Average Error: 0.6 → 0.6
Time: 51.8s
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 r8046020 = 1.0;
        double r8046021 = /* ERROR: no posit support in C */;
        double r8046022 = x;
        double r8046023 = sqrt(r8046022);
        double r8046024 = r8046021 / r8046023;
        double r8046025 = r8046022 + r8046021;
        double r8046026 = sqrt(r8046025);
        double r8046027 = r8046021 / r8046026;
        double r8046028 = r8046024 - r8046027;
        return r8046028;
}
↓
double f(double x) {
        double r8046029 = 1.0;
        double r8046030 = x;
        double r8046031 = sqrt(r8046030);
        double r8046032 = r8046029 / r8046031;
        double r8046033 = r8046030 + r8046029;
        double r8046034 = sqrt(r8046033);
        double r8046035 = r8046029 / r8046034;
        double r8046036 = r8046032 - r8046035;
        return r8046036;
}

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 2019132 +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))))))