x / (x^2 + 1)

Average Error: 15.0 → 0.0

Time: 21.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -487177471.4868003:\\ \;\;\;\;\left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 402.2628134253315:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{x}\\ \end{array}\]

C

double f(double x) {
        double r2171956 = x;
        double r2171957 = r2171956 * r2171956;
        double r2171958 = 1.0;
        double r2171959 = r2171957 + r2171958;
        double r2171960 = r2171956 / r2171959;
        return r2171960;
}

↓

double f(double x) {
        double r2171961 = x;
        double r2171962 = -487177471.4868003;
        bool r2171963 = r2171961 <= r2171962;
        double r2171964 = 1.0;
        double r2171965 = r2171961 * r2171961;
        double r2171966 = r2171965 * r2171961;
        double r2171967 = r2171965 * r2171966;
        double r2171968 = r2171964 / r2171967;
        double r2171969 = r2171964 / r2171966;
        double r2171970 = r2171968 - r2171969;
        double r2171971 = r2171964 / r2171961;
        double r2171972 = r2171970 + r2171971;
        double r2171973 = 402.2628134253315;
        bool r2171974 = r2171961 <= r2171973;
        double r2171975 = r2171965 - r2171964;
        double r2171976 = r2171965 * r2171965;
        double r2171977 = -1.0;
        double r2171978 = r2171976 + r2171977;
        double r2171979 = r2171961 / r2171978;
        double r2171980 = r2171975 * r2171979;
        double r2171981 = r2171974 ? r2171980 : r2171972;
        double r2171982 = r2171963 ? r2171972 : r2171981;
        return r2171982;
}

Error

Bits error versus x

Target

Original	15.0
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -487177471.4868003 or 402.2628134253315 < x

   1. Initial program 30.5

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -487177471.4868003 < x < 402.2628134253315

   1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
   2. Using strategy rm

   3. Applied flip-+ 0.0

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

   4. Applied associate-/r/ 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -487177471.4868003:\\ \;\;\;\;\left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 402.2628134253315:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{1}{\left(x \cdot x\right) \cdot x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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