Average Error: 14.5 → 0.4
Time: 1.6m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{(x \cdot x + x)_*}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-1}{(x \cdot x + x)_*}
double f(double x) {
        double r4616277 = 1.0;
        double r4616278 = x;
        double r4616279 = r4616278 + r4616277;
        double r4616280 = r4616277 / r4616279;
        double r4616281 = r4616277 / r4616278;
        double r4616282 = r4616280 - r4616281;
        return r4616282;
}
↓
double f(double x) {
        double r4616283 = -1.0;
        double r4616284 = x;
        double r4616285 = fma(r4616284, r4616284, r4616284);
        double r4616286 = r4616283 / r4616285;
        return r4616286;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.5

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

  3. Applied frac-sub13.8

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

    \[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))