Average Error: 21.9 → 7.9
Time: 5.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.67125714385780667 \cdot 10^{76} \lor \neg \left(y \le 1.64838597728570229 \cdot 10^{83}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.67125714385780667 \cdot 10^{76} \lor \neg \left(y \le 1.64838597728570229 \cdot 10^{83}\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 r702123 = 1.0;
        double r702124 = x;
        double r702125 = r702123 - r702124;
        double r702126 = y;
        double r702127 = r702125 * r702126;
        double r702128 = r702126 + r702123;
        double r702129 = r702127 / r702128;
        double r702130 = r702123 - r702129;
        return r702130;
}
↓
double f(double x, double y) {
        double r702131 = y;
        double r702132 = -4.6712571438578067e+76;
        bool r702133 = r702131 <= r702132;
        double r702134 = 1.6483859772857023e+83;
        bool r702135 = r702131 <= r702134;
        double r702136 = !r702135;
        bool r702137 = r702133 || r702136;
        double r702138 = x;
        double r702139 = r702138 / r702131;
        double r702140 = 1.0;
        double r702141 = r702140 / r702131;
        double r702142 = r702141 - r702140;
        double r702143 = fma(r702139, r702142, r702138);
        double r702144 = r702131 + r702140;
        double r702145 = r702131 / r702144;
        double r702146 = r702138 - r702140;
        double r702147 = fma(r702145, r702146, r702140);
        double r702148 = r702137 ? r702143 : r702147;
        return r702148;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 2 regimes

  2. if y < -4.6712571438578067e+76 or 1.6483859772857023e+83 < y

    1. Initial program 49.8

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

    2. Simplified29.8

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

    3. Taylor expanded around inf 12.7

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

    4. Simplified12.7

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

    if -4.6712571438578067e+76 < y < 1.6483859772857023e+83

    1. Initial program 6.1

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

    2. Simplified5.2

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.67125714385780667 \cdot 10^{76} \lor \neg \left(y \le 1.64838597728570229 \cdot 10^{83}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))