Average Error: 15.1 → 0.4
Time: 21.4s
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 r1528854 = 1.0;
        double r1528855 = x;
        double r1528856 = r1528855 + r1528854;
        double r1528857 = r1528854 / r1528856;
        double r1528858 = r1528854 / r1528855;
        double r1528859 = r1528857 - r1528858;
        return r1528859;
}
↓
double f(double x) {
        double r1528860 = x;
        double r1528861 = r1528860 - r1528860;
        double r1528862 = 1.0;
        double r1528863 = r1528861 - r1528862;
        double r1528864 = fma(r1528860, r1528860, r1528860);
        double r1528865 = r1528863 / r1528864;
        return r1528865;
}

Error

Bits error versus x

Derivation

  1. Initial program 15.1

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

  3. Applied frac-sub14.5

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

  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 2019139 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))