Average Error: 22.5 → 7.7
Time: 3.9s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\

\end{array}
double f(double x, double y) {
        double r681715 = 1.0;
        double r681716 = x;
        double r681717 = r681715 - r681716;
        double r681718 = y;
        double r681719 = r681717 * r681718;
        double r681720 = r681718 + r681715;
        double r681721 = r681719 / r681720;
        double r681722 = r681715 - r681721;
        return r681722;
}
↓
double f(double x, double y) {
        double r681723 = y;
        double r681724 = -1.8933009259790781e+18;
        bool r681725 = r681723 <= r681724;
        double r681726 = 1.02337014478886e+20;
        bool r681727 = r681723 <= r681726;
        double r681728 = !r681727;
        bool r681729 = r681725 || r681728;
        double r681730 = x;
        double r681731 = r681730 / r681723;
        double r681732 = 1.0;
        double r681733 = r681732 / r681723;
        double r681734 = r681733 - r681732;
        double r681735 = fma(r681731, r681734, r681730);
        double r681736 = 3.0;
        double r681737 = pow(r681723, r681736);
        double r681738 = pow(r681732, r681736);
        double r681739 = r681737 + r681738;
        double r681740 = r681723 / r681739;
        double r681741 = r681723 * r681723;
        double r681742 = r681732 * r681732;
        double r681743 = r681723 * r681732;
        double r681744 = r681742 - r681743;
        double r681745 = r681741 + r681744;
        double r681746 = r681740 * r681745;
        double r681747 = r681730 - r681732;
        double r681748 = fma(r681746, r681747, r681732);
        double r681749 = r681729 ? r681735 : r681748;
        return r681749;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.8933009259790781e+18 or 1.02337014478886e+20 < y

    1. Initial program 46.7

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

    2. Simplified29.5

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

    3. Taylor expanded around inf 15.2

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

    4. Simplified15.2

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

    if -1.8933009259790781e+18 < y < 1.02337014478886e+20

    1. Initial program 0.9

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

    2. Simplified0.9

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

    4. Applied flip3-+0.9

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}, x - 1, 1\right)\]

    5. Applied associate-/r/0.9

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1893300925979078144 \lor \neg \left(y \le 102337014478885994496\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))