Average Error: 14.7 → 0.0
Time: 7.2s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2077704617075050.2:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 3907.4508450000635:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2077704617075050.2:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 3907.4508450000635:\\
\;\;\;\;\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 r981100 = x;
        double r981101 = r981100 * r981100;
        double r981102 = 1.0;
        double r981103 = r981101 + r981102;
        double r981104 = r981100 / r981103;
        return r981104;
}
↓
double f(double x) {
        double r981105 = x;
        double r981106 = -2077704617075050.2;
        bool r981107 = r981105 <= r981106;
        double r981108 = 1.0;
        double r981109 = 5.0;
        double r981110 = pow(r981105, r981109);
        double r981111 = r981108 / r981110;
        double r981112 = r981108 / r981105;
        double r981113 = r981105 * r981105;
        double r981114 = r981105 * r981113;
        double r981115 = r981108 / r981114;
        double r981116 = r981112 - r981115;
        double r981117 = r981111 + r981116;
        double r981118 = 3907.4508450000635;
        bool r981119 = r981105 <= r981118;
        double r981120 = fma(r981105, r981105, r981108);
        double r981121 = r981105 / r981120;
        double r981122 = r981119 ? r981121 : r981117;
        double r981123 = r981107 ? r981117 : r981122;
        return r981123;
}

Error

Bits error versus x

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2077704617075050.2 or 3907.4508450000635 < x

    1. Initial program 30.4

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

    2. Simplified30.4

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

    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 -2077704617075050.2 < x < 3907.4508450000635

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2077704617075050.2:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 3907.4508450000635:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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