Average Error: 0.6 → 0.9
Time: 11.5s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}\]
double f(double x) {
        double r3182957 = 1.0;
        double r3182958 = x;
        double r3182959 = r3182958 + r3182957;
        double r3182960 = r3182957 / r3182959;
        double r3182961 = r3182957 / r3182958;
        double r3182962 = r3182960 - r3182961;
        return r3182962;
}
↓
double f(double x) {
        double r3182963 = 1.0;
        double r3182964 = x;
        double r3182965 = r3182964 + r3182963;
        double r3182966 = r3182963 / r3182965;
        double r3182967 = r3182963 / r3182964;
        double r3182968 = r3182966 + r3182967;
        double r3182969 = r3182966 - r3182967;
        double r3182970 = r3182968 / r3182969;
        double r3182971 = r3182968 / r3182970;
        return r3182971;
}


\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{\frac{1}{x + 1} + \frac{1}{x}}{\frac{1}{x + 1} - \frac{1}{x}}}

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 2019101 +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)))