Average Error: 22.2 → 7.3
Time: 15.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.694902778609720771805097257206126856758 \cdot 10^{47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y} - \frac{x}{y}, 1, x\right)\\ \mathbf{elif}\;y \le 62033795.03298564255237579345703125:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y} - \frac{x}{y}, 1, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.694902778609720771805097257206126856758 \cdot 10^{47}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y} - \frac{x}{y}, 1, x\right)\\

\mathbf{elif}\;y \le 62033795.03298564255237579345703125:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\

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

\end{array}
double f(double x, double y) {
        double r28274981 = 1.0;
        double r28274982 = x;
        double r28274983 = r28274981 - r28274982;
        double r28274984 = y;
        double r28274985 = r28274983 * r28274984;
        double r28274986 = r28274984 + r28274981;
        double r28274987 = r28274985 / r28274986;
        double r28274988 = r28274981 - r28274987;
        return r28274988;
}
↓
double f(double x, double y) {
        double r28274989 = y;
        double r28274990 = -6.694902778609721e+47;
        bool r28274991 = r28274989 <= r28274990;
        double r28274992 = x;
        double r28274993 = r28274992 / r28274989;
        double r28274994 = r28274993 / r28274989;
        double r28274995 = r28274994 - r28274993;
        double r28274996 = 1.0;
        double r28274997 = fma(r28274995, r28274996, r28274992);
        double r28274998 = 62033795.03298564;
        bool r28274999 = r28274989 <= r28274998;
        double r28275000 = r28274989 + r28274996;
        double r28275001 = r28274989 / r28275000;
        double r28275002 = r28274992 - r28274996;
        double r28275003 = fma(r28275001, r28275002, r28274996);
        double r28275004 = r28274999 ? r28275003 : r28274997;
        double r28275005 = r28274991 ? r28274997 : r28275004;
        return r28275005;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.2
Target0.2
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.694902778609721e+47 or 62033795.03298564 < y

    1. Initial program 46.5

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

    2. Simplified29.4

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

    3. Taylor expanded around inf 14.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{x}{{y}^{2}} + x\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified14.0

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

    if -6.694902778609721e+47 < y < 62033795.03298564

    1. Initial program 1.8

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

    2. Simplified1.7

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.694902778609720771805097257206126856758 \cdot 10^{47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y} - \frac{x}{y}, 1, x\right)\\ \mathbf{elif}\;y \le 62033795.03298564255237579345703125:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y} - \frac{x}{y}, 1, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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))))