x / (x^2 + 1)

Average Error: 14.4 → 0.1

Time: 23.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x) {
        double r70598 = x;
        double r70599 = r70598 * r70598;
        double r70600 = 1.0;
        double r70601 = r70599 + r70600;
        double r70602 = r70598 / r70601;
        return r70602;
}

↓

double f(double x) {
        double r70603 = 1.0;
        double r70604 = 1.0;
        double r70605 = x;
        double r70606 = r70604 / r70605;
        double r70607 = r70606 + r70605;
        double r70608 = r70603 / r70607;
        return r70608;
}

Error

Bits error versus x

Target

Original	14.4
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 14.4

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

2. Simplified 14.4

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

4. Applied add-sqr-sqrt 14.4

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

5. Applied associate-/r* 14.3

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

7. Applied *-un-lft-identity 14.3

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

8. Applied *-un-lft-identity 14.3

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

9. Applied times-frac 14.3

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

10. Applied associate-/l* 14.4

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

11. Simplified 14.4

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

12. Taylor expanded around 0 0.1

    \[\leadsto \frac{\frac{1}{1}}{\color{blue}{x + 1 \cdot \frac{1}{x}}}\]

13. Simplified 0.1

    \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{1}{x} + x}}\]

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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