Average Error: 21.9 → 0.1
Time: 33.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -150725849.0902552902698516845703125:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 180419841.1371018588542938232421875:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \frac{y}{1 + y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -150725849.0902552902698516845703125:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 180419841.1371018588542938232421875:\\
\;\;\;\;\mathsf{fma}\left(x - 1, \frac{y}{1 + y}, 1\right)\\

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

\end{array}
double f(double x, double y) {
        double r24676816 = 1.0;
        double r24676817 = x;
        double r24676818 = r24676816 - r24676817;
        double r24676819 = y;
        double r24676820 = r24676818 * r24676819;
        double r24676821 = r24676819 + r24676816;
        double r24676822 = r24676820 / r24676821;
        double r24676823 = r24676816 - r24676822;
        return r24676823;
}
↓
double f(double x, double y) {
        double r24676824 = y;
        double r24676825 = -150725849.0902553;
        bool r24676826 = r24676824 <= r24676825;
        double r24676827 = 1.0;
        double r24676828 = 1.0;
        double r24676829 = r24676828 / r24676824;
        double r24676830 = x;
        double r24676831 = r24676830 / r24676824;
        double r24676832 = r24676829 - r24676831;
        double r24676833 = fma(r24676827, r24676832, r24676830);
        double r24676834 = 180419841.13710186;
        bool r24676835 = r24676824 <= r24676834;
        double r24676836 = r24676830 - r24676827;
        double r24676837 = r24676827 + r24676824;
        double r24676838 = r24676824 / r24676837;
        double r24676839 = fma(r24676836, r24676838, r24676827);
        double r24676840 = r24676835 ? r24676839 : r24676833;
        double r24676841 = r24676826 ? r24676833 : r24676840;
        return r24676841;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.9
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 < -150725849.0902553 or 180419841.13710186 < y

    1. Initial program 45.6

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

    2. Simplified29.3

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

    if -150725849.0902553 < y < 180419841.13710186

    1. Initial program 0.2

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

    2. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150725849.0902552902698516845703125:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 180419841.1371018588542938232421875:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \frac{y}{1 + y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))