Average Error: 15.1 → 0.0
Time: 15.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\
\;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\

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

\end{array}
double f(double x) {
        double r2450496 = x;
        double r2450497 = r2450496 * r2450496;
        double r2450498 = 1.0;
        double r2450499 = r2450497 + r2450498;
        double r2450500 = r2450496 / r2450499;
        return r2450500;
}
↓
double f(double x) {
        double r2450501 = x;
        double r2450502 = -1.3342569343764892e+154;
        bool r2450503 = r2450501 <= r2450502;
        double r2450504 = 1.0;
        double r2450505 = 5.0;
        double r2450506 = pow(r2450501, r2450505);
        double r2450507 = r2450504 / r2450506;
        double r2450508 = 1.0;
        double r2450509 = r2450508 / r2450501;
        double r2450510 = r2450504 / r2450501;
        double r2450511 = r2450501 * r2450501;
        double r2450512 = r2450510 / r2450511;
        double r2450513 = r2450509 - r2450512;
        double r2450514 = r2450507 + r2450513;
        double r2450515 = 217598.07123922362;
        bool r2450516 = r2450501 <= r2450515;
        double r2450517 = fma(r2450501, r2450501, r2450504);
        double r2450518 = sqrt(r2450517);
        double r2450519 = r2450501 / r2450518;
        double r2450520 = r2450519 / r2450518;
        double r2450521 = r2450516 ? r2450520 : r2450514;
        double r2450522 = r2450503 ? r2450514 : r2450521;
        return r2450522;
}

Error

Bits error versus x

Target

Original15.1
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3342569343764892e+154 or 217598.07123922362 < x

    1. Initial program 40.5

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

    2. Simplified40.5

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

    4. Applied add-sqr-sqrt40.5

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

    5. Applied associate-/r*40.5

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -1.3342569343764892e+154 < x < 217598.07123922362

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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