Average Error: 0.6 → 0.6
Time: 1.1m
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 r9715099 = 1.0;
        double r9715100 = /* ERROR: no posit support in C */;
        double r9715101 = x;
        double r9715102 = sqrt(r9715101);
        double r9715103 = r9715100 / r9715102;
        double r9715104 = r9715101 + r9715100;
        double r9715105 = sqrt(r9715104);
        double r9715106 = r9715100 / r9715105;
        double r9715107 = r9715103 - r9715106;
        return r9715107;
}
↓
double f(double x) {
        double r9715108 = 1.0;
        double r9715109 = x;
        double r9715110 = sqrt(r9715109);
        double r9715111 = r9715108 / r9715110;
        double r9715112 = r9715109 + r9715108;
        double r9715113 = sqrt(r9715112);
        double r9715114 = r9715108 / r9715113;
        double r9715115 = r9715111 - r9715114;
        return r9715115;
}

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 2019130 +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))))))