x / (x^2 + 1)

Average Error: 15.4 → 0.0

Time: 23.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3765989384353292 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 393.77334187402573:\\ \;\;\;\;\frac{x \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r1789139 = x;
        double r1789140 = r1789139 * r1789139;
        double r1789141 = 1.0;
        double r1789142 = r1789140 + r1789141;
        double r1789143 = r1789139 / r1789142;
        return r1789143;
}

↓

double f(double x) {
        double r1789144 = x;
        double r1789145 = -1.3765989384353292e+154;
        bool r1789146 = r1789144 <= r1789145;
        double r1789147 = 1.0;
        double r1789148 = r1789147 / r1789144;
        double r1789149 = r1789144 * r1789144;
        double r1789150 = r1789148 / r1789149;
        double r1789151 = r1789148 - r1789150;
        double r1789152 = 5.0;
        double r1789153 = pow(r1789144, r1789152);
        double r1789154 = r1789147 / r1789153;
        double r1789155 = r1789151 + r1789154;
        double r1789156 = 393.77334187402573;
        bool r1789157 = r1789144 <= r1789156;
        double r1789158 = fma(r1789144, r1789144, r1789147);
        double r1789159 = sqrt(r1789158);
        double r1789160 = r1789147 / r1789159;
        double r1789161 = r1789144 * r1789160;
        double r1789162 = r1789161 / r1789159;
        double r1789163 = r1789157 ? r1789162 : r1789155;
        double r1789164 = r1789146 ? r1789155 : r1789163;
        return r1789164;
}

Error

Bits error versus x

Target

Original	15.4
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3765989384353292e+154 or 393.77334187402573 < x

   1. Initial program 41.4

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

   2. Simplified 41.4

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -1.3765989384353292e+154 < x < 393.77334187402573

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.1

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

   5. Applied associate-/r* 0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\]
   6. Using strategy rm

   7. Applied div-inv 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3765989384353292 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 393.77334187402573:\\ \;\;\;\;\frac{x \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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