Average Error: 2.8 → 0.9
Time: 2.2s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -959124237.32905316:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \le -959124237.32905316:\\
\;\;\;\;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 r349169 = x;
        double r349170 = y;
        double r349171 = 1.1283791670955126;
        double r349172 = z;
        double r349173 = exp(r349172);
        double r349174 = r349171 * r349173;
        double r349175 = r349169 * r349170;
        double r349176 = r349174 - r349175;
        double r349177 = r349170 / r349176;
        double r349178 = r349169 + r349177;
        return r349178;
}
↓
double f(double x, double y, double z) {
        double r349179 = z;
        double r349180 = -959124237.3290532;
        bool r349181 = r349179 <= r349180;
        double r349182 = x;
        double r349183 = 1.0;
        double r349184 = r349183 / r349182;
        double r349185 = r349182 - r349184;
        double r349186 = y;
        double r349187 = 1.1283791670955126;
        double r349188 = exp(r349179);
        double r349189 = r349187 * r349188;
        double r349190 = r349182 * r349186;
        double r349191 = r349189 - r349190;
        double r349192 = r349186 / r349191;
        double r349193 = r349182 + r349192;
        double r349194 = r349181 ? r349185 : r349193;
        return r349194;
}

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
Target0.1
Herbie0.9
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -959124237.3290532

    1. Initial program 7.9

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 0.0

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

    if -959124237.3290532 < z

    1. Initial program 1.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -959124237.32905316:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))))