x / (x^2 + 1)

Average Error: 14.9 → 0.3

Time: 17.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1.02083405594772891333832376403734087944:\\ \;\;\;\;1 \cdot \left(\left({x}^{5} - x \cdot \left(x \cdot x\right)\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r2321215 = x;
        double r2321216 = r2321215 * r2321215;
        double r2321217 = 1.0;
        double r2321218 = r2321216 + r2321217;
        double r2321219 = r2321215 / r2321218;
        return r2321219;
}

↓

double f(double x) {
        double r2321220 = x;
        double r2321221 = -1.0026376323582022;
        bool r2321222 = r2321220 <= r2321221;
        double r2321223 = 1.0;
        double r2321224 = 5.0;
        double r2321225 = pow(r2321220, r2321224);
        double r2321226 = r2321223 / r2321225;
        double r2321227 = 1.0;
        double r2321228 = r2321227 / r2321220;
        double r2321229 = r2321220 * r2321220;
        double r2321230 = r2321220 * r2321229;
        double r2321231 = r2321223 / r2321230;
        double r2321232 = r2321228 - r2321231;
        double r2321233 = r2321226 + r2321232;
        double r2321234 = 1.020834055947729;
        bool r2321235 = r2321220 <= r2321234;
        double r2321236 = r2321225 - r2321230;
        double r2321237 = r2321236 + r2321220;
        double r2321238 = r2321223 * r2321237;
        double r2321239 = r2321235 ? r2321238 : r2321233;
        double r2321240 = r2321222 ? r2321233 : r2321239;
        return r2321240;
}

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0026376323582022 or 1.020834055947729 < x

   1. Initial program 29.6

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

   2. Simplified 29.6

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

   3. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

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

   if -1.0026376323582022 < x < 1.020834055947729

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   3. Taylor expanded around 0 0.3

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{1 \cdot \left(x + \left({x}^{5} - x \cdot \left(x \cdot x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1.02083405594772891333832376403734087944:\\ \;\;\;\;1 \cdot \left(\left({x}^{5} - x \cdot \left(x \cdot x\right)\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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