Average Error: 0.5 → 0.5
Time: 1.0m
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 r10017050 = 1.0;
        double r10017051 = /* ERROR: no posit support in C */;
        double r10017052 = x;
        double r10017053 = sqrt(r10017052);
        double r10017054 = r10017051 / r10017053;
        double r10017055 = r10017052 + r10017051;
        double r10017056 = sqrt(r10017055);
        double r10017057 = r10017051 / r10017056;
        double r10017058 = r10017054 - r10017057;
        return r10017058;
}
↓
double f(double x) {
        double r10017059 = 1.0;
        double r10017060 = x;
        double r10017061 = sqrt(r10017060);
        double r10017062 = r10017059 / r10017061;
        double r10017063 = r10017060 + r10017059;
        double r10017064 = sqrt(r10017063);
        double r10017065 = r10017059 / r10017064;
        double r10017066 = r10017062 - r10017065;
        return r10017066;
}

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 +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))))))