Average Error: 14.5 → 0.3
Time: 1.6m
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 r6278532 = 1.0;
        double r6278533 = x;
        double r6278534 = r6278533 + r6278532;
        double r6278535 = r6278532 / r6278534;
        double r6278536 = r6278532 / r6278533;
        double r6278537 = r6278535 - r6278536;
        return r6278537;
}
↓
double f(double x) {
        double r6278538 = -1.0;
        double r6278539 = x;
        double r6278540 = fma(r6278539, r6278539, r6278539);
        double r6278541 = r6278538 / r6278540;
        return r6278541;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.5

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

  3. Applied frac-sub13.9

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))