Average Error: 0.8 → 0.3
Time: 9.6s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r1450605 = x;
        double r1450606 = 1.0;
        double r1450607 = /* ERROR: no posit support in C */;
        double r1450608 = r1450605 + r1450607;
        double r1450609 = sqrt(r1450608);
        double r1450610 = sqrt(r1450605);
        double r1450611 = r1450609 - r1450610;
        return r1450611;
}
↓
double f(double x) {
        double r1450612 = x;
        double r1450613 = 1.0;
        double r1450614 = r1450613 - r1450612;
        double r1450615 = r1450612 + r1450614;
        double r1450616 = r1450612 + r1450613;
        double r1450617 = sqrt(r1450616);
        double r1450618 = sqrt(r1450612);
        double r1450619 = r1450617 + r1450618;
        double r1450620 = r1450615 / r1450619;
        return r1450620;
}

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. Using strategy rm

  5. Applied sqrt-sqrd.p160.5

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

  7. Applied sqrt-sqrd.p160.4

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

  9. Applied associate--l+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))