Average Error: 22.3 → 0.2
Time: 24.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -184222856.9532372057437896728515625 \lor \neg \left(y \le 105820959.87149597704410552978515625\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\left(y - 1\right)\right) + \left(y - 1\right)\right) \cdot \frac{1 - x}{\frac{\left(y - 1\right) \cdot \left(1 + y\right)}{y}} + \mathsf{fma}\left(-\frac{1 - x}{\frac{1 + y}{\frac{y}{y - 1}}}, y - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -184222856.9532372057437896728515625 \lor \neg \left(y \le 105820959.87149597704410552978515625\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r656639 = 1.0;
        double r656640 = x;
        double r656641 = r656639 - r656640;
        double r656642 = y;
        double r656643 = r656641 * r656642;
        double r656644 = r656642 + r656639;
        double r656645 = r656643 / r656644;
        double r656646 = r656639 - r656645;
        return r656646;
}
↓
double f(double x, double y) {
        double r656647 = y;
        double r656648 = -184222856.9532372;
        bool r656649 = r656647 <= r656648;
        double r656650 = 105820959.87149598;
        bool r656651 = r656647 <= r656650;
        double r656652 = !r656651;
        bool r656653 = r656649 || r656652;
        double r656654 = 1.0;
        double r656655 = 1.0;
        double r656656 = r656655 / r656647;
        double r656657 = x;
        double r656658 = r656657 / r656647;
        double r656659 = r656656 - r656658;
        double r656660 = fma(r656654, r656659, r656657);
        double r656661 = r656647 - r656654;
        double r656662 = -r656661;
        double r656663 = r656662 + r656661;
        double r656664 = r656654 - r656657;
        double r656665 = r656654 + r656647;
        double r656666 = r656661 * r656665;
        double r656667 = r656666 / r656647;
        double r656668 = r656664 / r656667;
        double r656669 = r656663 * r656668;
        double r656670 = r656647 / r656661;
        double r656671 = r656665 / r656670;
        double r656672 = r656664 / r656671;
        double r656673 = -r656672;
        double r656674 = fma(r656673, r656661, r656654);
        double r656675 = r656669 + r656674;
        double r656676 = r656653 ? r656660 : r656675;
        return r656676;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
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 < -184222856.9532372 or 105820959.87149598 < y

    1. Initial program 45.9

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -184222856.9532372 < y < 105820959.87149598

    1. Initial program 0.1

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied prod-diff0.1

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

    7. Simplified0.2

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

    8. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -184222856.9532372057437896728515625 \lor \neg \left(y \le 105820959.87149597704410552978515625\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\left(y - 1\right)\right) + \left(y - 1\right)\right) \cdot \frac{1 - x}{\frac{\left(y - 1\right) \cdot \left(1 + y\right)}{y}} + \mathsf{fma}\left(-\frac{1 - x}{\frac{1 + y}{\frac{y}{y - 1}}}, y - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))))