Average Error: 0.6 → 1.0
Time: 51.3s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\left(\frac{1}{x + 1} + \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}{\frac{1}{x + 1} + \frac{1}{x}}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\left(\frac{1}{x + 1} + \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}{\frac{1}{x + 1} + \frac{1}{x}}
double f(double x) {
        double r9626161 = 1.0;
        double r9626162 = /* ERROR: no posit support in C */;
        double r9626163 = x;
        double r9626164 = r9626163 + r9626162;
        double r9626165 = r9626162 / r9626164;
        double r9626166 = r9626162 / r9626163;
        double r9626167 = r9626165 - r9626166;
        return r9626167;
}
↓
double f(double x) {
        double r9626168 = 1.0;
        double r9626169 = x;
        double r9626170 = r9626169 + r9626168;
        double r9626171 = r9626168 / r9626170;
        double r9626172 = r9626168 / r9626169;
        double r9626173 = r9626171 + r9626172;
        double r9626174 = r9626171 - r9626172;
        double r9626175 = r9626173 * r9626174;
        double r9626176 = r9626175 / r9626173;
        return r9626176;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.6

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

  3. Applied p16-flip--1.3

    \[\leadsto \color{blue}{\frac{\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)\right) - \left(\left(\frac{\left(1\right)}{x}\right) \cdot \left(\frac{\left(1\right)}{x}\right)\right)\right)}{\left(\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\frac{\left(1\right)}{x}\right)}\right)}}\]
  4. Using strategy rm

  5. Applied difference-of-squares1.0

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

  6. Final simplification1.0

    \[\leadsto \frac{\left(\frac{1}{x + 1} + \frac{1}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{1}{x}\right)}{\frac{1}{x + 1} + \frac{1}{x}}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))