Average Error: 22.8 → 0.2
Time: 3.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -913328192986.92358 \lor \neg \left(y \le 239862300.49767068\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -913328192986.92358 \lor \neg \left(y \le 239862300.49767068\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r895267 = 1.0;
        double r895268 = x;
        double r895269 = r895267 - r895268;
        double r895270 = y;
        double r895271 = r895269 * r895270;
        double r895272 = r895270 + r895267;
        double r895273 = r895271 / r895272;
        double r895274 = r895267 - r895273;
        return r895274;
}
↓
double f(double x, double y) {
        double r895275 = y;
        double r895276 = -913328192986.9236;
        bool r895277 = r895275 <= r895276;
        double r895278 = 239862300.49767068;
        bool r895279 = r895275 <= r895278;
        double r895280 = !r895279;
        bool r895281 = r895277 || r895280;
        double r895282 = 1.0;
        double r895283 = 1.0;
        double r895284 = r895283 / r895275;
        double r895285 = x;
        double r895286 = r895285 / r895275;
        double r895287 = r895284 - r895286;
        double r895288 = r895282 * r895287;
        double r895289 = r895288 + r895285;
        double r895290 = r895282 - r895285;
        double r895291 = r895275 * r895275;
        double r895292 = r895282 * r895282;
        double r895293 = r895291 - r895292;
        double r895294 = r895275 / r895293;
        double r895295 = r895275 - r895282;
        double r895296 = r895294 * r895295;
        double r895297 = r895290 * r895296;
        double r895298 = r895282 - r895297;
        double r895299 = r895281 ? r895289 : r895298;
        return r895299;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.8
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -913328192986.9236 or 239862300.49767068 < 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 -913328192986.9236 < y < 239862300.49767068

    1. Initial program 0.3

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

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

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

    4. Applied times-frac0.2

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

    5. Simplified0.2

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

    7. Applied flip-+0.3

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

    8. Applied associate-/r/0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -913328192986.92358 \lor \neg \left(y \le 239862300.49767068\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

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