Average Error: 13.9 → 0.4
Time: 27.6s
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 r1597452 = 1.0;
        double r1597453 = x;
        double r1597454 = r1597453 + r1597452;
        double r1597455 = r1597452 / r1597454;
        double r1597456 = r1597452 / r1597453;
        double r1597457 = r1597455 - r1597456;
        return r1597457;
}
↓
double f(double x) {
        double r1597458 = -1.0;
        double r1597459 = x;
        double r1597460 = fma(r1597459, r1597459, r1597459);
        double r1597461 = r1597458 / r1597460;
        return r1597461;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.9

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

  3. Applied frac-sub13.3

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

  4. Simplified13.3

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

  5. Simplified13.3

    \[\leadsto \frac{\left(x - 1\right) - x}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity13.3

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

  8. Applied associate-/r*13.3

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))