Average Error: 0.0 → 0.0
Time: 13.3s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Initial simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]

  3. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))

Details

Time bar (total: 8.9s)Debug log

start62.0ms

Algorithm
intervals

setup476.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 4 expressions with local error:

3.3b
(- 2 (/ 2 (+ 1 t)))
3.3b
(- 2 (/ 2 (+ 1 t)))
0.0b
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
0.0b
(/ 2 (+ 1 t))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

8.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
1.0ms
(/ 2 (+ 1 t))
1.0ms
(- 2 (/ 2 (+ 1 t)))

series49.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

21.0ms
(- 2 (/ 2 (+ 1 t)))
12.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
9.0ms
(- 2 (/ 2 (+ 1 t)))
7.0ms
(/ 2 (+ 1 t))

simplify7.3s

Counts
44 → 81
Calls

44 calls. Slowest were:

775.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)))
644.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- 2 (/ 2 (+ 1 t))))
588.0ms
(* (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))) (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))))

prune938.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b