Average Error: 22.6 → 7.6
Time: 5.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r703618 = 1.0;
        double r703619 = x;
        double r703620 = r703618 - r703619;
        double r703621 = y;
        double r703622 = r703620 * r703621;
        double r703623 = r703621 + r703618;
        double r703624 = r703622 / r703623;
        double r703625 = r703618 - r703624;
        return r703625;
}
↓
double f(double x, double y) {
        double r703626 = y;
        double r703627 = -1.210662606949831e+16;
        bool r703628 = r703626 <= r703627;
        double r703629 = 1.6618012416610568e+32;
        bool r703630 = r703626 <= r703629;
        double r703631 = !r703630;
        bool r703632 = r703628 || r703631;
        double r703633 = x;
        double r703634 = r703633 / r703626;
        double r703635 = 1.0;
        double r703636 = r703635 / r703626;
        double r703637 = r703636 - r703635;
        double r703638 = fma(r703634, r703637, r703633);
        double r703639 = r703626 + r703635;
        double r703640 = r703626 / r703639;
        double r703641 = r703633 - r703635;
        double r703642 = fma(r703640, r703641, r703635);
        double r703643 = r703632 ? r703638 : r703642;
        return r703643;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.210662606949831e+16 or 1.6618012416610568e+32 < y

    1. Initial program 47.7

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

    2. Simplified29.6

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

    3. Taylor expanded around inf 15.0

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

    4. Simplified15.0

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

    if -1.210662606949831e+16 < y < 1.6618012416610568e+32

    1. Initial program 1.3

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

    2. Simplified1.2

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -12106626069498310 \lor \neg \left(y \le 166180124166105681923912094449664\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))