Average Error: 14.9 → 0.0
Time: 21.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.182351251673564960881259272718336657084 \cdot 10^{51}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 268164.477104543824680149555206298828125:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.182351251673564960881259272718336657084 \cdot 10^{51}:\\
\;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 268164.477104543824680149555206298828125:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r2704522 = x;
        double r2704523 = r2704522 * r2704522;
        double r2704524 = 1.0;
        double r2704525 = r2704523 + r2704524;
        double r2704526 = r2704522 / r2704525;
        return r2704526;
}
↓
double f(double x) {
        double r2704527 = x;
        double r2704528 = -2.182351251673565e+51;
        bool r2704529 = r2704527 <= r2704528;
        double r2704530 = 1.0;
        double r2704531 = r2704530 / r2704527;
        double r2704532 = 1.0;
        double r2704533 = r2704527 * r2704527;
        double r2704534 = r2704533 * r2704527;
        double r2704535 = r2704532 / r2704534;
        double r2704536 = 5.0;
        double r2704537 = pow(r2704527, r2704536);
        double r2704538 = r2704532 / r2704537;
        double r2704539 = r2704535 - r2704538;
        double r2704540 = r2704531 - r2704539;
        double r2704541 = 268164.4771045438;
        bool r2704542 = r2704527 <= r2704541;
        double r2704543 = fma(r2704527, r2704527, r2704532);
        double r2704544 = r2704527 / r2704543;
        double r2704545 = r2704542 ? r2704544 : r2704540;
        double r2704546 = r2704529 ? r2704540 : r2704545;
        return r2704546;
}

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 < -2.182351251673565e+51 or 268164.4771045438 < x

    1. Initial program 32.5

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

    2. Simplified32.5

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

    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

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

    if -2.182351251673565e+51 < x < 268164.4771045438

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.182351251673564960881259272718336657084 \cdot 10^{51}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 268164.477104543824680149555206298828125:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{\left(x \cdot x\right) \cdot x} - \frac{1}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))