Average Error: 14.8 → 0.4
Time: 41.6s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\left(x - x\right) - 1}{\mathsf{fma}\left(x, x, x\right)}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\left(x - x\right) - 1}{\mathsf{fma}\left(x, x, x\right)}
double f(double x) {
        double r1683363 = 1.0;
        double r1683364 = x;
        double r1683365 = r1683364 + r1683363;
        double r1683366 = r1683363 / r1683365;
        double r1683367 = r1683363 / r1683364;
        double r1683368 = r1683366 - r1683367;
        return r1683368;
}
↓
double f(double x) {
        double r1683369 = x;
        double r1683370 = r1683369 - r1683369;
        double r1683371 = 1.0;
        double r1683372 = r1683370 - r1683371;
        double r1683373 = fma(r1683369, r1683369, r1683369);
        double r1683374 = r1683372 / r1683373;
        return r1683374;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.8

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  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. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))