Average Error: 22.5 → 7.0
Time: 4.0s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.6268126955223299 \cdot 10^{49} \lor \neg \left(y \le 3781485819941662700\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.6268126955223299 \cdot 10^{49} \lor \neg \left(y \le 3781485819941662700\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r2157416 = 1.0;
        double r2157417 = x;
        double r2157418 = r2157416 - r2157417;
        double r2157419 = y;
        double r2157420 = r2157418 * r2157419;
        double r2157421 = r2157419 + r2157416;
        double r2157422 = r2157420 / r2157421;
        double r2157423 = r2157416 - r2157422;
        return r2157423;
}
↓
double f(double x, double y) {
        double r2157424 = y;
        double r2157425 = -2.62681269552233e+49;
        bool r2157426 = r2157424 <= r2157425;
        double r2157427 = 3.7814858199416627e+18;
        bool r2157428 = r2157424 <= r2157427;
        double r2157429 = !r2157428;
        bool r2157430 = r2157426 || r2157429;
        double r2157431 = x;
        double r2157432 = r2157431 / r2157424;
        double r2157433 = 1.0;
        double r2157434 = r2157433 / r2157424;
        double r2157435 = r2157434 - r2157433;
        double r2157436 = fma(r2157432, r2157435, r2157431);
        double r2157437 = r2157424 * r2157424;
        double r2157438 = r2157433 * r2157433;
        double r2157439 = r2157437 - r2157438;
        double r2157440 = r2157424 / r2157439;
        double r2157441 = r2157424 - r2157433;
        double r2157442 = r2157440 * r2157441;
        double r2157443 = r2157431 - r2157433;
        double r2157444 = fma(r2157442, r2157443, r2157433);
        double r2157445 = r2157430 ? r2157436 : r2157444;
        return r2157445;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.62681269552233e+49 or 3.7814858199416627e+18 < y

    1. Initial program 47.5

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

    2. Simplified29.7

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

    3. Taylor expanded around inf 13.1

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

    4. Simplified13.1

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

    if -2.62681269552233e+49 < y < 3.7814858199416627e+18

    1. Initial program 2.2

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

    2. Simplified2.0

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

    4. Applied flip-+2.0

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

    5. Applied associate-/r/2.0

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.6268126955223299 \cdot 10^{49} \lor \neg \left(y \le 3781485819941662700\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))