Average Error: 0.0 → 0.0

Time: 8.1s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `t`

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 0.0
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
Initial simplification0.0
\[\leadsto \frac{1 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]
Final simplification0.0
\[\leadsto \frac{1 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]

Runtime

start59.0ms

Algorithm: intervals

setup145.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

3.8b	(- 2 (/ 2 (+ 1 t)))
3.8b	(- 2 (/ 2 (+ 1 t)))
3.8b	(- 2 (/ 2 (+ 1 t)))
3.8b	(- 2 (/ 2 (+ 1 t)))

rewrite2.0ms

Algorithm

rewrite-expression-head

Counts

4 → 44

Calls

4 calls. Slowest were:

0.0ms	(- 2 (/ 2 (+ 1 t)))
0.0ms	(- 2 (/ 2 (+ 1 t)))
0.0ms	(- 2 (/ 2 (+ 1 t)))

series41.0ms

Counts

4 → 12

Calls

4 calls. Slowest were:

12.0ms	(- 2 (/ 2 (+ 1 t)))
10.0ms	(- 2 (/ 2 (+ 1 t)))
10.0ms	(- 2 (/ 2 (+ 1 t)))

simplify3.1s

Counts

12 → 56

Calls

12 calls. Slowest were:

549.0ms	(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
502.0ms	(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
500.0ms	(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune620.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018359 
(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))))))))