Average Error: 2.8 → 1.2
Time: 8.3s
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 r2244 = x;
        double r2245 = y;
        double r2246 = 1.1283791670955126;
        double r2247 = z;
        double r2248 = exp(r2247);
        double r2249 = r2246 * r2248;
        double r2250 = r2244 * r2245;
        double r2251 = r2249 - r2250;
        double r2252 = r2245 / r2251;
        double r2253 = r2244 + r2252;
        return r2253;
}
↓
double f(double x, double y, double z) {
        double r2254 = z;
        double r2255 = exp(r2254);
        double r2256 = 0.9995763772490712;
        bool r2257 = r2255 <= r2256;
        double r2258 = x;
        double r2259 = 1.0;
        double r2260 = r2259 / r2258;
        double r2261 = r2258 - r2260;
        double r2262 = y;
        double r2263 = 1.1283791670955126;
        double r2264 = r2263 * r2255;
        double r2265 = r2258 * r2262;
        double r2266 = r2264 - r2265;
        double r2267 = r2259 / r2266;
        double r2268 = r2262 * r2267;
        double r2269 = r2258 + r2268;
        double r2270 = r2257 ? r2261 : r2269;
        return r2270;
}

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)))))