2frac (problem 3.3.1)

Average Error: 14.4 → 0.1

Time: 6.6min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 14.4

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

3. Applied frac-sub 13.7

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

4. Simplified 0.4

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

6. Applied associate-/r* 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))