Average Error: 2.8 → 1.2
Time: 7.4s
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 r2165 = x;
        double r2166 = y;
        double r2167 = 1.1283791670955126;
        double r2168 = z;
        double r2169 = exp(r2168);
        double r2170 = r2167 * r2169;
        double r2171 = r2165 * r2166;
        double r2172 = r2170 - r2171;
        double r2173 = r2166 / r2172;
        double r2174 = r2165 + r2173;
        return r2174;
}
↓
double f(double x, double y, double z) {
        double r2175 = z;
        double r2176 = exp(r2175);
        double r2177 = 0.9995763772490712;
        bool r2178 = r2176 <= r2177;
        double r2179 = x;
        double r2180 = 1.0;
        double r2181 = r2180 / r2179;
        double r2182 = r2179 - r2181;
        double r2183 = y;
        double r2184 = 1.1283791670955126;
        double r2185 = r2184 * r2176;
        double r2186 = r2179 * r2183;
        double r2187 = r2185 - r2186;
        double r2188 = r2183 / r2187;
        double r2189 = r2179 + r2188;
        double r2190 = r2178 ? r2182 : r2189;
        return r2190;
}

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)))))