Average Error: 14.3 → 0.4
Time: 1.5m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{(x \cdot x + x)_*}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-1}{(x \cdot x + x)_*}
double f(double x) {
        double r4397508 = 1.0;
        double r4397509 = x;
        double r4397510 = r4397509 + r4397508;
        double r4397511 = r4397508 / r4397510;
        double r4397512 = r4397508 / r4397509;
        double r4397513 = r4397511 - r4397512;
        return r4397513;
}
↓
double f(double x) {
        double r4397514 = -1.0;
        double r4397515 = x;
        double r4397516 = fma(r4397515, r4397515, r4397515);
        double r4397517 = r4397514 / r4397516;
        return r4397517;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.3

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  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

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

  5. Simplified0.4

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

  6. Final simplification0.4

    \[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))