Average Error: 22.8 → 0.2
Time: 12.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -222263426.7177731096744537353515625 \lor \neg \left(y \le 309589145.77208232879638671875\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -222263426.7177731096744537353515625 \lor \neg \left(y \le 309589145.77208232879638671875\right):\\
\;\;\;\;\left(\frac{1}{y} - 1 \cdot \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 r586436 = 1.0;
        double r586437 = x;
        double r586438 = r586436 - r586437;
        double r586439 = y;
        double r586440 = r586438 * r586439;
        double r586441 = r586439 + r586436;
        double r586442 = r586440 / r586441;
        double r586443 = r586436 - r586442;
        return r586443;
}
↓
double f(double x, double y) {
        double r586444 = y;
        double r586445 = -222263426.7177731;
        bool r586446 = r586444 <= r586445;
        double r586447 = 309589145.7720823;
        bool r586448 = r586444 <= r586447;
        double r586449 = !r586448;
        bool r586450 = r586446 || r586449;
        double r586451 = 1.0;
        double r586452 = r586451 / r586444;
        double r586453 = x;
        double r586454 = r586453 / r586444;
        double r586455 = r586451 * r586454;
        double r586456 = r586452 - r586455;
        double r586457 = r586456 + r586453;
        double r586458 = r586451 - r586453;
        double r586459 = r586444 + r586451;
        double r586460 = r586444 / r586459;
        double r586461 = r586458 * r586460;
        double r586462 = r586451 - r586461;
        double r586463 = r586450 ? r586457 : r586462;
        return r586463;
}

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
Target0.2
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 < -222263426.7177731 or 309589145.7720823 < y

    1. Initial program 46.4

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

    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

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

    if -222263426.7177731 < y < 309589145.7720823

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    4. Applied times-frac0.2

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

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -222263426.7177731096744537353515625 \lor \neg \left(y \le 309589145.77208232879638671875\right):\\ \;\;\;\;\left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))