Average Error: 14.7 → 0.4
Time: 1.7m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{\mathsf{fma}\left(x, x, x\right)}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-1}{\mathsf{fma}\left(x, x, x\right)}
double f(double x) {
        double r4587925 = 1.0;
        double r4587926 = x;
        double r4587927 = r4587926 + r4587925;
        double r4587928 = r4587925 / r4587927;
        double r4587929 = r4587925 / r4587926;
        double r4587930 = r4587928 - r4587929;
        return r4587930;
}
↓
double f(double x) {
        double r4587931 = -1.0;
        double r4587932 = x;
        double r4587933 = fma(r4587932, r4587932, r4587932);
        double r4587934 = r4587931 / r4587933;
        return r4587934;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.7

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

  3. Applied frac-sub14.0

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

  4. Simplified0.4

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

  5. Simplified0.4

    \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}\]

  6. Final simplification0.4

    \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x, x\right)}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))