Average Error: 14.7 → 0.3
Time: 1.5m
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 r4389492 = 1.0;
        double r4389493 = x;
        double r4389494 = r4389493 + r4389492;
        double r4389495 = r4389492 / r4389494;
        double r4389496 = r4389492 / r4389493;
        double r4389497 = r4389495 - r4389496;
        return r4389497;
}
↓
double f(double x) {
        double r4389498 = -1.0;
        double r4389499 = x;
        double r4389500 = fma(r4389499, r4389499, r4389499);
        double r4389501 = r4389498 / r4389500;
        return r4389501;
}

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-sub14.1

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))