x / (x^2 + 1)

Average Error: 14.9 → 0.0

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -340871036306084751321343440453632 \lor \neg \left(x \le 194132570.4950101375579833984375\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r39196 = x;
        double r39197 = r39196 * r39196;
        double r39198 = 1.0;
        double r39199 = r39197 + r39198;
        double r39200 = r39196 / r39199;
        return r39200;
}

↓

double f(double x) {
        double r39201 = x;
        double r39202 = -3.4087103630608475e+32;
        bool r39203 = r39201 <= r39202;
        double r39204 = 194132570.49501014;
        bool r39205 = r39201 <= r39204;
        double r39206 = !r39205;
        bool r39207 = r39203 || r39206;
        double r39208 = 1.0;
        double r39209 = r39208 / r39201;
        double r39210 = 1.0;
        double r39211 = 5.0;
        double r39212 = pow(r39201, r39211);
        double r39213 = r39210 / r39212;
        double r39214 = 3.0;
        double r39215 = pow(r39201, r39214);
        double r39216 = r39210 / r39215;
        double r39217 = r39213 - r39216;
        double r39218 = r39209 + r39217;
        double r39219 = r39201 * r39201;
        double r39220 = r39219 + r39210;
        double r39221 = r39201 / r39220;
        double r39222 = r39207 ? r39218 : r39221;
        return r39222;
}

Error

Bits error versus x

Target

Original	14.9
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -3.4087103630608475e+32 or 194132570.49501014 < x

   1. Initial program 31.7

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

   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. Simplified 0.0

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

   if -3.4087103630608475e+32 < x < 194132570.49501014

   1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -340871036306084751321343440453632 \lor \neg \left(x \le 194132570.4950101375579833984375\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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