Average Error: 15.1 → 0.0
Time: 16.8s
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 r46307 = x;
        double r46308 = r46307 * r46307;
        double r46309 = 1.0;
        double r46310 = r46308 + r46309;
        double r46311 = r46307 / r46310;
        return r46311;
}
↓
double f(double x) {
        double r46312 = x;
        double r46313 = -853997951174.7607;
        bool r46314 = r46312 <= r46313;
        double r46315 = 529.597454464039;
        bool r46316 = r46312 <= r46315;
        double r46317 = !r46316;
        bool r46318 = r46314 || r46317;
        double r46319 = 1.0;
        double r46320 = r46319 / r46312;
        double r46321 = 1.0;
        double r46322 = 5.0;
        double r46323 = pow(r46312, r46322);
        double r46324 = r46321 / r46323;
        double r46325 = r46320 + r46324;
        double r46326 = 3.0;
        double r46327 = pow(r46312, r46326);
        double r46328 = r46321 / r46327;
        double r46329 = r46325 - r46328;
        double r46330 = fma(r46312, r46312, r46321);
        double r46331 = r46312 / r46330;
        double r46332 = r46318 ? r46329 : r46331;
        return r46332;
}

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 *-un-lft-identity30.5

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

    6. Applied times-frac30.4

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

    7. Taylor expanded around inf 0.0

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

    8. 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)))