Average Error: 14.4 → 0.4
Time: 17.0s
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 r1436372 = 1.0;
        double r1436373 = x;
        double r1436374 = r1436373 + r1436372;
        double r1436375 = r1436372 / r1436374;
        double r1436376 = r1436372 / r1436373;
        double r1436377 = r1436375 - r1436376;
        return r1436377;
}
↓
double f(double x) {
        double r1436378 = x;
        double r1436379 = r1436378 - r1436378;
        double r1436380 = 1.0;
        double r1436381 = r1436379 - r1436380;
        double r1436382 = fma(r1436378, r1436378, r1436378);
        double r1436383 = r1436381 / r1436382;
        return r1436383;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.4

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

  3. Applied frac-sub13.8

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

  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 2019141 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))