Average Error: 2.7 → 1.2
Time: 3.0s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\
\;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{1}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r456856 = x;
        double r456857 = y;
        double r456858 = 1.1283791670955126;
        double r456859 = z;
        double r456860 = exp(r456859);
        double r456861 = r456858 * r456860;
        double r456862 = r456856 * r456857;
        double r456863 = r456861 - r456862;
        double r456864 = r456857 / r456863;
        double r456865 = r456856 + r456864;
        return r456865;
}
↓
double f(double x, double y, double z) {
        double r456866 = x;
        double r456867 = y;
        double r456868 = 1.1283791670955126;
        double r456869 = z;
        double r456870 = exp(r456869);
        double r456871 = r456868 * r456870;
        double r456872 = r456866 * r456867;
        double r456873 = r456871 - r456872;
        double r456874 = r456867 / r456873;
        double r456875 = r456866 + r456874;
        double r456876 = 7.375980088346895e+198;
        bool r456877 = r456875 <= r456876;
        double r456878 = 1.0;
        double r456879 = r456878 / r456866;
        double r456880 = r456866 - r456879;
        double r456881 = r456877 ? r456875 : r456880;
        return r456881;
}

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

Original2.7
Target0.0
Herbie1.2
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 7.375980088346895e+198

    1. Initial program 1.0

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

    if 7.375980088346895e+198 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 13.5

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

    2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{x - \frac{1}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y} \le 7.375980088346894603541968916273812509677 \cdot 10^{198}:\\ \;\;\;\;x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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