Average Error: 0.6 → 0.6
Time: 16.8s
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 r4166796 = 1.0;
        double r4166797 = /* ERROR: no posit support in C */;
        double r4166798 = x;
        double r4166799 = sqrt(r4166798);
        double r4166800 = r4166797 / r4166799;
        double r4166801 = r4166798 + r4166797;
        double r4166802 = sqrt(r4166801);
        double r4166803 = r4166797 / r4166802;
        double r4166804 = r4166800 - r4166803;
        return r4166804;
}
↓
double f(double x) {
        double r4166805 = 1.0;
        double r4166806 = x;
        double r4166807 = sqrt(r4166806);
        double r4166808 = r4166805 / r4166807;
        double r4166809 = r4166806 + r4166805;
        double r4166810 = sqrt(r4166809);
        double r4166811 = r4166805 / r4166810;
        double r4166812 = r4166808 - r4166811;
        return r4166812;
}

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 2019124 +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))))))