Average Error: 15.1 → 0.0
Time: 19.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -853997951174.7607421875 \lor \neg \left(x \le 529.5974544640389467531349509954452514648\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -853997951174.7607421875 \lor \neg \left(x \le 529.5974544640389467531349509954452514648\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r70348 = x;
        double r70349 = r70348 * r70348;
        double r70350 = 1.0;
        double r70351 = r70349 + r70350;
        double r70352 = r70348 / r70351;
        return r70352;
}
↓
double f(double x) {
        double r70353 = x;
        double r70354 = -853997951174.7607;
        bool r70355 = r70353 <= r70354;
        double r70356 = 529.597454464039;
        bool r70357 = r70353 <= r70356;
        double r70358 = !r70357;
        bool r70359 = r70355 || r70358;
        double r70360 = 1.0;
        double r70361 = 5.0;
        double r70362 = pow(r70353, r70361);
        double r70363 = r70360 / r70362;
        double r70364 = 1.0;
        double r70365 = r70364 / r70353;
        double r70366 = 3.0;
        double r70367 = pow(r70353, r70366);
        double r70368 = r70360 / r70367;
        double r70369 = r70365 - r70368;
        double r70370 = r70363 + r70369;
        double r70371 = fma(r70353, r70353, r70360);
        double r70372 = r70353 / r70371;
        double r70373 = r70359 ? r70370 : r70372;
        return r70373;
}

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 < -853997951174.7607 or 529.597454464039 < x

    1. Initial program 30.5

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

    2. Simplified30.5

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

    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

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

    if -853997951174.7607 < x < 529.597454464039

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -853997951174.7607421875 \lor \neg \left(x \le 529.5974544640389467531349509954452514648\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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