Average Error: 3.4 → 1.3
Time: 10.5s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.0:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r294994 = x;
        double r294995 = y;
        double r294996 = 1.1283791670955126;
        double r294997 = z;
        double r294998 = exp(r294997);
        double r294999 = r294996 * r294998;
        double r295000 = r294994 * r294995;
        double r295001 = r294999 - r295000;
        double r295002 = r294995 / r295001;
        double r295003 = r294994 + r295002;
        return r295003;
}
↓
double f(double x, double y, double z) {
        double r295004 = z;
        double r295005 = exp(r295004);
        double r295006 = 0.0;
        bool r295007 = r295005 <= r295006;
        double r295008 = x;
        double r295009 = 1.0;
        double r295010 = r295009 / r295008;
        double r295011 = r295008 - r295010;
        double r295012 = y;
        double r295013 = 1.1283791670955126;
        double r295014 = r295005 * r295013;
        double r295015 = r295008 * r295012;
        double r295016 = r295014 - r295015;
        double r295017 = r295012 / r295016;
        double r295018 = r295008 + r295017;
        double r295019 = r295007 ? r295011 : r295018;
        return r295019;
}

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

Original3.4
Target0.1
Herbie1.3
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.0

    1. Initial program 8.7

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 0.0

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

    if 0.0 < (exp z)

    1. Initial program 1.7

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

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

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