Average Error: 15.3 → 0.0
Time: 6.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3379998661029618 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1626101.3987142143:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3379998661029618 \cdot 10^{+154}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 1626101.3987142143:\\
\;\;\;\;\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 r718071 = x;
        double r718072 = r718071 * r718071;
        double r718073 = 1.0;
        double r718074 = r718072 + r718073;
        double r718075 = r718071 / r718074;
        return r718075;
}
↓
double f(double x) {
        double r718076 = x;
        double r718077 = -1.3379998661029618e+154;
        bool r718078 = r718076 <= r718077;
        double r718079 = 1.0;
        double r718080 = r718079 / r718076;
        double r718081 = r718076 * r718076;
        double r718082 = r718080 / r718081;
        double r718083 = r718080 - r718082;
        double r718084 = 5.0;
        double r718085 = pow(r718076, r718084);
        double r718086 = r718079 / r718085;
        double r718087 = r718083 + r718086;
        double r718088 = 1626101.3987142143;
        bool r718089 = r718076 <= r718088;
        double r718090 = fma(r718076, r718076, r718079);
        double r718091 = sqrt(r718090);
        double r718092 = r718076 / r718091;
        double r718093 = r718092 / r718091;
        double r718094 = r718089 ? r718093 : r718087;
        double r718095 = r718078 ? r718087 : r718094;
        return r718095;
}

Error

Bits error versus x

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3379998661029618e+154 or 1626101.3987142143 < x

    1. Initial program 40.6

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

    2. Simplified40.6

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

    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 -1.3379998661029618e+154 < x < 1626101.3987142143

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3379998661029618 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1626101.3987142143:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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