Average Error: 22.5 → 0.2
Time: 15.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -99997189.22729729115962982177734375 \lor \neg \left(y \le 176788432.6068610250949859619140625\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(y \cdot \frac{1}{y + 1}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -99997189.22729729115962982177734375 \lor \neg \left(y \le 176788432.6068610250949859619140625\right):\\
\;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r464422 = 1.0;
        double r464423 = x;
        double r464424 = r464422 - r464423;
        double r464425 = y;
        double r464426 = r464424 * r464425;
        double r464427 = r464425 + r464422;
        double r464428 = r464426 / r464427;
        double r464429 = r464422 - r464428;
        return r464429;
}
↓
double f(double x, double y) {
        double r464430 = y;
        double r464431 = -99997189.22729729;
        bool r464432 = r464430 <= r464431;
        double r464433 = 176788432.60686103;
        bool r464434 = r464430 <= r464433;
        double r464435 = !r464434;
        bool r464436 = r464432 || r464435;
        double r464437 = x;
        double r464438 = 1.0;
        double r464439 = r464438 / r464430;
        double r464440 = r464437 / r464430;
        double r464441 = r464438 * r464440;
        double r464442 = r464439 - r464441;
        double r464443 = r464437 + r464442;
        double r464444 = r464438 - r464437;
        double r464445 = 1.0;
        double r464446 = r464430 + r464438;
        double r464447 = r464445 / r464446;
        double r464448 = r464430 * r464447;
        double r464449 = r464444 * r464448;
        double r464450 = r464438 - r464449;
        double r464451 = r464436 ? r464443 : r464450;
        return r464451;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
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 < -99997189.22729729 or 176788432.60686103 < y

    1. Initial program 46.1

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

    3. Applied *-un-lft-identity46.1

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

    4. Applied times-frac29.4

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

    5. Simplified29.4

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

    7. Applied div-inv29.5

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

    8. Taylor expanded around inf 0.1

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

    9. Simplified0.1

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

    if -99997189.22729729 < y < 176788432.60686103

    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

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

    7. Applied div-inv0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -99997189.22729729115962982177734375 \lor \neg \left(y \le 176788432.6068610250949859619140625\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \left(y \cdot \frac{1}{y + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))))