Average Error: 22.2 → 0.2
Time: 12.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -389622710888582.6875 \lor \neg \left(y \le 164787377.2666540443897247314453125\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -389622710888582.6875 \lor \neg \left(y \le 164787377.2666540443897247314453125\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

\mathbf{else}:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\

\end{array}
double f(double x, double y) {
        double r509244 = 1.0;
        double r509245 = x;
        double r509246 = r509244 - r509245;
        double r509247 = y;
        double r509248 = r509246 * r509247;
        double r509249 = r509247 + r509244;
        double r509250 = r509248 / r509249;
        double r509251 = r509244 - r509250;
        return r509251;
}
↓
double f(double x, double y) {
        double r509252 = y;
        double r509253 = -389622710888582.7;
        bool r509254 = r509252 <= r509253;
        double r509255 = 164787377.26665404;
        bool r509256 = r509252 <= r509255;
        double r509257 = !r509256;
        bool r509258 = r509254 || r509257;
        double r509259 = 1.0;
        double r509260 = 1.0;
        double r509261 = r509260 / r509252;
        double r509262 = x;
        double r509263 = r509262 / r509252;
        double r509264 = r509261 - r509263;
        double r509265 = r509259 * r509264;
        double r509266 = r509265 + r509262;
        double r509267 = r509259 - r509262;
        double r509268 = r509252 + r509259;
        double r509269 = r509252 / r509268;
        double r509270 = r509267 * r509269;
        double r509271 = r509259 - r509270;
        double r509272 = r509258 ? r509266 : r509271;
        return r509272;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -389622710888582.7 or 164787377.26665404 < y

    1. Initial program 46.1

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x}\]

    if -389622710888582.7 < y < 164787377.26665404

    1. Initial program 0.4

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.4

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1\right)}}\]

    4. Applied times-frac0.4

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

    5. Simplified0.4

      \[\leadsto 1 - \color{blue}{\left(1 - x\right)} \cdot \frac{y}{y + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -389622710888582.6875 \lor \neg \left(y \le 164787377.2666540443897247314453125\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))