x / (x^2 + 1)

Average Error: 14.3 → 0.0

Time: 18.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -267060974.6176845133304595947265625:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 508.8749887332332946243695914745330810547:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \end{array}\]

C

double f(double x) {
        double r2469198 = x;
        double r2469199 = r2469198 * r2469198;
        double r2469200 = 1.0;
        double r2469201 = r2469199 + r2469200;
        double r2469202 = r2469198 / r2469201;
        return r2469202;
}

↓

double f(double x) {
        double r2469203 = x;
        double r2469204 = -267060974.6176845;
        bool r2469205 = r2469203 <= r2469204;
        double r2469206 = 1.0;
        double r2469207 = 5.0;
        double r2469208 = pow(r2469203, r2469207);
        double r2469209 = r2469206 / r2469208;
        double r2469210 = r2469206 / r2469203;
        double r2469211 = r2469203 * r2469203;
        double r2469212 = r2469210 / r2469211;
        double r2469213 = r2469209 - r2469212;
        double r2469214 = 1.0;
        double r2469215 = r2469214 / r2469203;
        double r2469216 = r2469213 + r2469215;
        double r2469217 = 508.8749887332333;
        bool r2469218 = r2469203 <= r2469217;
        double r2469219 = fma(r2469203, r2469203, r2469206);
        double r2469220 = r2469203 / r2469219;
        double r2469221 = r2469218 ? r2469220 : r2469216;
        double r2469222 = r2469205 ? r2469216 : r2469221;
        return r2469222;
}

Error

Bits error versus x

Target

Original	14.3
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -267060974.6176845 or 508.8749887332333 < x

   1. Initial program 29.6

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

   2. Simplified 29.6

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -267060974.6176845 < x < 508.8749887332333

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -267060974.6176845133304595947265625:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 508.8749887332332946243695914745330810547:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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