Average Error: 3.0 → 1.2
Time: 3.5s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\
\;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r419804 = x;
        double r419805 = y;
        double r419806 = 1.1283791670955126;
        double r419807 = z;
        double r419808 = exp(r419807);
        double r419809 = r419806 * r419808;
        double r419810 = r419804 * r419805;
        double r419811 = r419809 - r419810;
        double r419812 = r419805 / r419811;
        double r419813 = r419804 + r419812;
        return r419813;
}
↓
double f(double x, double y, double z) {
        double r419814 = x;
        double r419815 = y;
        double r419816 = 1.1283791670955126;
        double r419817 = z;
        double r419818 = exp(r419817);
        double r419819 = r419816 * r419818;
        double r419820 = r419814 * r419815;
        double r419821 = r419819 - r419820;
        double r419822 = r419815 / r419821;
        double r419823 = r419814 + r419822;
        double r419824 = 6.392865387640042e+200;
        bool r419825 = r419823 <= r419824;
        double r419826 = 1.0;
        double r419827 = r419826 / r419814;
        double r419828 = r419814 - r419827;
        double r419829 = r419825 ? r419823 : r419828;
        return r419829;
}

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

Original3.0
Target0.0
Herbie1.2
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 6.392865387640042e+200

    1. Initial program 1.1

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    if 6.392865387640042e+200 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 15.2

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 1.9

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y} \le 6.3928653876400416 \cdot 10^{200}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))))