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

Percentage Accurate: 95.5% → 99.5%

Time: 11.8s

Alternatives: 14

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: 95.5% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right)\\ \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000000000002:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(t\_0, t\_0, -1.2732395447351628\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), -1.1283791670955126\right)}, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (fma z 0.18806319451591877 0.5641895835477563))))
   (if (<= (exp z) 0.0)
     (+ x (/ -1.0 x))
     (if (<= (exp z) 1.00000000000002)
       (+
        x
        (/
         y
         (-
          (fma
           z
           (/
            (fma t_0 t_0 -1.2732395447351628)
            (fma
             z
             (fma z 0.18806319451591877 0.5641895835477563)
             -1.1283791670955126))
           1.1283791670955126)
          (* x y))))
       (fma (exp (- z)) (* y 0.8862269254527579) x)))))

C

double code(double x, double y, double z) {
	double t_0 = z * fma(z, 0.18806319451591877, 0.5641895835477563);
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.00000000000002) {
		tmp = x + (y / (fma(z, (fma(t_0, t_0, -1.2732395447351628) / fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), -1.1283791670955126)), 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma(exp(-z), (y * 0.8862269254527579), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * fma(z, 0.18806319451591877, 0.5641895835477563))
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.00000000000002)
		tmp = Float64(x + Float64(y / Float64(fma(z, Float64(fma(t_0, t_0, -1.2732395447351628) / fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), -1.1283791670955126)), 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(exp(Float64(-z)), Float64(y * 0.8862269254527579), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.00000000000002], N[(x + N[(y / N[(N[(z * N[(N[(t$95$0 * t$95$0 + -1.2732395447351628), $MachinePrecision] / N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + -1.1283791670955126), $MachinePrecision]), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-z)], $MachinePrecision] * N[(y * 0.8862269254527579), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 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) < 1.00000000000002

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), z \cdot \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), -1.2732395447351628\right)}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), -1.1283791670955126\right)}}, 1.1283791670955126\right) - x \cdot y} \]

   if 1.00000000000002 < (exp.f64 z)

   1. Initial program 95.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. rec-exp N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{-z}, 0.8862269254527579 \cdot y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.00000000000002:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), z \cdot \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), -1.2732395447351628\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), -1.1283791670955126\right)}, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-157}:\\ \;\;\;\;x + y \cdot \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.7853981633974483\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_1 -10.0)
     t_0
     (if (<= t_1 -1e-157)
       (+ x (* y (fma y (* x 0.7853981633974483) 0.8862269254527579)))
       (if (<= t_1 4e-7) (+ x (* (* y (* x y)) 0.7853981633974483)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= -1e-157) {
		tmp = x + (y * fma(y, (x * 0.7853981633974483), 0.8862269254527579));
	} else if (t_1 <= 4e-7) {
		tmp = x + ((y * (x * y)) * 0.7853981633974483);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= -1e-157)
		tmp = Float64(x + Float64(y * fma(y, Float64(x * 0.7853981633974483), 0.8862269254527579)));
	elseif (t_1 <= 4e-7)
		tmp = Float64(x + Float64(Float64(y * Float64(x * y)) * 0.7853981633974483));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$0, If[LessEqual[t$95$1, -1e-157], N[(x + N[(y * N[(y * N[(x * 0.7853981633974483), $MachinePrecision] + 0.8862269254527579), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(x + N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * 0.7853981633974483), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -10 or 3.9999999999999998e-7 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if -10 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -9.99999999999999943e-158

   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

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{5000000000000000}{5641895835477563} + \left(\frac{-5000000000000000}{5641895835477563} \cdot z + y \cdot \left(\frac{-50000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot z\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.3%

      \[\leadsto x + y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 0.7853981633974483, -1.5707963267948966 \cdot \left(z \cdot x\right)\right), \mathsf{fma}\left(z, -0.8862269254527579, 0.8862269254527579\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto x + y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \color{blue}{\left(x \cdot y\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 75.8%

       \[\leadsto x + y \cdot \mathsf{fma}\left(y, x \cdot \color{blue}{0.7853981633974483}, 0.8862269254527579\right) \]

   if -9.99999999999999943e-158 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 3.9999999999999998e-7

   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

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{5000000000000000}{5641895835477563} + \left(\frac{-5000000000000000}{5641895835477563} \cdot z + y \cdot \left(\frac{-50000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot z\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

      \[\leadsto x + y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 0.7853981633974483, -1.5707963267948966 \cdot \left(z \cdot x\right)\right), \mathsf{fma}\left(z, -0.8862269254527579, 0.8862269254527579\right)\right)} \]

   9. Taylor expanded in x around -inf

      \[\leadsto x + -1 \cdot \left(x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{50000000000000000000000000000000}{31830988618379068626528276418969} \cdot z - \frac{25000000000000000000000000000000}{31830988618379068626528276418969}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.7%

       \[\leadsto x + \left(y \cdot \left(y \cdot x\right)\right) \cdot \left(-\mathsf{fma}\left(z, 1.5707963267948966, -0.7853981633974483\right)\right) \]

   12. Taylor expanded in z around 0

       \[\leadsto x + \left(y \cdot \left(y \cdot x\right)\right) \cdot \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \]
   13. Step-by-step derivation

   14. Applied rewrites 51.1%

       \[\leadsto x + \left(y \cdot \left(y \cdot x\right)\right) \cdot 0.7853981633974483 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -10:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -1 \cdot 10^{-157}:\\ \;\;\;\;x + y \cdot \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right)\\ \mathbf{elif}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.7853981633974483\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
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 2024228 
(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)))))