Average Error: 14.6 → 0.1
Time: 32.6s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{1}{-1 - x}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{1}{-1 - x}}{x}
double f(double x) {
        double r764316 = 1.0;
        double r764317 = x;
        double r764318 = r764317 + r764316;
        double r764319 = r764316 / r764318;
        double r764320 = r764316 / r764317;
        double r764321 = r764319 - r764320;
        return r764321;
}
↓
double f(double x) {
        double r764322 = 1.0;
        double r764323 = -1.0;
        double r764324 = x;
        double r764325 = r764323 - r764324;
        double r764326 = r764322 / r764325;
        double r764327 = r764326 / r764324;
        return r764327;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.6

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  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. Simplified13.9

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

  5. Simplified13.9

    \[\leadsto \frac{\left(x - 1\right) - x}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}\]
  6. Using strategy rm

  7. Applied frac-2neg13.9

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

  8. Simplified0.4

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

  9. Simplified0.4

    \[\leadsto \frac{1}{\color{blue}{\left(-1 - x\right) \cdot x}}\]
  10. Using strategy rm

  11. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{-1 - x}}{x}}\]

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))