x / (x^2 + 1)

Average Error: 15.2 → 0.1

Time: 2.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r68932 = x;
        double r68933 = r68932 * r68932;
        double r68934 = 1.0;
        double r68935 = r68933 + r68934;
        double r68936 = r68932 / r68935;
        return r68936;
}

↓

double f(double x) {
        double r68937 = 1.0;
        double r68938 = 1.0;
        double r68939 = x;
        double r68940 = r68937 / r68939;
        double r68941 = fma(r68938, r68940, r68939);
        double r68942 = r68937 / r68941;
        return r68942;
}

Error

Bits error versus x

Target

Original	15.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 15.2

   \[\frac{x}{x \cdot x + 1}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 15.2

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

4. Applied associate-/r* 15.1

   \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 15.1

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

7. Applied *-un-lft-identity 15.1

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

8. Applied times-frac 15.1

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

9. Applied associate-/l* 15.2

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

10. Simplified 15.2

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

11. Taylor expanded around 0 0.1

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

12. Simplified 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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