Average Error: 14.4 → 0.4
Time: 19.2s
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 r1158800 = 1.0;
        double r1158801 = x;
        double r1158802 = r1158801 + r1158800;
        double r1158803 = r1158800 / r1158802;
        double r1158804 = r1158800 / r1158801;
        double r1158805 = r1158803 - r1158804;
        return r1158805;
}
↓
double f(double x) {
        double r1158806 = x;
        double r1158807 = r1158806 - r1158806;
        double r1158808 = 1.0;
        double r1158809 = r1158807 - r1158808;
        double r1158810 = fma(r1158806, r1158806, r1158806);
        double r1158811 = r1158809 / r1158810;
        return r1158811;
}

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-sub13.7

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

  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 2019138 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))