Average Error: 3.4 → 1.3
Time: 17.2s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r19886153 = x;
        double r19886154 = y;
        double r19886155 = 1.1283791670955126;
        double r19886156 = z;
        double r19886157 = exp(r19886156);
        double r19886158 = r19886155 * r19886157;
        double r19886159 = r19886153 * r19886154;
        double r19886160 = r19886158 - r19886159;
        double r19886161 = r19886154 / r19886160;
        double r19886162 = r19886153 + r19886161;
        return r19886162;
}
↓
double f(double x, double y, double z) {
        double r19886163 = z;
        double r19886164 = exp(r19886163);
        double r19886165 = 2.2891660016159315e-308;
        bool r19886166 = r19886164 <= r19886165;
        double r19886167 = x;
        double r19886168 = 1.0;
        double r19886169 = r19886168 / r19886167;
        double r19886170 = r19886167 - r19886169;
        double r19886171 = y;
        double r19886172 = 1.1283791670955126;
        double r19886173 = r19886164 * r19886172;
        double r19886174 = r19886167 * r19886171;
        double r19886175 = r19886173 - r19886174;
        double r19886176 = r19886171 / r19886175;
        double r19886177 = r19886167 + r19886176;
        double r19886178 = r19886166 ? r19886170 : r19886177;
        return r19886178;
}

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

Original3.4
Target0.1
Herbie1.3
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 2.2891660016159315e-308

    1. Initial program 8.7

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 0.0

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

    if 2.2891660016159315e-308 < (exp z)

    1. Initial program 1.7

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 2.289166001615931520939634610774789889377 \cdot 10^{-308}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.128379167095512558560699289955664426088 - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))