Average Error: 0.6 → 0.6
Time: 20.6s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\]
↓
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
\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{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
double f(double x) {
        double r1678338 = 1.0;
        double r1678339 = /* ERROR: no posit support in C */;
        double r1678340 = x;
        double r1678341 = sqrt(r1678340);
        double r1678342 = r1678339 / r1678341;
        double r1678343 = r1678340 + r1678339;
        double r1678344 = sqrt(r1678343);
        double r1678345 = r1678339 / r1678344;
        double r1678346 = r1678342 - r1678345;
        return r1678346;
}
↓
double f(double x) {
        double r1678347 = 1.0;
        double r1678348 = x;
        double r1678349 = sqrt(r1678348);
        double r1678350 = r1678347 / r1678349;
        double r1678351 = r1678348 + r1678347;
        double r1678352 = sqrt(r1678351);
        double r1678353 = r1678347 / r1678352;
        double r1678354 = r1678350 - r1678353;
        return r1678354;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.6

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

  2. Final simplification0.6

    \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]

Reproduce

herbie shell --seed 2019154 +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))))))