Average Error: 15.3 → 0.0
Time: 19.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -294452774.063504397869110107421875 \lor \neg \left(x \le 476.3344512926516358675144147127866744995\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -294452774.063504397869110107421875 \lor \neg \left(x \le 476.3344512926516358675144147127866744995\right):\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r55195 = x;
        double r55196 = r55195 * r55195;
        double r55197 = 1.0;
        double r55198 = r55196 + r55197;
        double r55199 = r55195 / r55198;
        return r55199;
}
↓
double f(double x) {
        double r55200 = x;
        double r55201 = -294452774.0635044;
        bool r55202 = r55200 <= r55201;
        double r55203 = 476.33445129265164;
        bool r55204 = r55200 <= r55203;
        double r55205 = !r55204;
        bool r55206 = r55202 || r55205;
        double r55207 = 1.0;
        double r55208 = 5.0;
        double r55209 = pow(r55200, r55208);
        double r55210 = r55207 / r55209;
        double r55211 = 3.0;
        double r55212 = pow(r55200, r55211);
        double r55213 = r55207 / r55212;
        double r55214 = r55210 - r55213;
        double r55215 = 1.0;
        double r55216 = r55215 / r55200;
        double r55217 = r55214 + r55216;
        double r55218 = fma(r55200, r55200, r55207);
        double r55219 = r55200 / r55218;
        double r55220 = r55206 ? r55217 : r55219;
        return r55220;
}

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 < -294452774.0635044 or 476.33445129265164 < x

    1. Initial program 30.7

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

    2. Simplified30.7

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

    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

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

    if -294452774.0635044 < x < 476.33445129265164

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -294452774.063504397869110107421875 \lor \neg \left(x \le 476.3344512926516358675144147127866744995\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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