Average Error: 14.6 → 0.1
Time: 2.1s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{1 \cdot \frac{-1}{x}}{1 + x}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.6

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

  3. Applied frac-sub14.0

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

  4. Simplified0.3

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

  5. Simplified0.3

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

  7. Applied associate-/r*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))