Average Error: 15.2 → 0.0
Time: 50.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6258382897202.597:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 580.2919375789389:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\mathsf{fma}\left(x \cdot x, x \cdot x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6258382897202.597:\\
\;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\

\mathbf{elif}\;x \le 580.2919375789389:\\
\;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\mathsf{fma}\left(x \cdot x, x \cdot x, -1\right)}\\

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

\end{array}
double f(double x) {
        double r1774393 = x;
        double r1774394 = r1774393 * r1774393;
        double r1774395 = 1.0;
        double r1774396 = r1774394 + r1774395;
        double r1774397 = r1774393 / r1774396;
        return r1774397;
}
↓
double f(double x) {
        double r1774398 = x;
        double r1774399 = -6258382897202.597;
        bool r1774400 = r1774398 <= r1774399;
        double r1774401 = 1.0;
        double r1774402 = r1774398 * r1774398;
        double r1774403 = r1774402 * r1774402;
        double r1774404 = r1774403 * r1774398;
        double r1774405 = r1774401 / r1774404;
        double r1774406 = r1774401 / r1774398;
        double r1774407 = r1774406 / r1774398;
        double r1774408 = r1774407 / r1774398;
        double r1774409 = r1774406 - r1774408;
        double r1774410 = r1774405 + r1774409;
        double r1774411 = 580.2919375789389;
        bool r1774412 = r1774398 <= r1774411;
        double r1774413 = r1774402 - r1774401;
        double r1774414 = -1.0;
        double r1774415 = fma(r1774402, r1774402, r1774414);
        double r1774416 = r1774398 / r1774415;
        double r1774417 = r1774413 * r1774416;
        double r1774418 = r1774412 ? r1774417 : r1774410;
        double r1774419 = r1774400 ? r1774410 : r1774418;
        return r1774419;
}

Error

Bits error versus x

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6258382897202.597 or 580.2919375789389 < x

    1. Initial program 30.6

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

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

    if -6258382897202.597 < x < 580.2919375789389

    1. Initial program 0.0

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

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6258382897202.597:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \le 580.2919375789389:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\mathsf{fma}\left(x \cdot x, x \cdot x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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