Average Error: 17.3 → 0.6
Time: 25.4s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(J \cdot \ell\right)\right)\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.3

    \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  2. Initial simplification17.3

    \[\leadsto U + \left(J \cdot e^{\ell} - \frac{J}{e^{\ell}}\right) \cdot \cos \left(\frac{K}{2}\right)\]

  3. Taylor expanded around 0 0.6

    \[\leadsto U + \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right)\]

  4. Final simplification0.6

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot \left(J \cdot \ell\right)\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 10.5s)Debug log

start375.0ms

Algorithm
intervals

setup207.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 20.0b

localize52.0ms

Local error

Found 3 expressions with local error:

5.9b
(- (* J (exp l)) (/ J (exp l)))
0.0b
(* (- (* J (exp l)) (/ J (exp l))) (cos (/ K 2)))
0.0b
(/ J (exp l))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

20.0ms
(* (- (* J (exp l)) (/ J (exp l))) (cos (/ K 2)))
7.0ms
(- (* J (exp l)) (/ J (exp l)))
1.0ms
(/ J (exp l))

series224.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(* (- (* J (exp l)) (/ J (exp l))) (cos (/ K 2)))
85.0ms
(- (* J (exp l)) (/ J (exp l)))
47.0ms
(/ J (exp l))

simplify1.9s

Counts
24 → 55
Calls

24 calls. Slowest were:

358.0ms
(* -1 (* (cos (* 1/2 K)) (* J (- (/ 1 (exp l)) (exp l)))))
340.0ms
(* -1 (* J (- (/ 1 (exp l)) (exp l))))
303.0ms
(* (- (* (* J (exp l)) (* J (exp l))) (* (/ J (exp l)) (/ J (exp l)))) (cos (/ K 2)))

prune782.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize10.0ms

Local error

Found 3 expressions with local error:

0.1b
(* (* 2 (* J l)) (cos (/ K 2)))
0.0b
(* 2 (* J l))
0.0b
(+ U (* (* 2 (* J l)) (cos (/ K 2))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 34
Calls

3 calls. Slowest were:

6.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
3.0ms
(+ U (* (* 2 (* J l)) (cos (/ K 2))))
1.0ms
(* 2 (* J l))

series156.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

71.0ms
(+ U (* (* 2 (* J l)) (cos (/ K 2))))
66.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
18.0ms
(* 2 (* J l))

simplify1.2s

Counts
15 → 43
Calls

15 calls. Slowest were:

790.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
58.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)
57.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)

prune613.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.1b

localize14.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))
0.2b
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series691.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

253.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
172.0ms
(cbrt (cos (/ K 2)))
138.0ms
(cbrt (cos (/ K 2)))
128.0ms
(cbrt (cos (/ K 2)))

simplify346.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

86.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
72.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
65.0ms
(- 1 (+ (* 1/25920 (pow K 6)) (* 1/12 (pow K 2))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.1b

localize41.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))
0.3b
(cbrt (cos (/ K 2)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series685.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

203.0ms
(cbrt (cos (/ K 2)))
182.0ms
(cbrt (cos (/ K 2)))
163.0ms
(cbrt (cos (/ K 2)))
137.0ms
(cbrt (cos (/ K 2)))

simplify316.0ms

Counts
24 → 56
Calls

24 calls. Slowest were:

69.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
69.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
67.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))

prune1.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.1b

regimes540.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch1.0ms