lowering-fma.f64 75.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]

   8. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.8862269254527579, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (fma
      y
      (- x)
      (fma
       z
       (fma z 0.5641895835477563 1.1283791670955126)
       1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / fma(y, -x, fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(y * (-x) + N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 87.4%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 96.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) - x \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\mathsf{neg}\left(\color{blue}{y \cdot x}\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{y \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto x + \frac{y}{y \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \color{blue}{\mathsf{neg}\left(x\right)}, \frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}\right)} \]

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

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \color{blue}{\mathsf{fma}\left(z, \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, \mathsf{neg}\left(x\right), \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{5641895835477563}{10000000000000000}} + \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)} \]

      13. accelerator-lowering-fma.f64 97.9

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right)}, 1.1283791670955126\right)\right)} \]

   5. Simplified 97.9%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-209}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-152}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.95e-209) x (if (<= x 4.5e-152) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.95e-209) {
		tmp = x;
	} else if (x <= 4.5e-152) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.95d-209)) then
        tmp = x
    else if (x <= 4.5d-152) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.95e-209) {
		tmp = x;
	} else if (x <= 4.5e-152) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.95e-209:
		tmp = x
	elif x <= 4.5e-152:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.95e-209)
		tmp = x;
	elseif (x <= 4.5e-152)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.95e-209)
		tmp = x;
	elseif (x <= 4.5e-152)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.95e-209], x, If[LessEqual[x, 4.5e-152], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.94999999999999979e-209 or 4.5000000000000004e-152 < x

   1. Initial program 95.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 78.2%

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

   if -2.94999999999999979e-209 < x < 4.5000000000000004e-152

   1. Initial program 90.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \color{blue}{\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} - x \cdot y}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} - \color{blue}{y \cdot x}} \]

      5. *-lowering-*.f64 72.1

         \[\leadsto x + \frac{y}{1.1283791670955126 - \color{blue}{y \cdot x}} \]

   5. Simplified 72.1%

      \[\leadsto \color{blue}{x + \frac{y}{1.1283791670955126 - y \cdot x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. *-lowering-*.f64 55.6

         \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]

   8. Simplified 55.6%

      \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.5% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 94.2%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 68.5%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))