Average Error: 0.6 → 0.9
Time: 24.3s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}
double f(double x) {
        double r5915889 = 1.0;
        double r5915890 = /* ERROR: no posit support in C */;
        double r5915891 = x;
        double r5915892 = r5915891 + r5915890;
        double r5915893 = r5915890 / r5915892;
        double r5915894 = r5915890 / r5915891;
        double r5915895 = r5915893 - r5915894;
        return r5915895;
}
↓
double f(double x) {
        double r5915896 = 1.0;
        double r5915897 = x;
        double r5915898 = r5915897 + r5915896;
        double r5915899 = r5915896 / r5915898;
        double r5915900 = r5915896 / r5915897;
        double r5915901 = r5915899 + r5915900;
        double r5915902 = r5915899 - r5915900;
        double r5915903 = r5915901 / r5915902;
        double r5915904 = r5915901 / r5915903;
        return r5915904;
}

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. Applied associate-/l*0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2019128 +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)))