Average Error: 0.0 → 0.0
Time: 8.0s
Precision: 64
Internal Precision: 128
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 1)_*}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}\]

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))

Details

Time bar (total: 7.8s)Debug log

sample82.0ms

Algorithm
intervals

simplify135.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
135.0ms
(/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))
2.3b
(- 2 (/ 2 (+ 1 t)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×flip--
4×log1p-expm1-u
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×flip3--
4×*-un-lft-identity
4×sub-neg
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
Counts
4 → 52
Calls
4 calls:
Slowest
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))
1.0ms
(- 2 (/ 2 (+ 1 t)))

series59.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
15.0ms
(- 2 (/ 2 (+ 1 t)))
15.0ms
(- 2 (/ 2 (+ 1 t)))
15.0ms
(- 2 (/ 2 (+ 1 t)))
13.0ms
(- 2 (/ 2 (+ 1 t)))

simplify4.8s

Counts
12 → 64
Calls
12 calls:
Slowest
683.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
682.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
676.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
640.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
290.0ms
(- (+ (* 2 (/ 1 (pow t 2))) 2) (* 2 (/ 1 t)))

prune736.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0.0b

end0.0ms

sample1.9s

Algorithm
intervals