Average Error: 14.7 → 0.0
Time: 14.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6289590436010291770888814592:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 446.3237971224295392858039122074842453003:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} \cdot \frac{x}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6289590436010291770888814592:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\

\mathbf{elif}\;x \le 446.3237971224295392858039122074842453003:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} \cdot \frac{x}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\

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

\end{array}
double f(double x) {
        double r2901045 = x;
        double r2901046 = r2901045 * r2901045;
        double r2901047 = 1.0;
        double r2901048 = r2901046 + r2901047;
        double r2901049 = r2901045 / r2901048;
        return r2901049;
}
↓
double f(double x) {
        double r2901050 = x;
        double r2901051 = -6.289590436010292e+27;
        bool r2901052 = r2901050 <= r2901051;
        double r2901053 = 1.0;
        double r2901054 = 5.0;
        double r2901055 = pow(r2901050, r2901054);
        double r2901056 = r2901053 / r2901055;
        double r2901057 = r2901050 * r2901050;
        double r2901058 = r2901053 / r2901057;
        double r2901059 = r2901058 / r2901050;
        double r2901060 = r2901056 - r2901059;
        double r2901061 = 1.0;
        double r2901062 = r2901061 / r2901050;
        double r2901063 = r2901060 + r2901062;
        double r2901064 = 446.32379712242954;
        bool r2901065 = r2901050 <= r2901064;
        double r2901066 = fma(r2901050, r2901050, r2901053);
        double r2901067 = sqrt(r2901066);
        double r2901068 = sqrt(r2901067);
        double r2901069 = r2901061 / r2901068;
        double r2901070 = r2901050 / r2901068;
        double r2901071 = r2901069 * r2901070;
        double r2901072 = r2901071 / r2901067;
        double r2901073 = r2901065 ? r2901072 : r2901063;
        double r2901074 = r2901052 ? r2901063 : r2901073;
        return r2901074;
}

Error

Bits error versus x

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.289590436010292e+27 or 446.32379712242954 < x

    1. Initial program 30.9

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

    2. Simplified30.9

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

    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

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

    if -6.289590436010292e+27 < x < 446.32379712242954

    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

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.0

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

    8. Applied sqrt-prod0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied times-frac0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6289590436010291770888814592:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 446.3237971224295392858039122074842453003:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} \cdot \frac{x}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))