Average Error: 0.6 → 0.6
Time: 37.6s
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 r2355299 = 1.0;
        double r2355300 = /* ERROR: no posit support in C */;
        double r2355301 = x;
        double r2355302 = r2355301 + r2355300;
        double r2355303 = r2355300 / r2355302;
        double r2355304 = r2355300 / r2355301;
        double r2355305 = r2355303 - r2355304;
        return r2355305;
}
↓
double f(double x) {
        double r2355306 = 1.0;
        double r2355307 = x;
        double r2355308 = r2355307 + r2355306;
        double r2355309 = r2355306 / r2355308;
        double r2355310 = r2355306 / r2355307;
        double r2355311 = r2355309 - r2355310;
        double r2355312 = r2355310 + r2355309;
        double r2355313 = r2355311 / r2355312;
        double r2355314 = 1.0;
        double r2355315 = r2355312 / r2355314;
        double r2355316 = r2355313 * r2355315;
        return r2355316;
}

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-frac0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))