Average Error: 3.0 → 1.0
Time: 7.2s
Precision: 64
\[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.0:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{1}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\

\end{array}
double f(double x, double y, double z) {
        double r300020 = x;
        double r300021 = y;
        double r300022 = 1.1283791670955126;
        double r300023 = z;
        double r300024 = exp(r300023);
        double r300025 = r300022 * r300024;
        double r300026 = r300020 * r300021;
        double r300027 = r300025 - r300026;
        double r300028 = r300021 / r300027;
        double r300029 = r300020 + r300028;
        return r300029;
}
↓
double f(double x, double y, double z) {
        double r300030 = z;
        double r300031 = exp(r300030);
        double r300032 = 0.0;
        bool r300033 = r300031 <= r300032;
        double r300034 = x;
        double r300035 = 1.0;
        double r300036 = r300035 / r300034;
        double r300037 = r300034 - r300036;
        double r300038 = y;
        double r300039 = 1.1283791670955126;
        double r300040 = r300039 * r300031;
        double r300041 = r300034 * r300038;
        double r300042 = r300040 - r300041;
        double r300043 = r300035 / r300042;
        double r300044 = r300038 * r300043;
        double r300045 = r300034 + r300044;
        double r300046 = r300033 ? r300037 : r300045;
        return r300046;
}

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

Original3.0
Target0.0
Herbie1.0
\[x + \frac{1}{\frac{1.128379167095512558560699289955664426088}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.0

    1. Initial program 8.2

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

    2. Taylor expanded around inf 0.0

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

    if 0.0 < (exp z)

    1. Initial program 1.3

      \[x + \frac{y}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\]
    2. Using strategy rm

    3. Applied div-inv1.3

      \[\leadsto x + \color{blue}{y \cdot \frac{1}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{1.128379167095512558560699289955664426088 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.12837916709551256 y) (exp z)) x)))

  (+ x (/ y (- (* 1.12837916709551256 (exp z)) (* x y)))))