Average Error: 22.6 → 0.1
Time: 4.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -86808837.13671295344829559326171875 \lor \neg \left(y \le 99278016.7060630321502685546875\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right) \cdot \left(y - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -86808837.13671295344829559326171875 \lor \neg \left(y \le 99278016.7060630321502685546875\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r748382 = 1.0;
        double r748383 = x;
        double r748384 = r748382 - r748383;
        double r748385 = y;
        double r748386 = r748384 * r748385;
        double r748387 = r748385 + r748382;
        double r748388 = r748386 / r748387;
        double r748389 = r748382 - r748388;
        return r748389;
}
↓
double f(double x, double y) {
        double r748390 = y;
        double r748391 = -86808837.13671295;
        bool r748392 = r748390 <= r748391;
        double r748393 = 99278016.70606303;
        bool r748394 = r748390 <= r748393;
        double r748395 = !r748394;
        bool r748396 = r748392 || r748395;
        double r748397 = 1.0;
        double r748398 = 1.0;
        double r748399 = r748398 / r748390;
        double r748400 = x;
        double r748401 = r748400 / r748390;
        double r748402 = r748399 - r748401;
        double r748403 = r748397 * r748402;
        double r748404 = r748403 + r748400;
        double r748405 = r748397 - r748400;
        double r748406 = r748390 * r748390;
        double r748407 = r748397 * r748397;
        double r748408 = r748406 - r748407;
        double r748409 = r748390 / r748408;
        double r748410 = r748405 * r748409;
        double r748411 = r748390 - r748397;
        double r748412 = r748410 * r748411;
        double r748413 = r748397 - r748412;
        double r748414 = r748396 ? r748404 : r748413;
        return r748414;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -86808837.13671295 or 99278016.70606303 < y

    1. Initial program 46.0

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

    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 -86808837.13671295 < y < 99278016.70606303

    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 flip-+0.2

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

    8. Applied associate-/r/0.2

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

    9. Applied associate-*r*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -86808837.13671295344829559326171875 \lor \neg \left(y \le 99278016.7060630321502685546875\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right) \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))))