Average Error: 22.3 → 0.2
Time: 13.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -92380677.54763953387737274169921875 \lor \neg \left(y \le 97640792.591452300548553466796875\right):\\ \;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{1 \cdot 1 - y \cdot y} \cdot \left(\left(1 - y\right) \cdot \left(1 - x\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -92380677.54763953387737274169921875 \lor \neg \left(y \le 97640792.591452300548553466796875\right):\\
\;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r473296 = 1.0;
        double r473297 = x;
        double r473298 = r473296 - r473297;
        double r473299 = y;
        double r473300 = r473298 * r473299;
        double r473301 = r473299 + r473296;
        double r473302 = r473300 / r473301;
        double r473303 = r473296 - r473302;
        return r473303;
}
↓
double f(double x, double y) {
        double r473304 = y;
        double r473305 = -92380677.54763953;
        bool r473306 = r473304 <= r473305;
        double r473307 = 97640792.5914523;
        bool r473308 = r473304 <= r473307;
        double r473309 = !r473308;
        bool r473310 = r473306 || r473309;
        double r473311 = x;
        double r473312 = 1.0;
        double r473313 = r473312 / r473304;
        double r473314 = r473304 / r473311;
        double r473315 = r473312 / r473314;
        double r473316 = r473313 - r473315;
        double r473317 = r473311 + r473316;
        double r473318 = r473312 * r473312;
        double r473319 = r473304 * r473304;
        double r473320 = r473318 - r473319;
        double r473321 = r473304 / r473320;
        double r473322 = r473312 - r473304;
        double r473323 = r473312 - r473311;
        double r473324 = r473322 * r473323;
        double r473325 = r473321 * r473324;
        double r473326 = r473312 - r473325;
        double r473327 = r473310 ? r473317 : r473326;
        return r473327;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.3
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 < -92380677.54763953 or 97640792.5914523 < y

    1. Initial program 45.9

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

    2. Simplified29.3

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -92380677.54763953 < y < 97640792.5914523

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied associate-*l*0.1

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

    6. Simplified0.1

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

    8. Applied flip-+0.1

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

    9. Applied associate-/r/0.1

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

    10. Applied associate-*l*0.1

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

    11. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -92380677.54763953387737274169921875 \lor \neg \left(y \le 97640792.591452300548553466796875\right):\\ \;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{1 \cdot 1 - y \cdot y} \cdot \left(\left(1 - y\right) \cdot \left(1 - x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))