x / (x^2 + 1)

Average Error: 15.1 → 0.0

Time: 16.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -957088378446.058:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 8137.871959944802:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r2135014 = x;
        double r2135015 = r2135014 * r2135014;
        double r2135016 = 1.0;
        double r2135017 = r2135015 + r2135016;
        double r2135018 = r2135014 / r2135017;
        return r2135018;
}

↓

double f(double x) {
        double r2135019 = x;
        double r2135020 = -957088378446.058;
        bool r2135021 = r2135019 <= r2135020;
        double r2135022 = 1.0;
        double r2135023 = r2135022 / r2135019;
        double r2135024 = r2135019 * r2135019;
        double r2135025 = r2135023 / r2135024;
        double r2135026 = r2135023 - r2135025;
        double r2135027 = 5.0;
        double r2135028 = pow(r2135019, r2135027);
        double r2135029 = r2135022 / r2135028;
        double r2135030 = r2135026 + r2135029;
        double r2135031 = 8137.871959944802;
        bool r2135032 = r2135019 <= r2135031;
        double r2135033 = fma(r2135019, r2135019, r2135022);
        double r2135034 = r2135019 / r2135033;
        double r2135035 = r2135032 ? r2135034 : r2135030;
        double r2135036 = r2135021 ? r2135030 : r2135035;
        return r2135036;
}

Error

Bits error versus x

Target

Original	15.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -957088378446.058 or 8137.871959944802 < x

   1. Initial program 31.2

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

   2. Simplified 31.2

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

   4. Applied add-sqr-sqrt 31.2

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

   5. Applied associate-/r* 31.1

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

   7. Applied div-inv 31.1

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

   8. Applied associate-/l* 31.2

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

   9. Simplified 31.2

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

   10. Taylor expanded around -inf 0.0

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

   11. Simplified 0.0

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

   if -957088378446.058 < x < 8137.871959944802

   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. Using strategy rm

   4. Applied add-sqr-sqrt 0.0

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

   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

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

   8. Applied associate-/l* 0.0

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

   9. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -957088378446.058:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 8137.871959944802:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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