Average Error: 0.8 → 0.2
Time: 29.4s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r9150040 = x;
        double r9150041 = 1.0;
        double r9150042 = /* ERROR: no posit support in C */;
        double r9150043 = r9150040 + r9150042;
        double r9150044 = sqrt(r9150043);
        double r9150045 = sqrt(r9150040);
        double r9150046 = r9150044 - r9150045;
        return r9150046;
}
↓
double f(double x) {
        double r9150047 = 1.0;
        double r9150048 = x;
        double r9150049 = r9150048 + r9150047;
        double r9150050 = sqrt(r9150049);
        double r9150051 = sqrt(r9150048);
        double r9150052 = r9150050 + r9150051;
        double r9150053 = r9150047 / r9150052;
        return r9150053;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.8

    \[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
  2. Using strategy rm

  3. Applied p16-flip--0.6

    \[\leadsto \color{blue}{\frac{\left(\left(\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)\right) - \left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right)}}\]

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{\left(1\right)}}{\left(\frac{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))