Average Error: 0.8 → 0.3
Time: 8.6s
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 r1217306 = x;
        double r1217307 = 1.0;
        double r1217308 = /* ERROR: no posit support in C */;
        double r1217309 = r1217306 + r1217308;
        double r1217310 = sqrt(r1217309);
        double r1217311 = sqrt(r1217306);
        double r1217312 = r1217310 - r1217311;
        return r1217312;
}
↓
double f(double x) {
        double r1217313 = x;
        double r1217314 = 1.0;
        double r1217315 = r1217314 - r1217313;
        double r1217316 = r1217313 + r1217315;
        double r1217317 = r1217313 + r1217314;
        double r1217318 = sqrt(r1217317);
        double r1217319 = sqrt(r1217313);
        double r1217320 = r1217318 + r1217319;
        double r1217321 = r1217316 / r1217320;
        return r1217321;
}

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 2019119 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))