Average Error: 15.1 → 0.0
Time: 16.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2678435.1094305497:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 11907527.731338572:\\
\;\;\;\;\frac{x}{\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 r969029 = x;
        double r969030 = r969029 * r969029;
        double r969031 = 1.0;
        double r969032 = r969030 + r969031;
        double r969033 = r969029 / r969032;
        return r969033;
}
↓
double f(double x) {
        double r969034 = x;
        double r969035 = -2678435.1094305497;
        bool r969036 = r969034 <= r969035;
        double r969037 = 1.0;
        double r969038 = r969037 / r969034;
        double r969039 = r969034 * r969034;
        double r969040 = r969038 / r969039;
        double r969041 = r969038 - r969040;
        double r969042 = 5.0;
        double r969043 = pow(r969034, r969042);
        double r969044 = r969037 / r969043;
        double r969045 = r969041 + r969044;
        double r969046 = 11907527.731338572;
        bool r969047 = r969034 <= r969046;
        double r969048 = fma(r969034, r969034, r969037);
        double r969049 = r969034 / r969048;
        double r969050 = r969047 ? r969049 : r969045;
        double r969051 = r969036 ? r969045 : r969050;
        return r969051;
}

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 < -2678435.1094305497 or 11907527.731338572 < 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 -2678435.1094305497 < x < 11907527.731338572

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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