Average Error: 14.4 → 0.4
Time: 9.8m
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 r22788934 = 1.0;
        double r22788935 = x;
        double r22788936 = r22788935 + r22788934;
        double r22788937 = r22788934 / r22788936;
        double r22788938 = r22788934 / r22788935;
        double r22788939 = r22788937 - r22788938;
        return r22788939;
}
↓
double f(double x) {
        double r22788940 = -1.0;
        double r22788941 = x;
        double r22788942 = fma(r22788941, r22788941, r22788941);
        double r22788943 = r22788940 / r22788942;
        return r22788943;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.4

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

  3. Applied frac-sub13.7

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

  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 2019112 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))