Average Error: 21.9 → 7.5
Time: 3.7s
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 r597114 = 1.0;
        double r597115 = x;
        double r597116 = r597114 - r597115;
        double r597117 = y;
        double r597118 = r597116 * r597117;
        double r597119 = r597117 + r597114;
        double r597120 = r597118 / r597119;
        double r597121 = r597114 - r597120;
        return r597121;
}
↓
double f(double x, double y) {
        double r597122 = y;
        double r597123 = -5.729793113088247e+21;
        bool r597124 = r597122 <= r597123;
        double r597125 = 1.1941330995112353e+55;
        bool r597126 = r597122 <= r597125;
        double r597127 = !r597126;
        bool r597128 = r597124 || r597127;
        double r597129 = x;
        double r597130 = r597129 / r597122;
        double r597131 = 1.0;
        double r597132 = r597131 / r597122;
        double r597133 = r597132 - r597131;
        double r597134 = fma(r597130, r597133, r597129);
        double r597135 = 1.0;
        double r597136 = r597122 + r597131;
        double r597137 = r597135 / r597136;
        double r597138 = r597122 * r597137;
        double r597139 = r597129 - r597131;
        double r597140 = fma(r597138, r597139, r597131);
        double r597141 = r597128 ? r597134 : r597140;
        return r597141;
}

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))))