Average Error: 14.7 → 0.0
Time: 39.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7870171189038026 \cdot 10^{+75}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 1244.1662554758816:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.7870171189038026 \cdot 10^{+75}:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 1244.1662554758816:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r16066906 = x;
        double r16066907 = r16066906 * r16066906;
        double r16066908 = 1.0;
        double r16066909 = r16066907 + r16066908;
        double r16066910 = r16066906 / r16066909;
        return r16066910;
}
↓
double f(double x) {
        double r16066911 = x;
        double r16066912 = -1.7870171189038026e+75;
        bool r16066913 = r16066911 <= r16066912;
        double r16066914 = 1.0;
        double r16066915 = 5.0;
        double r16066916 = pow(r16066911, r16066915);
        double r16066917 = r16066914 / r16066916;
        double r16066918 = r16066914 / r16066911;
        double r16066919 = r16066917 + r16066918;
        double r16066920 = r16066911 * r16066911;
        double r16066921 = r16066920 * r16066911;
        double r16066922 = r16066914 / r16066921;
        double r16066923 = r16066919 - r16066922;
        double r16066924 = 1244.1662554758816;
        bool r16066925 = r16066911 <= r16066924;
        double r16066926 = fma(r16066911, r16066911, r16066914);
        double r16066927 = r16066911 / r16066926;
        double r16066928 = r16066925 ? r16066927 : r16066923;
        double r16066929 = r16066913 ? r16066923 : r16066928;
        return r16066929;
}

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 < -1.7870171189038026e+75 or 1244.1662554758816 < x

    1. Initial program 33.5

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

    2. Simplified33.5

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

    4. Applied add-cube-cbrt34.0

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}\]

    5. Applied *-un-lft-identity34.0

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}\]

    6. Applied times-frac34.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}} \cdot \frac{x}{\sqrt[3]{\mathsf{fma}\left(x, x, 1\right)}}}\]

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

    if -1.7870171189038026e+75 < x < 1244.1662554758816

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.7870171189038026 \cdot 10^{+75}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 1244.1662554758816:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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