Average Error: 0.8 → 0.3
Time: 8.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\]
double f(double x) {
        double r1600502 = x;
        double r1600503 = 1.0;
        double r1600504 = r1600502 + r1600503;
        double r1600505 = sqrt(r1600504);
        double r1600506 = sqrt(r1600502);
        double r1600507 = r1600505 - r1600506;
        return r1600507;
}
↓
double f(double x) {
        double r1600508 = x;
        double r1600509 = 1.0;
        double r1600510 = r1600509 - r1600508;
        double r1600511 = r1600508 + r1600510;
        double r1600512 = r1600508 + r1600509;
        double r1600513 = sqrt(r1600512);
        double r1600514 = sqrt(r1600508);
        double r1600515 = r1600513 + r1600514;
        double r1600516 = r1600511 / r1600515;
        return r1600516;
}


\sqrt{x + 1} - \sqrt{x}
↓
\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}

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 2019101 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))