Average Error: 2.8 → 1.2
Time: 3.9s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.999576377249071224:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
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 + y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r605258 = x;
        double r605259 = y;
        double r605260 = 1.1283791670955126;
        double r605261 = z;
        double r605262 = exp(r605261);
        double r605263 = r605260 * r605262;
        double r605264 = r605258 * r605259;
        double r605265 = r605263 - r605264;
        double r605266 = r605259 / r605265;
        double r605267 = r605258 + r605266;
        return r605267;
}
↓
double f(double x, double y, double z) {
        double r605268 = z;
        double r605269 = exp(r605268);
        double r605270 = 0.9995763772490712;
        bool r605271 = r605269 <= r605270;
        double r605272 = x;
        double r605273 = 1.0;
        double r605274 = r605273 / r605272;
        double r605275 = r605272 - r605274;
        double r605276 = y;
        double r605277 = 1.1283791670955126;
        double r605278 = r605277 * r605269;
        double r605279 = r605272 * r605276;
        double r605280 = r605278 - r605279;
        double r605281 = r605273 / r605280;
        double r605282 = r605276 * r605281;
        double r605283 = r605272 + r605282;
        double r605284 = r605271 ? r605275 : r605283;
        return r605284;
}

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

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
    2. Using strategy rm

    3. Applied div-inv1.4

      \[\leadsto x + \color{blue}{y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))))