Average Error: 0.5 → 0.7
Time: 26.0s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\]
↓
\[\left(\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{1.0}{\frac{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}}}\]
\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)
↓
\left(\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{1.0}{\frac{\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}}}
double f(double x) {
        double r3778598 = 1.0;
        double r3778599 = /* ERROR: no posit support in C */;
        double r3778600 = x;
        double r3778601 = sqrt(r3778600);
        double r3778602 = r3778599 / r3778601;
        double r3778603 = r3778600 + r3778599;
        double r3778604 = sqrt(r3778603);
        double r3778605 = r3778599 / r3778604;
        double r3778606 = r3778602 - r3778605;
        return r3778606;
}
↓
double f(double x) {
        double r3778607 = 1.0;
        double r3778608 = x;
        double r3778609 = r3778607 + r3778608;
        double r3778610 = sqrt(r3778609);
        double r3778611 = r3778607 / r3778610;
        double r3778612 = sqrt(r3778608);
        double r3778613 = r3778607 / r3778612;
        double r3778614 = r3778611 + r3778613;
        double r3778615 = 1.0;
        double r3778616 = r3778613 - r3778611;
        double r3778617 = r3778614 / r3778616;
        double r3778618 = r3778615 / r3778617;
        double r3778619 = r3778614 * r3778618;
        return r3778619;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.5

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

  3. Applied p16-flip--0.7

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

  5. Applied p16-*-un-lft-identity0.7

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

  6. Applied p16-*-un-lft-identity0.7

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

  7. Applied p16-times-frac0.7

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

  8. Applied p16-*-un-lft-identity0.7

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

  9. Applied p16-*-un-lft-identity0.7

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

  10. Applied p16-times-frac0.7

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

  11. Applied distribute-lft-out0.7

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

  12. Applied difference-of-squares0.6

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

  13. Applied p16-times-frac0.6

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

  14. Simplified0.6

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

  15. Simplified0.6

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

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

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

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

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

  19. Applied distribute-lft-out--0.6

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

  20. Applied associate-/l*0.7

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

  21. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  (-.p16 (/.p16 (real->posit16 1) (sqrt.p16 x)) (/.p16 (real->posit16 1) (sqrt.p16 (+.p16 x (real->posit16 1))))))