x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -491659454900747.938 \lor \neg \left(x \le 5758.0035822516511\right):\\ \;\;\;\;\frac{1}{{x}^{5}} - \left(\frac{1}{{x}^{3}} - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r54168 = x;
        double r54169 = r54168 * r54168;
        double r54170 = 1.0;
        double r54171 = r54169 + r54170;
        double r54172 = r54168 / r54171;
        return r54172;
}

↓

double f(double x) {
        double r54173 = x;
        double r54174 = -491659454900747.94;
        bool r54175 = r54173 <= r54174;
        double r54176 = 5758.003582251651;
        bool r54177 = r54173 <= r54176;
        double r54178 = !r54177;
        bool r54179 = r54175 || r54178;
        double r54180 = 1.0;
        double r54181 = 5.0;
        double r54182 = pow(r54173, r54181);
        double r54183 = r54180 / r54182;
        double r54184 = 3.0;
        double r54185 = pow(r54173, r54184);
        double r54186 = r54180 / r54185;
        double r54187 = 1.0;
        double r54188 = r54187 / r54173;
        double r54189 = r54186 - r54188;
        double r54190 = r54183 - r54189;
        double r54191 = r54173 * r54173;
        double r54192 = r54191 + r54180;
        double r54193 = r54173 / r54192;
        double r54194 = r54179 ? r54190 : r54193;
        return r54194;
}

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 < -491659454900747.94 or 5758.003582251651 < x

   1. Initial program 30.7

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

   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

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

   if -491659454900747.94 < x < 5758.003582251651

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -491659454900747.938 \lor \neg \left(x \le 5758.0035822516511\right):\\ \;\;\;\;\frac{1}{{x}^{5}} - \left(\frac{1}{{x}^{3}} - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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