Average Error: 2.9 → 0.4
Time: 3.4s
Precision: binary64
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 7.235366708299216 \cdot 10^{-68}:\\ \;\;\;\;x + \frac{1}{-x}\\ \mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.128379167095513:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\ \end{array}\]
x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 7.235366708299216 \cdot 10^{-68}:\\
\;\;\;\;x + \frac{1}{-x}\\

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

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\

\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)) 7.235366708299216e-68)
   (+ x (/ 1.0 (- x)))
   (if (<= (* 1.1283791670955126 (exp z)) 1.128379167095513)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (+ x (/ y (* 1.1283791670955126 (exp z)))))))
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)) <= 7.235366708299216e-68) {
		tmp = x + (1.0 / -x);
	} else if ((1.1283791670955126 * exp(z)) <= 1.128379167095513) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / (1.1283791670955126 * exp(z)));
	}
	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
Herbie0.4
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 5081767996463981/4503599627370496 (exp.f64 z)) < 7.23536670829921633e-68

    1. Initial program 7.5

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

    3. Applied clear-num_binary647.4

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.1283791670955126 \cdot e^{z} - x \cdot y}{y}}}\]

    4. Simplified7.4

      \[\leadsto x + \frac{1}{\color{blue}{\frac{e^{z} \cdot 1.1283791670955126 - y \cdot x}{y}}}\]

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if 7.23536670829921633e-68 < (*.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. Taylor expanded around 0 0.3

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

    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 0 0.9

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126 \cdot e^{z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 7.235366708299216 \cdot 10^{-68}:\\ \;\;\;\;x + \frac{1}{-x}\\ \mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.128379167095513:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\ \end{array}\]

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)))))