Average Error: 0.6 → 0.6
Time: 15.3s
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 r3822777 = 1.0;
        double r3822778 = /* ERROR: no posit support in C */;
        double r3822779 = x;
        double r3822780 = sqrt(r3822779);
        double r3822781 = r3822778 / r3822780;
        double r3822782 = r3822779 + r3822778;
        double r3822783 = sqrt(r3822782);
        double r3822784 = r3822778 / r3822783;
        double r3822785 = r3822781 - r3822784;
        return r3822785;
}
↓
double f(double x) {
        double r3822786 = 1.0;
        double r3822787 = x;
        double r3822788 = sqrt(r3822787);
        double r3822789 = r3822786 / r3822788;
        double r3822790 = r3822787 + r3822786;
        double r3822791 = sqrt(r3822790);
        double r3822792 = r3822786 / r3822791;
        double r3822793 = r3822789 - r3822792;
        return r3822793;
}

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 2019119 +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))))))