Average Error: 2.9 → 0.3
Time: 3.4s
Precision: binary64
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 6.5402550362961 \cdot 10^{-306}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.128379167095513:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 6.5402550362961 \cdot 10^{-306}:\\
\;\;\;\;x + \frac{-1}{x}\\

\mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.128379167095513:\\
\;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* 1.1283791670955126 (exp z)) 6.5402550362961e-306)
   (+ x (/ -1.0 x))
   (if (<= (* 1.1283791670955126 (exp z)) 1.128379167095513)
     (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))
     x)))
double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((1.1283791670955126 * exp(z)) <= 6.5402550362961e-306) {
		tmp = x + (-1.0 / x);
	} else if ((1.1283791670955126 * exp(z)) <= 1.128379167095513) {
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.9
Target0.0
Herbie0.3
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 5081767996463981/4503599627370496 (exp.f64 z)) < 6.5402550362961002e-306

    1. Initial program 7.5

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

    2. Taylor expanded around inf 0.0

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

    if 6.5402550362961002e-306 < (*.f64 5081767996463981/4503599627370496 (exp.f64 z)) < 1.128379167095513

    1. Initial program 0.0

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
    2. Using strategy rm

    3. Applied sub-neg_binary640.0

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

    4. Simplified0.0

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

    if 1.128379167095513 < (*.f64 5081767996463981/4503599627370496 (exp.f64 z))

    1. Initial program 4.0

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 6.5402550362961 \cdot 10^{-306}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.128379167095513:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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