Average Error: 0.8 → 0.3
Time: 7.3s
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 r1112277 = x;
        double r1112278 = 1.0;
        double r1112279 = /* ERROR: no posit support in C */;
        double r1112280 = r1112277 + r1112279;
        double r1112281 = sqrt(r1112280);
        double r1112282 = sqrt(r1112277);
        double r1112283 = r1112281 - r1112282;
        return r1112283;
}
↓
double f(double x) {
        double r1112284 = x;
        double r1112285 = 1.0;
        double r1112286 = r1112285 - r1112284;
        double r1112287 = r1112284 + r1112286;
        double r1112288 = r1112284 + r1112285;
        double r1112289 = sqrt(r1112288);
        double r1112290 = sqrt(r1112284);
        double r1112291 = r1112289 + r1112290;
        double r1112292 = r1112287 / r1112291;
        return r1112292;
}

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)))