Average Error: 14.4 → 0.0
Time: 8.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7672388198.064882:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 433.55902965551593:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7672388198.064882:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 433.55902965551593:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\end{array}
double f(double x) {
        double r1593047 = x;
        double r1593048 = r1593047 * r1593047;
        double r1593049 = 1.0;
        double r1593050 = r1593048 + r1593049;
        double r1593051 = r1593047 / r1593050;
        return r1593051;
}
↓
double f(double x) {
        double r1593052 = x;
        double r1593053 = -7672388198.064882;
        bool r1593054 = r1593052 <= r1593053;
        double r1593055 = 1.0;
        double r1593056 = 5.0;
        double r1593057 = pow(r1593052, r1593056);
        double r1593058 = r1593055 / r1593057;
        double r1593059 = r1593055 / r1593052;
        double r1593060 = r1593052 * r1593052;
        double r1593061 = r1593052 * r1593060;
        double r1593062 = r1593055 / r1593061;
        double r1593063 = r1593059 - r1593062;
        double r1593064 = r1593058 + r1593063;
        double r1593065 = 433.55902965551593;
        bool r1593066 = r1593052 <= r1593065;
        double r1593067 = fma(r1593052, r1593052, r1593055);
        double r1593068 = r1593052 / r1593067;
        double r1593069 = r1593066 ? r1593068 : r1593064;
        double r1593070 = r1593054 ? r1593064 : r1593069;
        return r1593070;
}

Error

Bits error versus x

Target

Original14.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7672388198.064882 or 433.55902965551593 < x

    1. Initial program 29.5

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

    2. Simplified29.5

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

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

    if -7672388198.064882 < x < 433.55902965551593

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7672388198.064882:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 433.55902965551593:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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