Average Error: 14.9 → 0.0
Time: 9.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -31939.41578853163:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 450.35164288801093:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -31939.41578853163:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 450.35164288801093:\\
\;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r816202 = x;
        double r816203 = r816202 * r816202;
        double r816204 = 1.0;
        double r816205 = r816203 + r816204;
        double r816206 = r816202 / r816205;
        return r816206;
}
↓
double f(double x) {
        double r816207 = x;
        double r816208 = -31939.41578853163;
        bool r816209 = r816207 <= r816208;
        double r816210 = 1.0;
        double r816211 = r816210 / r816207;
        double r816212 = r816207 * r816207;
        double r816213 = r816211 / r816212;
        double r816214 = r816211 - r816213;
        double r816215 = 5.0;
        double r816216 = pow(r816207, r816215);
        double r816217 = r816210 / r816216;
        double r816218 = r816214 + r816217;
        double r816219 = 450.35164288801093;
        bool r816220 = r816207 <= r816219;
        double r816221 = fma(r816207, r816207, r816210);
        double r816222 = sqrt(r816221);
        double r816223 = r816207 / r816222;
        double r816224 = r816223 / r816222;
        double r816225 = r816220 ? r816224 : r816218;
        double r816226 = r816209 ? r816218 : r816225;
        return r816226;
}

Error

Bits error versus x

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -31939.41578853163 or 450.35164288801093 < x

    1. Initial program 30.4

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

    2. Simplified30.4

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -31939.41578853163 < x < 450.35164288801093

    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. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -31939.41578853163:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 450.35164288801093:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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