Average Error: 0.8 → 0.6
Time: 31.2s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\left(\frac{\left(1.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\left(\frac{\left(1.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)
double f(double x) {
        double r6499444 = x;
        double r6499445 = 1.0;
        double r6499446 = /* ERROR: no posit support in C */;
        double r6499447 = r6499444 + r6499446;
        double r6499448 = sqrt(r6499447);
        double r6499449 = sqrt(r6499444);
        double r6499450 = r6499448 - r6499449;
        return r6499450;
}
↓
double f(double x) {
        double r6499451 = 1.0;
        double r6499452 = /* ERROR: no posit support in C */;
        double r6499453 = 1.0;
        double r6499454 = /* ERROR: no posit support in C */;
        double r6499455 = x;
        double r6499456 = r6499454 + r6499455;
        double r6499457 = sqrt(r6499456);
        double r6499458 = sqrt(r6499455);
        double r6499459 = r6499457 + r6499458;
        double r6499460 = r6499452 / r6499459;
        double r6499461 = r6499457 * r6499457;
        double r6499462 = -r6499458;
        double r6499463 = r6499462 * r6499458;
        double r6499464 = r6499461 + r6499463;
        double r6499465 = r6499460 * r6499464;
        return r6499465;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.8

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

  3. Applied p16-flip--0.6

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

  4. Simplified0.8

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

  5. Simplified0.8

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

  7. Applied +p16-rgt-identity-expand0.8

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\color{blue}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)} - \left(\sqrt{x}\right)\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  8. Applied associate--l+0.8

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\left(0.0\right) - \left(\sqrt{x}\right)\right)}\right)}\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  9. Applied distribute-rgt-in0.6

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)\right)}{\left(\left(\left(0.0\right) - \left(\sqrt{x}\right)\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)\right)}\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  10. Simplified0.6

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

  12. Applied distribute-lft-in0.7

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

  13. Applied associate-+r+0.9

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

  14. Simplified0.6

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)\right)}}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]
  15. Using strategy rm

  16. Applied *p16-rgt-identity-expand0.6

    \[\leadsto \frac{\left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)}{\color{blue}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(1.0\right)\right)}}\]

  17. Applied p16-*-un-lft-identity0.6

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)\right)}}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(1.0\right)\right)}\]

  18. Applied p16-times-frac0.6

    \[\leadsto \color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  19. Simplified0.6

    \[\leadsto \left(\frac{\left(1.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)}\]

  20. Final simplification0.6

    \[\leadsto \left(\frac{\left(1.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)\right)}{\left(\left(-\left(\sqrt{x}\right)\right) \cdot \left(\sqrt{x}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))