Average Error: 22.4 → 0.2
Time: 4.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119383534.4574892520904541015625 \lor \neg \left(y \le 240320796.851219654083251953125\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right) \cdot \left(y - 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -119383534.4574892520904541015625 \lor \neg \left(y \le 240320796.851219654083251953125\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 r717906 = 1.0;
        double r717907 = x;
        double r717908 = r717906 - r717907;
        double r717909 = y;
        double r717910 = r717908 * r717909;
        double r717911 = r717909 + r717906;
        double r717912 = r717910 / r717911;
        double r717913 = r717906 - r717912;
        return r717913;
}
↓
double f(double x, double y) {
        double r717914 = y;
        double r717915 = -119383534.45748925;
        bool r717916 = r717914 <= r717915;
        double r717917 = 240320796.85121965;
        bool r717918 = r717914 <= r717917;
        double r717919 = !r717918;
        bool r717920 = r717916 || r717919;
        double r717921 = 1.0;
        double r717922 = 1.0;
        double r717923 = r717922 / r717914;
        double r717924 = x;
        double r717925 = r717924 / r717914;
        double r717926 = r717923 - r717925;
        double r717927 = r717921 * r717926;
        double r717928 = r717927 + r717924;
        double r717929 = r717921 - r717924;
        double r717930 = r717914 * r717914;
        double r717931 = r717921 * r717921;
        double r717932 = r717930 - r717931;
        double r717933 = r717914 / r717932;
        double r717934 = r717929 * r717933;
        double r717935 = r717914 - r717921;
        double r717936 = r717934 * r717935;
        double r717937 = r717921 - r717936;
        double r717938 = r717920 ? r717928 : r717937;
        return r717938;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.4
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 < -119383534.45748925 or 240320796.85121965 < y

    1. Initial program 46.1

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -119383534.45748925 < y < 240320796.85121965

    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119383534.4574892520904541015625 \lor \neg \left(y \le 240320796.851219654083251953125\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(1 - x\right) \cdot \frac{y}{y \cdot y - 1 \cdot 1}\right) \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))