x / (x^2 + 1)

Average Error: 14.3 → 0.0

Time: 13.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2330149 = x;
        double r2330150 = r2330149 * r2330149;
        double r2330151 = 1.0;
        double r2330152 = r2330150 + r2330151;
        double r2330153 = r2330149 / r2330152;
        return r2330153;
}

↓

double f(double x) {
        double r2330154 = x;
        double r2330155 = -267060974.6176845;
        bool r2330156 = r2330154 <= r2330155;
        double r2330157 = 1.0;
        double r2330158 = 5.0;
        double r2330159 = pow(r2330154, r2330158);
        double r2330160 = r2330157 / r2330159;
        double r2330161 = r2330154 * r2330154;
        double r2330162 = r2330157 / r2330161;
        double r2330163 = r2330162 / r2330154;
        double r2330164 = r2330160 - r2330163;
        double r2330165 = 1.0;
        double r2330166 = r2330165 / r2330154;
        double r2330167 = r2330164 + r2330166;
        double r2330168 = 508.8749887332333;
        bool r2330169 = r2330154 <= r2330168;
        double r2330170 = fma(r2330154, r2330154, r2330157);
        double r2330171 = r2330154 / r2330170;
        double r2330172 = r2330169 ? r2330171 : r2330167;
        double r2330173 = r2330156 ? r2330167 : r2330172;
        return r2330173;
}

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

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

   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

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

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)))