Average Error: 0.6 → 0.6
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 r10732076 = 1.0;
        double r10732077 = /* ERROR: no posit support in C */;
        double r10732078 = x;
        double r10732079 = sqrt(r10732078);
        double r10732080 = r10732077 / r10732079;
        double r10732081 = r10732078 + r10732077;
        double r10732082 = sqrt(r10732081);
        double r10732083 = r10732077 / r10732082;
        double r10732084 = r10732080 - r10732083;
        return r10732084;
}
↓
double f(double x) {
        double r10732085 = 1.0;
        double r10732086 = x;
        double r10732087 = sqrt(r10732086);
        double r10732088 = r10732085 / r10732087;
        double r10732089 = r10732086 + r10732085;
        double r10732090 = sqrt(r10732089);
        double r10732091 = r10732085 / r10732090;
        double r10732092 = r10732088 - r10732091;
        return r10732092;
}

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 
(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))))))