Average Error: 2.8 → 1.2
Time: 4.5s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.999576377249071224:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.999576377249071224:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r427693 = x;
        double r427694 = y;
        double r427695 = 1.1283791670955126;
        double r427696 = z;
        double r427697 = exp(r427696);
        double r427698 = r427695 * r427697;
        double r427699 = r427693 * r427694;
        double r427700 = r427698 - r427699;
        double r427701 = r427694 / r427700;
        double r427702 = r427693 + r427701;
        return r427702;
}
↓
double f(double x, double y, double z) {
        double r427703 = z;
        double r427704 = exp(r427703);
        double r427705 = 0.9995763772490712;
        bool r427706 = r427704 <= r427705;
        double r427707 = x;
        double r427708 = 1.0;
        double r427709 = r427708 / r427707;
        double r427710 = r427707 - r427709;
        double r427711 = y;
        double r427712 = 1.1283791670955126;
        double r427713 = r427712 * r427704;
        double r427714 = r427707 * r427711;
        double r427715 = r427713 - r427714;
        double r427716 = r427711 / r427715;
        double r427717 = r427707 + r427716;
        double r427718 = r427706 ? r427710 : r427717;
        return r427718;
}

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

Original2.8
Target0.0
Herbie1.2
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.9995763772490712

    1. Initial program 6.7

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

    2. Taylor expanded around inf 0.5

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

    if 0.9995763772490712 < (exp z)

    1. Initial program 1.4

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.999576377249071224:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))))