x / (x^2 + 1)

Average Error: 14.6 → 0.0

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.19500256338616044 \cdot 10^{62} \lor \neg \left(x \le 507.01009039307485\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r38207 = x;
        double r38208 = r38207 * r38207;
        double r38209 = 1.0;
        double r38210 = r38208 + r38209;
        double r38211 = r38207 / r38210;
        return r38211;
}

↓

double f(double x) {
        double r38212 = x;
        double r38213 = -3.1950025633861604e+62;
        bool r38214 = r38212 <= r38213;
        double r38215 = 507.01009039307485;
        bool r38216 = r38212 <= r38215;
        double r38217 = !r38216;
        bool r38218 = r38214 || r38217;
        double r38219 = 1.0;
        double r38220 = r38219 / r38212;
        double r38221 = 1.0;
        double r38222 = 5.0;
        double r38223 = pow(r38212, r38222);
        double r38224 = r38221 / r38223;
        double r38225 = 3.0;
        double r38226 = pow(r38212, r38225);
        double r38227 = r38221 / r38226;
        double r38228 = r38224 - r38227;
        double r38229 = r38220 + r38228;
        double r38230 = r38212 * r38212;
        double r38231 = r38230 + r38221;
        double r38232 = r38212 / r38231;
        double r38233 = r38218 ? r38229 : r38232;
        return r38233;
}

Error

Bits error versus x

Target

Original	14.6
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -3.1950025633861604e+62 or 507.01009039307485 < x

   1. Initial program 33.3

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

   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 -3.1950025633861604e+62 < x < 507.01009039307485

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.19500256338616044 \cdot 10^{62} \lor \neg \left(x \le 507.01009039307485\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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