3frac (problem 3.3.3)

Average Error: 10.6 → 0.1

Time: 8.7s

Precision: 64

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

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{x}\]

C

double f(double x) {
        double r101446 = 1.0;
        double r101447 = x;
        double r101448 = r101447 + r101446;
        double r101449 = r101446 / r101448;
        double r101450 = 2.0;
        double r101451 = r101450 / r101447;
        double r101452 = r101449 - r101451;
        double r101453 = r101447 - r101446;
        double r101454 = r101446 / r101453;
        double r101455 = r101452 + r101454;
        return r101455;
}

↓

double f(double x) {
        double r101456 = 2.0;
        double r101457 = x;
        double r101458 = r101457 * r101457;
        double r101459 = 1.0;
        double r101460 = r101459 * r101459;
        double r101461 = r101458 - r101460;
        double r101462 = r101456 / r101461;
        double r101463 = r101462 / r101457;
        return r101463;
}

Error

Bits error versus x

Target

Original	10.6
Target	0.3
Herbie	0.1

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Initial program 10.6

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied frac-sub 26.2

   \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]

4. Applied frac-add 25.6

   \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

5. Simplified 26.0

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

6. Taylor expanded around 0 0.3

   \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
7. Using strategy rm

8. Applied *-un-lft-identity 0.3

   \[\leadsto \frac{\color{blue}{1 \cdot 2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

9. Applied times-frac 0.1

   \[\leadsto \color{blue}{\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}}\]
10. Using strategy rm

11. Applied pow1 0.1

    \[\leadsto \frac{1}{\left(x + 1\right) \cdot x} \cdot \color{blue}{{\left(\frac{2}{x - 1}\right)}^{1}}\]

12. Applied pow1 0.1

    \[\leadsto \color{blue}{{\left(\frac{1}{\left(x + 1\right) \cdot x}\right)}^{1}} \cdot {\left(\frac{2}{x - 1}\right)}^{1}\]

13. Applied pow-prod-down 0.1

    \[\leadsto \color{blue}{{\left(\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}\right)}^{1}}\]

14. Simplified 0.1

    \[\leadsto {\color{blue}{\left(\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{x}\right)}}^{1}\]

15. Final simplification 0.1

    \[\leadsto \frac{\frac{2}{x \cdot x - 1 \cdot 1}}{x}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))