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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\ \;\;\;\;x + \sqrt{e^{-1 \cdot z}} \cdot \frac{\sqrt{e^{-1 \cdot z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\
\;\;\;\;x + \sqrt{e^{-1 \cdot z}} \cdot \frac{\sqrt{e^{-1 \cdot z}}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

\end{array}

double f(double x, double y, double z) {
        double r416340 = x;
        double r416341 = y;
        double r416342 = z;
        double r416343 = r416342 + r416341;
        double r416344 = r416341 / r416343;
        double r416345 = log(r416344);
        double r416346 = r416341 * r416345;
        double r416347 = exp(r416346);
        double r416348 = r416347 / r416341;
        double r416349 = r416340 + r416348;
        return r416349;
}

↓

double f(double x, double y, double z) {
        double r416350 = y;
        double r416351 = -1.1161118209519008e+42;
        bool r416352 = r416350 <= r416351;
        double r416353 = 1.2927406694831862;
        bool r416354 = r416350 <= r416353;
        double r416355 = !r416354;
        bool r416356 = r416352 || r416355;
        double r416357 = x;
        double r416358 = -1.0;
        double r416359 = z;
        double r416360 = r416358 * r416359;
        double r416361 = exp(r416360);
        double r416362 = sqrt(r416361);
        double r416363 = r416362 / r416350;
        double r416364 = r416362 * r416363;
        double r416365 = r416357 + r416364;
        double r416366 = exp(r416350);
        double r416367 = r416359 + r416350;
        double r416368 = r416350 / r416367;
        double r416369 = log(r416368);
        double r416370 = pow(r416366, r416369);
        double r416371 = r416370 / r416350;
        double r416372 = r416357 + r416371;
        double r416373 = r416356 ? r416365 : r416372;
        return r416373;
}

Original	6.1
Target	1.3
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -1.1161118209519008e+42 or 1.2927406694831862 < y
1. Initial program 2.4
  \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
2. Taylor expanded around inf 0.0
  \[\leadsto x + \frac{\color{blue}{e^{-1 \cdot z}}}{y}\]
3. Using strategy rm
4. Applied *-un-lft-identity0.0
  \[\leadsto x + \frac{e^{-1 \cdot z}}{\color{blue}{1 \cdot y}}\]
5. Applied add-sqr-sqrt0.0
  \[\leadsto x + \frac{\color{blue}{\sqrt{e^{-1 \cdot z}} \cdot \sqrt{e^{-1 \cdot z}}}}{1 \cdot y}\]
6. Applied times-frac0.0
  \[\leadsto x + \color{blue}{\frac{\sqrt{e^{-1 \cdot z}}}{1} \cdot \frac{\sqrt{e^{-1 \cdot z}}}{y}}\]
7. Simplified0.0
  \[\leadsto x + \color{blue}{\sqrt{e^{-1 \cdot z}}} \cdot \frac{\sqrt{e^{-1 \cdot z}}}{y}\]
if -1.1161118209519008e+42 < y < 1.2927406694831862
1. Initial program 9.8
  \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
2. Using strategy rm
3. Applied add-log-exp14.1
  \[\leadsto x + \frac{e^{\color{blue}{\log \left(e^{y}\right)} \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
4. Applied exp-to-pow0.1
  \[\leadsto x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}}{y}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1116111820951900757028957439299745235337000 \lor \neg \left(y \le 1.292740669483186222166182233195286244154\right):\\ \;\;\;\;x + \sqrt{e^{-1 \cdot z}} \cdot \frac{\sqrt{e^{-1 \cdot z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))

Error

Try it out

Target

Derivation

`if y < -1.1161118209519008e+42 or 1.2927406694831862 < y`

`if -1.1161118209519008e+42 < y < 1.2927406694831862`

Reproduce

In
Out