Average Error: 15.1 → 0.0
Time: 2.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -24215928678.961597442626953125 \lor \neg \left(x \le 3500.846768719505689659854397177696228027\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -24215928678.961597442626953125 \lor \neg \left(x \le 3500.846768719505689659854397177696228027\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \frac{x}{x \cdot x + 1}\\

\end{array}
double f(double x) {
        double r73461 = x;
        double r73462 = r73461 * r73461;
        double r73463 = 1.0;
        double r73464 = r73462 + r73463;
        double r73465 = r73461 / r73464;
        return r73465;
}
↓
double f(double x) {
        double r73466 = x;
        double r73467 = -24215928678.961597;
        bool r73468 = r73466 <= r73467;
        double r73469 = 3500.8467687195057;
        bool r73470 = r73466 <= r73469;
        double r73471 = !r73470;
        bool r73472 = r73468 || r73471;
        double r73473 = 1.0;
        double r73474 = 1.0;
        double r73475 = 5.0;
        double r73476 = pow(r73466, r73475);
        double r73477 = r73474 / r73476;
        double r73478 = 3.0;
        double r73479 = pow(r73466, r73478);
        double r73480 = r73474 / r73479;
        double r73481 = r73477 - r73480;
        double r73482 = r73473 * r73481;
        double r73483 = r73474 / r73466;
        double r73484 = r73482 + r73483;
        double r73485 = r73466 * r73466;
        double r73486 = r73485 + r73473;
        double r73487 = r73466 / r73486;
        double r73488 = r73474 * r73487;
        double r73489 = r73472 ? r73484 : r73488;
        return r73489;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -24215928678.961597 or 3500.8467687195057 < x

    1. Initial program 30.8

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -24215928678.961597 < x < 3500.8467687195057

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.0

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

    7. Applied sqrt-prod0.0

      \[\leadsto \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\color{blue}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}}\]

    8. Applied *-un-lft-identity0.0

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

    9. Applied sqrt-prod0.0

      \[\leadsto \frac{\frac{x}{\color{blue}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}}}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}\]

    10. Applied *-un-lft-identity0.0

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot x}}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}\]

    11. Applied times-frac0.0

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}}}{\sqrt{1} \cdot \sqrt{x \cdot x + 1}}\]

    12. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{1}}}{\sqrt{1}} \cdot \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]

    13. Simplified0.0

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\]

    14. Simplified0.0

      \[\leadsto 1 \cdot \color{blue}{\frac{x}{x \cdot x + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -24215928678.961597442626953125 \lor \neg \left(x \le 3500.846768719505689659854397177696228027\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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