Average Error: 22.2 → 0.2
Time: 17.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -163886191.99208533763885498046875 \lor \neg \left(y \le 160471059.9958191215991973876953125\right):\\ \;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -163886191.99208533763885498046875 \lor \neg \left(y \le 160471059.9958191215991973876953125\right):\\
\;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\

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

\end{array}
double f(double x, double y) {
        double r773913 = 1.0;
        double r773914 = x;
        double r773915 = r773913 - r773914;
        double r773916 = y;
        double r773917 = r773915 * r773916;
        double r773918 = r773916 + r773913;
        double r773919 = r773917 / r773918;
        double r773920 = r773913 - r773919;
        return r773920;
}
↓
double f(double x, double y) {
        double r773921 = y;
        double r773922 = -163886191.99208534;
        bool r773923 = r773921 <= r773922;
        double r773924 = 160471059.99581912;
        bool r773925 = r773921 <= r773924;
        double r773926 = !r773925;
        bool r773927 = r773923 || r773926;
        double r773928 = x;
        double r773929 = 1.0;
        double r773930 = r773929 / r773921;
        double r773931 = 1.0;
        double r773932 = r773931 - r773928;
        double r773933 = r773930 * r773932;
        double r773934 = r773928 + r773933;
        double r773935 = r773928 - r773929;
        double r773936 = r773921 + r773929;
        double r773937 = r773935 / r773936;
        double r773938 = fma(r773937, r773921, r773929);
        double r773939 = r773927 ? r773934 : r773938;
        return r773939;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.2
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 < -163886191.99208534 or 160471059.99581912 < y

    1. Initial program 45.6

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

    2. Simplified30.3

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

    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

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

    if -163886191.99208534 < y < 160471059.99581912

    1. Initial program 0.2

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

    2. Simplified0.2

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

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

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

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

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

    6. Applied times-frac0.2

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

    7. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -163886191.99208533763885498046875 \lor \neg \left(y \le 160471059.9958191215991973876953125\right):\\ \;\;\;\;x + \frac{1}{y} \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))))