Average Error: 22.9 → 7.6
Time: 3.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\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 r633156 = 1.0;
        double r633157 = x;
        double r633158 = r633156 - r633157;
        double r633159 = y;
        double r633160 = r633158 * r633159;
        double r633161 = r633159 + r633156;
        double r633162 = r633160 / r633161;
        double r633163 = r633156 - r633162;
        return r633163;
}
↓
double f(double x, double y) {
        double r633164 = y;
        double r633165 = -2.202659579672187e+16;
        bool r633166 = r633164 <= r633165;
        double r633167 = 14000563147.599981;
        bool r633168 = r633164 <= r633167;
        double r633169 = !r633168;
        bool r633170 = r633166 || r633169;
        double r633171 = x;
        double r633172 = r633171 / r633164;
        double r633173 = 1.0;
        double r633174 = r633173 / r633164;
        double r633175 = r633174 - r633173;
        double r633176 = fma(r633172, r633175, r633171);
        double r633177 = r633164 + r633173;
        double r633178 = r633164 / r633177;
        double r633179 = r633171 - r633173;
        double r633180 = fma(r633178, r633179, r633173);
        double r633181 = r633170 ? r633176 : r633180;
        return r633181;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.9
Target0.2
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.202659579672187e+16 or 14000563147.599981 < y

    1. Initial program 46.4

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

    2. Simplified29.3

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

    3. Taylor expanded around inf 15.1

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

    4. Simplified15.1

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

    if -2.202659579672187e+16 < y < 14000563147.599981

    1. Initial program 0.5

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

    2. Simplified0.5

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -22026595796721872 \lor \neg \left(y \le 14000563147.5999813\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))