Average Error: 0.6 → 0.7
Time: 17.7s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}
double f(double x) {
        double r2678291 = 1.0;
        double r2678292 = /* ERROR: no posit support in C */;
        double r2678293 = x;
        double r2678294 = r2678293 + r2678292;
        double r2678295 = r2678292 / r2678294;
        double r2678296 = r2678292 / r2678293;
        double r2678297 = r2678295 - r2678296;
        return r2678297;
}
↓
double f(double x) {
        double r2678298 = 1.0;
        double r2678299 = x;
        double r2678300 = r2678299 + r2678298;
        double r2678301 = r2678298 / r2678300;
        double r2678302 = r2678298 / r2678299;
        double r2678303 = r2678301 - r2678302;
        double r2678304 = r2678302 + r2678301;
        double r2678305 = r2678303 / r2678304;
        double r2678306 = 1.0;
        double r2678307 = r2678304 / r2678306;
        double r2678308 = r2678305 * r2678307;
        return r2678308;
}

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. Simplified1.0

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

  5. Simplified1.0

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

  7. Applied *p16-rgt-identity-expand1.0

    \[\leadsto \frac{\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\right) \cdot \left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right) \cdot \left(1.0\right)\right)}}\]

  8. Applied p16-times-frac0.7

    \[\leadsto \color{blue}{\left(\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\right)}{\left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)}{\left(1.0\right)}\right)}\]

  9. Final simplification0.7

    \[\leadsto \frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))