Average Error: 14.9 → 0.0
Time: 10.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -340871036306084751321343440453632 \lor \neg \left(x \le 194132570.4950101375579833984375\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -340871036306084751321343440453632 \lor \neg \left(x \le 194132570.4950101375579833984375\right):\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\

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

\end{array}
double f(double x) {
        double r84708 = x;
        double r84709 = r84708 * r84708;
        double r84710 = 1.0;
        double r84711 = r84709 + r84710;
        double r84712 = r84708 / r84711;
        return r84712;
}
↓
double f(double x) {
        double r84713 = x;
        double r84714 = -3.4087103630608475e+32;
        bool r84715 = r84713 <= r84714;
        double r84716 = 194132570.49501014;
        bool r84717 = r84713 <= r84716;
        double r84718 = !r84717;
        bool r84719 = r84715 || r84718;
        double r84720 = 1.0;
        double r84721 = r84720 / r84713;
        double r84722 = 1.0;
        double r84723 = 3.0;
        double r84724 = pow(r84713, r84723);
        double r84725 = r84722 / r84724;
        double r84726 = r84721 - r84725;
        double r84727 = 5.0;
        double r84728 = pow(r84713, r84727);
        double r84729 = r84722 / r84728;
        double r84730 = r84726 + r84729;
        double r84731 = fma(r84713, r84713, r84722);
        double r84732 = r84713 / r84731;
        double r84733 = r84719 ? r84730 : r84732;
        return r84733;
}

Error

Bits error versus x

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.4087103630608475e+32 or 194132570.49501014 < x

    1. Initial program 31.7

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified31.7

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right) - 1 \cdot \frac{1}{{x}^{3}}}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}}\]

    if -3.4087103630608475e+32 < x < 194132570.49501014

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -340871036306084751321343440453632 \lor \neg \left(x \le 194132570.4950101375579833984375\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))