Average Error: 0.6 → 0.6
Time: 19.1s
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 r4145691 = 1.0;
        double r4145692 = /* ERROR: no posit support in C */;
        double r4145693 = x;
        double r4145694 = sqrt(r4145693);
        double r4145695 = r4145692 / r4145694;
        double r4145696 = r4145693 + r4145692;
        double r4145697 = sqrt(r4145696);
        double r4145698 = r4145692 / r4145697;
        double r4145699 = r4145695 - r4145698;
        return r4145699;
}
↓
double f(double x) {
        double r4145700 = 1.0;
        double r4145701 = x;
        double r4145702 = sqrt(r4145701);
        double r4145703 = r4145700 / r4145702;
        double r4145704 = r4145701 + r4145700;
        double r4145705 = sqrt(r4145704);
        double r4145706 = r4145700 / r4145705;
        double r4145707 = r4145703 - r4145706;
        return r4145707;
}

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 2019104 +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))))))