Average Error: 0.0 → 0.0
Time: 18.2s
Precision: 64
Internal Precision: 128
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2}\]

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. Using strategy rm

  4. Applied add-cube-cbrt0.0

    \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right)}}\]

  5. Final simplification0.0

    \[\leadsto 1 - \frac{1}{\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2}\]

Reproduce

herbie shell --seed 2018362 
(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: 14.0s)Debug log

start96.0ms

Algorithm
intervals

setup878.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
0.0b
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
0.0b
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

4.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
1.0ms
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
1.0ms
(- 2 (/ 2 (+ 1 t)))

series73.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

37.0ms
(/ 1 (+ 2 (* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))))
13.0ms
(* (- 2 (/ 2 (+ 1 t))) (- 2 (/ 2 (+ 1 t))))
12.0ms
(- 2 (/ 2 (+ 1 t)))
11.0ms
(- 2 (/ 2 (+ 1 t)))

simplify8.0s

Counts
45 → 84
Calls

45 calls. Slowest were:

685.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- 2 (/ 2 (+ 1 t))))
667.0ms
(* (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)) (- (pow 2 3) (pow (/ 2 (+ 1 t)) 3)))
647.0ms
(* (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))) (+ (* 2 2) (+ (* (/ 2 (+ 1 t)) (/ 2 (+ 1 t))) (* 2 (/ 2 (+ 1 t))))))

prune934.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize32.0ms

Local error

Found 4 expressions with local error:

4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))
4.8b
(- 2 (/ 2 (+ 1 t)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

1.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))
0.0ms
(- 2 (/ 2 (+ 1 t)))

series54.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

16.0ms
(- 2 (/ 2 (+ 1 t)))
15.0ms
(- 2 (/ 2 (+ 1 t)))
14.0ms
(- 2 (/ 2 (+ 1 t)))
9.0ms
(- 2 (/ 2 (+ 1 t)))

simplify3.0s

Counts
12 → 56
Calls

12 calls. Slowest were:

537.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
512.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
475.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune805.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes34.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch2.0ms