Average Error: 5.6 → 0.7
Time: 4.4s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 0.5563468047611133115992743114475160837173:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{e^{-1 \cdot z}}{\sqrt{y}}}{\sqrt{y}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 0.5563468047611133115992743114475160837173:\\
\;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r369096 = x;
        double r369097 = y;
        double r369098 = z;
        double r369099 = r369098 + r369097;
        double r369100 = r369097 / r369099;
        double r369101 = log(r369100);
        double r369102 = r369097 * r369101;
        double r369103 = exp(r369102);
        double r369104 = r369103 / r369097;
        double r369105 = r369096 + r369104;
        return r369105;
}
↓
double f(double x, double y, double z) {
        double r369106 = y;
        double r369107 = 0.5563468047611133;
        bool r369108 = r369106 <= r369107;
        double r369109 = x;
        double r369110 = 0.0;
        double r369111 = r369106 * r369110;
        double r369112 = exp(r369111);
        double r369113 = r369112 / r369106;
        double r369114 = r369109 + r369113;
        double r369115 = -1.0;
        double r369116 = z;
        double r369117 = r369115 * r369116;
        double r369118 = exp(r369117);
        double r369119 = sqrt(r369106);
        double r369120 = r369118 / r369119;
        double r369121 = r369120 / r369119;
        double r369122 = r369109 + r369121;
        double r369123 = r369108 ? r369114 : r369122;
        return r369123;
}

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

Original5.6
Target0.8
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 0.5563468047611133

    1. Initial program 7.2

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

    2. Taylor expanded around inf 0.9

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

    if 0.5563468047611133 < y

    1. Initial program 1.7

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

    2. Taylor expanded around inf 0.1

      \[\leadsto x + \color{blue}{\frac{e^{-1 \cdot z}}{y}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto x + \frac{e^{-1 \cdot z}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\]

    5. Applied associate-/r*0.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 0.5563468047611133115992743114475160837173:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{e^{-1 \cdot z}}{\sqrt{y}}}{\sqrt{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))