Average Error: 0.5 → 0.6
Time: 12.2s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\]
↓
\[\frac{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
\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)
↓
\frac{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}
double f(double x) {
        double r2878273 = 1.0;
        double r2878274 = /* ERROR: no posit support in C */;
        double r2878275 = x;
        double r2878276 = sqrt(r2878275);
        double r2878277 = r2878274 / r2878276;
        double r2878278 = r2878275 + r2878274;
        double r2878279 = sqrt(r2878278);
        double r2878280 = r2878274 / r2878279;
        double r2878281 = r2878277 - r2878280;
        return r2878281;
}
↓
double f(double x) {
        double r2878282 = 1.0;
        double r2878283 = x;
        double r2878284 = sqrt(r2878283);
        double r2878285 = r2878282 / r2878284;
        double r2878286 = r2878283 + r2878282;
        double r2878287 = sqrt(r2878286);
        double r2878288 = r2878282 / r2878287;
        double r2878289 = r2878285 + r2878288;
        double r2878290 = r2878285 - r2878288;
        double r2878291 = r2878289 * r2878290;
        double r2878292 = r2878291 / r2878289;
        return r2878292;
}

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 difference-of-squares0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))))))