Average Error: 14.9 → 0.0
Time: 1.3m
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -473716199.82409686:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 433.85008447300646:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -473716199.82409686:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}\\

\mathbf{elif}\;x \le 433.85008447300646:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r3859158 = x;
        double r3859159 = r3859158 * r3859158;
        double r3859160 = 1.0;
        double r3859161 = r3859159 + r3859160;
        double r3859162 = r3859158 / r3859161;
        return r3859162;
}
↓
double f(double x) {
        double r3859163 = x;
        double r3859164 = -473716199.82409686;
        bool r3859165 = r3859163 <= r3859164;
        double r3859166 = 1.0;
        double r3859167 = 5.0;
        double r3859168 = pow(r3859163, r3859167);
        double r3859169 = r3859166 / r3859168;
        double r3859170 = r3859166 / r3859163;
        double r3859171 = r3859170 / r3859163;
        double r3859172 = r3859171 / r3859163;
        double r3859173 = r3859169 - r3859172;
        double r3859174 = r3859173 + r3859170;
        double r3859175 = 433.85008447300646;
        bool r3859176 = r3859163 <= r3859175;
        double r3859177 = fma(r3859163, r3859163, r3859166);
        double r3859178 = r3859163 / r3859177;
        double r3859179 = r3859176 ? r3859178 : r3859174;
        double r3859180 = r3859165 ? r3859174 : r3859179;
        return r3859180;
}

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 < -473716199.82409686 or 433.85008447300646 < x

    1. Initial program 30.8

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

    2. Simplified30.8

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt30.8

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

    5. Applied associate-/r*30.7

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

    6. Taylor expanded around -inf 0.0

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

    7. Simplified0.0

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

    if -473716199.82409686 < x < 433.85008447300646

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -473716199.82409686:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 433.85008447300646:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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