x / (x^2 + 1)

Average Error: 15.4 → 0.0

Time: 21.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -243329265348.2015:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 517.3933217180979:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

C

double f(double x) {
        double r1687301 = x;
        double r1687302 = r1687301 * r1687301;
        double r1687303 = 1.0;
        double r1687304 = r1687302 + r1687303;
        double r1687305 = r1687301 / r1687304;
        return r1687305;
}

↓

double f(double x) {
        double r1687306 = x;
        double r1687307 = -243329265348.2015;
        bool r1687308 = r1687306 <= r1687307;
        double r1687309 = 1.0;
        double r1687310 = r1687306 * r1687306;
        double r1687311 = r1687310 * r1687310;
        double r1687312 = r1687311 * r1687306;
        double r1687313 = r1687309 / r1687312;
        double r1687314 = r1687309 / r1687306;
        double r1687315 = r1687314 / r1687310;
        double r1687316 = r1687314 - r1687315;
        double r1687317 = r1687313 + r1687316;
        double r1687318 = 517.3933217180979;
        bool r1687319 = r1687306 <= r1687318;
        double r1687320 = fma(r1687306, r1687306, r1687309);
        double r1687321 = r1687309 / r1687320;
        double r1687322 = r1687321 * r1687306;
        double r1687323 = r1687319 ? r1687322 : r1687317;
        double r1687324 = r1687308 ? r1687317 : r1687323;
        return r1687324;
}

Error

Bits error versus x

Target

Original	15.4
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -243329265348.2015 or 517.3933217180979 < x

   1. Initial program 31.6

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

   2. Simplified 31.6

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

   4. Applied div-inv 31.7

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

   5. Taylor expanded around inf 0.0

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

   6. Simplified 0.0

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

   if -243329265348.2015 < x < 517.3933217180979

   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. Using strategy rm

   4. Applied div-inv 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -243329265348.2015:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 517.3933217180979:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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