Average Error: 15.2 → 0.1
Time: 46.6s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{x}}{x + 1}\]
double f(double x) {
        double r4097283 = 1.0;
        double r4097284 = x;
        double r4097285 = r4097284 + r4097283;
        double r4097286 = r4097283 / r4097285;
        double r4097287 = r4097283 / r4097284;
        double r4097288 = r4097286 - r4097287;
        return r4097288;
}
↓
double f(double x) {
        double r4097289 = -1.0;
        double r4097290 = x;
        double r4097291 = r4097289 / r4097290;
        double r4097292 = 1.0;
        double r4097293 = r4097290 + r4097292;
        double r4097294 = r4097291 / r4097293;
        return r4097294;
}


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

Error

Bits error versus x

Derivation

  1. Initial program 15.2

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  2. Using strategy rm

  3. Applied frac-sub14.6

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

  4. Simplified0.3

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

  5. Simplified0.3

    \[\leadsto \frac{-1}{\color{blue}{x \cdot x + x}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.3

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

  8. Applied distribute-rgt-out0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))