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

Percentage Accurate: 95.6% → 99.9%

Time: 10.1s

Alternatives: 11

Speedup: 0.8×

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: 95.6% 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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, \mathsf{fma}\left(-x, y, 1.1283791670955126\right)\right)} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2e+163)
     (+
      (/
       y
       (fma
        (fma
         (fma 0.18806319451591877 z 0.5641895835477563)
         z
         1.1283791670955126)
        z
        (fma (- x) y 1.1283791670955126)))
      x)
     (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2e+163) {
		tmp = (y / fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, fma(-x, y, 1.1283791670955126))) + x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2e+163)
		tmp = Float64(Float64(y / fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, fma(Float64(-x), y, 1.1283791670955126))) + x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2e+163], N[(N[(y / N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + N[((-x) * y + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 1.9999999999999999e163

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5641895835477563}{30000000000000000}, z, \frac{5641895835477563}{10000000000000000}\right), z, \frac{5641895835477563}{5000000000000000}\right), z, \color{blue}{-1 \cdot \left(x \cdot y\right)} + \frac{5641895835477563}{5000000000000000}\right)} \]

      13. associate-*r* N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5641895835477563}{30000000000000000}, z, \frac{5641895835477563}{10000000000000000}\right), z, \frac{5641895835477563}{5000000000000000}\right), z, \color{blue}{\left(-1 \cdot x\right) \cdot y} + \frac{5641895835477563}{5000000000000000}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5641895835477563}{30000000000000000}, z, \frac{5641895835477563}{10000000000000000}\right), z, \frac{5641895835477563}{5000000000000000}\right), z, \color{blue}{\mathsf{fma}\left(-1 \cdot x, y, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 1.9999999999999999e163 < (exp.f64 z)

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{\color{blue}{1 \cdot y}}{e^{z}} + x \]

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. rec-exp N/A

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

      12. neg-mul-1 N/A

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

      13. lower-exp.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \mathsf{fma}\left(y \cdot 0.8862269254527579, e^{\color{blue}{-z}}, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 0.8862269254527579, e^{-z}, x\right)} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 50.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.2%

       \[\leadsto \mathsf{fma}\left(\frac{y}{x}, 0.8862269254527579, 1\right) \cdot x \]

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, \mathsf{fma}\left(-x, y, 1.1283791670955126\right)\right)} + x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. 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 2024230 
(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)))))