Average Error: 5.8 → 1.0
Time: 6.3s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 7.7540686399925493 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 7.7540686399925493 \cdot 10^{-32}:\\
\;\;\;\;x + \frac{e^{0}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r376972 = x;
        double r376973 = y;
        double r376974 = z;
        double r376975 = r376974 + r376973;
        double r376976 = r376973 / r376975;
        double r376977 = log(r376976);
        double r376978 = r376973 * r376977;
        double r376979 = exp(r376978);
        double r376980 = r376979 / r376973;
        double r376981 = r376972 + r376980;
        return r376981;
}
↓
double f(double x, double y, double z) {
        double r376982 = y;
        double r376983 = 7.754068639992549e-32;
        bool r376984 = r376982 <= r376983;
        double r376985 = x;
        double r376986 = 0.0;
        double r376987 = exp(r376986);
        double r376988 = r376987 / r376982;
        double r376989 = r376985 + r376988;
        double r376990 = -1.0;
        double r376991 = z;
        double r376992 = r376990 * r376991;
        double r376993 = exp(r376992);
        double r376994 = r376993 / r376982;
        double r376995 = r376985 + r376994;
        double r376996 = r376984 ? r376989 : r376995;
        return r376996;
}

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

Original5.8
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 7.754068639992549e-32

    1. Initial program 7.7

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

    2. Taylor expanded around inf 1.1

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

    if 7.754068639992549e-32 < y

    1. Initial program 1.5

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 7.7540686399925493 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))