Average Error: 14.2 → 0.1
Time: 1.8s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-\frac{1}{x + 1}}{\frac{1}{1} \cdot \frac{x}{1}}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-\frac{1}{x + 1}}{\frac{1}{1} \cdot \frac{x}{1}}
double code(double x) {
	return ((1.0 / (x + 1.0)) - (1.0 / x));
}
↓
double code(double x) {
	return (-(1.0 / (x + 1.0)) / ((1.0 / 1.0) * (x / 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

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

  3. Applied clear-num14.2

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

  4. Applied clear-num14.2

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

  5. Applied frac-sub13.6

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

  6. Simplified13.6

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{-1}}{\frac{x + 1}{1} \cdot \frac{x}{1}}\]
  8. Using strategy rm

  9. Applied div-inv0.3

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

  10. Applied associate-*l*0.3

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

  11. Applied associate-/r*0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020071 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))