Average Error: 15.4 → 0.0
Time: 2.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -599.2763266259171 \lor \neg \left(x \le 261301.49671835295\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-1, 1, {x}^{4}\right)} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -599.2763266259171 \lor \neg \left(x \le 261301.49671835295\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(-1, 1, {x}^{4}\right)} \cdot \left(x \cdot x - 1\right)\\

\end{array}
double f(double x) {
        double r95285 = x;
        double r95286 = r95285 * r95285;
        double r95287 = 1.0;
        double r95288 = r95286 + r95287;
        double r95289 = r95285 / r95288;
        return r95289;
}
↓
double f(double x) {
        double r95290 = x;
        double r95291 = -599.2763266259171;
        bool r95292 = r95290 <= r95291;
        double r95293 = 261301.49671835295;
        bool r95294 = r95290 <= r95293;
        double r95295 = !r95294;
        bool r95296 = r95292 || r95295;
        double r95297 = 1.0;
        double r95298 = 1.0;
        double r95299 = 5.0;
        double r95300 = pow(r95290, r95299);
        double r95301 = r95298 / r95300;
        double r95302 = 3.0;
        double r95303 = pow(r95290, r95302);
        double r95304 = r95298 / r95303;
        double r95305 = r95301 - r95304;
        double r95306 = r95298 / r95290;
        double r95307 = fma(r95297, r95305, r95306);
        double r95308 = -r95297;
        double r95309 = 4.0;
        double r95310 = pow(r95290, r95309);
        double r95311 = fma(r95308, r95297, r95310);
        double r95312 = r95290 / r95311;
        double r95313 = r95290 * r95290;
        double r95314 = r95313 - r95297;
        double r95315 = r95312 * r95314;
        double r95316 = r95296 ? r95307 : r95315;
        return r95316;
}

Error

Bits error versus x

Target

Original15.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -599.2763266259171 or 261301.49671835295 < x

    1. Initial program 31.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -599.2763266259171 < x < 261301.49671835295

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(-1, 1, {x}^{4}\right)}} \cdot \left(x \cdot x - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -599.2763266259171 \lor \neg \left(x \le 261301.49671835295\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-1, 1, {x}^{4}\right)} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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