Average Error: 15.1 → 0.0
Time: 8.2s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 2168.52278018713832\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 2168.52278018713832\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\

\end{array}
double f(double x) {
        double r83524 = x;
        double r83525 = r83524 * r83524;
        double r83526 = 1.0;
        double r83527 = r83525 + r83526;
        double r83528 = r83524 / r83527;
        return r83528;
}
↓
double f(double x) {
        double r83529 = x;
        double r83530 = -62379821099783.21;
        bool r83531 = r83529 <= r83530;
        double r83532 = 2168.5227801871383;
        bool r83533 = r83529 <= r83532;
        double r83534 = !r83533;
        bool r83535 = r83531 || r83534;
        double r83536 = 1.0;
        double r83537 = r83536 / r83529;
        double r83538 = 1.0;
        double r83539 = 5.0;
        double r83540 = pow(r83529, r83539);
        double r83541 = r83538 / r83540;
        double r83542 = r83537 + r83541;
        double r83543 = 3.0;
        double r83544 = pow(r83529, r83543);
        double r83545 = r83538 / r83544;
        double r83546 = r83542 - r83545;
        double r83547 = fma(r83529, r83529, r83538);
        double r83548 = r83536 / r83547;
        double r83549 = r83529 * r83548;
        double r83550 = r83535 ? r83546 : r83549;
        return r83550;
}

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 < -62379821099783.21 or 2168.5227801871383 < 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. Using strategy rm

    4. Applied div-inv31.0

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

    6. Applied add-cube-cbrt31.4

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

    7. Applied associate-*l*31.4

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

    8. Simplified31.4

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

    9. Taylor expanded around inf 0.0

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

    10. Simplified0.0

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

    if -62379821099783.21 < x < 2168.5227801871383

    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 div-inv0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 2168.52278018713832\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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