x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 13.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.90036294001547 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \le 415.7689475654002:\\ \;\;\;\;\frac{1}{1 + x \cdot x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r1853183 = x;
        double r1853184 = r1853183 * r1853183;
        double r1853185 = 1.0;
        double r1853186 = r1853184 + r1853185;
        double r1853187 = r1853183 / r1853186;
        return r1853187;
}

↓

double f(double x) {
        double r1853188 = x;
        double r1853189 = -8.90036294001547e+25;
        bool r1853190 = r1853188 <= r1853189;
        double r1853191 = 1.0;
        double r1853192 = r1853191 / r1853188;
        double r1853193 = r1853188 * r1853188;
        double r1853194 = r1853192 / r1853193;
        double r1853195 = r1853192 - r1853194;
        double r1853196 = r1853193 * r1853193;
        double r1853197 = r1853188 * r1853196;
        double r1853198 = r1853191 / r1853197;
        double r1853199 = r1853195 + r1853198;
        double r1853200 = 415.7689475654002;
        bool r1853201 = r1853188 <= r1853200;
        double r1853202 = r1853191 + r1853193;
        double r1853203 = r1853191 / r1853202;
        double r1853204 = r1853203 * r1853188;
        double r1853205 = r1853201 ? r1853204 : r1853199;
        double r1853206 = r1853190 ? r1853199 : r1853205;
        return r1853206;
}

Error

Bits error versus x

Target

Original	15.2
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -8.90036294001547e+25 or 415.7689475654002 < x

   1. Initial program 31.8

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

   3. Applied div-inv 31.8

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]

   4. Taylor expanded around inf 0.0

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

   5. Simplified 0.0

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

   if -8.90036294001547e+25 < x < 415.7689475654002

   1. Initial program 0.0

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

   3. Applied div-inv 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.90036294001547 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;x \le 415.7689475654002:\\ \;\;\;\;\frac{1}{1 + x \cdot x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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