Average Error: 0.6 → 0.6
Time: 16.6s
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 r2210370 = 1.0;
        double r2210371 = /* ERROR: no posit support in C */;
        double r2210372 = x;
        double r2210373 = r2210372 + r2210371;
        double r2210374 = r2210371 / r2210373;
        double r2210375 = r2210371 / r2210372;
        double r2210376 = r2210374 - r2210375;
        return r2210376;
}
↓
double f(double x) {
        double r2210377 = 1.0;
        double r2210378 = x;
        double r2210379 = r2210378 + r2210377;
        double r2210380 = r2210377 / r2210379;
        double r2210381 = r2210377 / r2210378;
        double r2210382 = r2210380 - r2210381;
        double r2210383 = r2210381 + r2210380;
        double r2210384 = r2210382 / r2210383;
        double r2210385 = 1.0;
        double r2210386 = r2210383 / r2210385;
        double r2210387 = r2210384 * r2210386;
        return r2210387;
}

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 +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)))