Average Error: 15.1 → 0.0
Time: 15.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -853997951174.7607421875 \lor \neg \left(x \le 529.5974544640389467531349509954452514648\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -853997951174.7607421875 \lor \neg \left(x \le 529.5974544640389467531349509954452514648\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r46830 = x;
        double r46831 = r46830 * r46830;
        double r46832 = 1.0;
        double r46833 = r46831 + r46832;
        double r46834 = r46830 / r46833;
        return r46834;
}
↓
double f(double x) {
        double r46835 = x;
        double r46836 = -853997951174.7607;
        bool r46837 = r46835 <= r46836;
        double r46838 = 529.597454464039;
        bool r46839 = r46835 <= r46838;
        double r46840 = !r46839;
        bool r46841 = r46837 || r46840;
        double r46842 = 1.0;
        double r46843 = r46842 / r46835;
        double r46844 = 1.0;
        double r46845 = 5.0;
        double r46846 = pow(r46835, r46845);
        double r46847 = r46844 / r46846;
        double r46848 = r46843 + r46847;
        double r46849 = 3.0;
        double r46850 = pow(r46835, r46849);
        double r46851 = r46844 / r46850;
        double r46852 = r46848 - r46851;
        double r46853 = fma(r46835, r46835, r46844);
        double r46854 = r46835 / r46853;
        double r46855 = r46841 ? r46852 : r46854;
        return r46855;
}

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. Using strategy rm

    4. Applied add-sqr-sqrt30.5

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

    5. Applied associate-/r*30.4

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

    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

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

    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

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

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)))