Average Error: 0.6 → 0.6
Time: 16.1s
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 r1890195 = 1.0;
        double r1890196 = /* ERROR: no posit support in C */;
        double r1890197 = x;
        double r1890198 = sqrt(r1890197);
        double r1890199 = r1890196 / r1890198;
        double r1890200 = r1890197 + r1890196;
        double r1890201 = sqrt(r1890200);
        double r1890202 = r1890196 / r1890201;
        double r1890203 = r1890199 - r1890202;
        return r1890203;
}
↓
double f(double x) {
        double r1890204 = 1.0;
        double r1890205 = x;
        double r1890206 = sqrt(r1890205);
        double r1890207 = r1890204 / r1890206;
        double r1890208 = r1890205 + r1890204;
        double r1890209 = sqrt(r1890208);
        double r1890210 = r1890204 / r1890209;
        double r1890211 = r1890207 - r1890210;
        return r1890211;
}

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 2019156 +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))))))