x / (x^2 + 1)

Average Error: 14.6 → 0.0

Time: 18.3s

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 r61851 = x;
        double r61852 = r61851 * r61851;
        double r61853 = 1.0;
        double r61854 = r61852 + r61853;
        double r61855 = r61851 / r61854;
        return r61855;
}

↓

double f(double x) {
        double r61856 = x;
        double r61857 = -3.1950025633861604e+62;
        bool r61858 = r61856 <= r61857;
        double r61859 = 507.01009039307485;
        bool r61860 = r61856 <= r61859;
        double r61861 = !r61860;
        bool r61862 = r61858 || r61861;
        double r61863 = 1.0;
        double r61864 = r61863 / r61856;
        double r61865 = 1.0;
        double r61866 = 5.0;
        double r61867 = pow(r61856, r61866);
        double r61868 = r61865 / r61867;
        double r61869 = 3.0;
        double r61870 = pow(r61856, r61869);
        double r61871 = r61865 / r61870;
        double r61872 = r61868 - r61871;
        double r61873 = r61864 + r61872;
        double r61874 = r61856 * r61856;
        double r61875 = r61874 + r61865;
        double r61876 = r61856 / r61875;
        double r61877 = r61862 ? r61873 : r61876;
        return r61877;
}

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 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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