Average Error: 21.9 → 7.5
Time: 3.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5729793113088246940000 \lor \neg \left(y \le 1.19413309951123531 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5729793113088246940000 \lor \neg \left(y \le 1.19413309951123531 \cdot 10^{55}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r548322 = 1.0;
        double r548323 = x;
        double r548324 = r548322 - r548323;
        double r548325 = y;
        double r548326 = r548324 * r548325;
        double r548327 = r548325 + r548322;
        double r548328 = r548326 / r548327;
        double r548329 = r548322 - r548328;
        return r548329;
}
↓
double f(double x, double y) {
        double r548330 = y;
        double r548331 = -5.729793113088247e+21;
        bool r548332 = r548330 <= r548331;
        double r548333 = 1.1941330995112353e+55;
        bool r548334 = r548330 <= r548333;
        double r548335 = !r548334;
        bool r548336 = r548332 || r548335;
        double r548337 = x;
        double r548338 = r548337 / r548330;
        double r548339 = 1.0;
        double r548340 = r548339 / r548330;
        double r548341 = r548340 - r548339;
        double r548342 = fma(r548338, r548341, r548337);
        double r548343 = 1.0;
        double r548344 = r548330 + r548339;
        double r548345 = r548343 / r548344;
        double r548346 = r548330 * r548345;
        double r548347 = r548337 - r548339;
        double r548348 = fma(r548346, r548347, r548339);
        double r548349 = r548336 ? r548342 : r548348;
        return r548349;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.9
Target0.2
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.729793113088247e+21 or 1.1941330995112353e+55 < y

    1. Initial program 46.9

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

    2. Simplified29.5

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

    3. Taylor expanded around inf 14.1

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

    4. Simplified14.1

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

    if -5.729793113088247e+21 < y < 1.1941330995112353e+55

    1. Initial program 2.7

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

    2. Simplified2.4

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

    4. Applied div-inv2.4

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5729793113088246940000 \lor \neg \left(y \le 1.19413309951123531 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))