Average Error: 6.1 → 0.9
Time: 5.4s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\
\;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r321696 = x;
        double r321697 = y;
        double r321698 = z;
        double r321699 = r321698 + r321697;
        double r321700 = r321697 / r321699;
        double r321701 = log(r321700);
        double r321702 = r321697 * r321701;
        double r321703 = exp(r321702);
        double r321704 = r321703 / r321697;
        double r321705 = r321696 + r321704;
        return r321705;
}
↓
double f(double x, double y, double z) {
        double r321706 = y;
        double r321707 = 6.718995110248579e-26;
        bool r321708 = r321706 <= r321707;
        double r321709 = x;
        double r321710 = 0.0;
        double r321711 = r321706 * r321710;
        double r321712 = exp(r321711);
        double r321713 = r321712 / r321706;
        double r321714 = r321709 + r321713;
        double r321715 = -1.0;
        double r321716 = z;
        double r321717 = r321715 * r321716;
        double r321718 = exp(r321717);
        double r321719 = r321718 / r321706;
        double r321720 = r321709 + r321719;
        double r321721 = r321708 ? r321714 : r321720;
        return r321721;
}

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

Original6.1
Target1.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 6.718995110248579e-26

    1. Initial program 8.3

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

    2. Taylor expanded around inf 0.9

      \[\leadsto x + \frac{e^{y \cdot \color{blue}{0}}}{y}\]

    if 6.718995110248579e-26 < y

    1. Initial program 1.6

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))