Average Error: 14.5 → 0.4
Time: 1.3m
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 r4161996 = 1.0;
        double r4161997 = x;
        double r4161998 = r4161997 + r4161996;
        double r4161999 = r4161996 / r4161998;
        double r4162000 = r4161996 / r4161997;
        double r4162001 = r4161999 - r4162000;
        return r4162001;
}
↓
double f(double x) {
        double r4162002 = -1.0;
        double r4162003 = x;
        double r4162004 = fma(r4162003, r4162003, r4162003);
        double r4162005 = r4162002 / r4162004;
        return r4162005;
}

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-sub13.9

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

  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 2019104 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))