Average Error: 17.6 → 0.4
Time: 30.3s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(\left(\ell \cdot 2 + \left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right)\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\]

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 17.6

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Final simplification0.4

    \[\leadsto U + \left(\left(\ell \cdot 2 + \left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right)\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\]

Reproduce

herbie shell --seed 2018360 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 16.7s)Debug log

start353.0ms

Algorithm
intervals

setup147.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 19.0b

localize37.0ms

Local error

Found 3 expressions with local error:

7.0b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

17.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
14.0ms
(* J (- (exp l) (exp (- l))))
4.0ms
(- (exp l) (exp (- l)))

series239.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

113.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
66.0ms
(* J (- (exp l) (exp (- l))))
60.0ms
(- (exp l) (exp (- l)))

simplify1.5s

Counts
32 → 59
Calls

32 calls. Slowest were:

301.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
169.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
153.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))

prune993.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize34.0ms

Local error

Found 4 expressions with local error:

0.1b
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
0.1b
(* 1/3 (pow l 3))
0.0b
(* 1/60 (pow l 5))
0.0b
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

33.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
7.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
2.0ms
(* 1/3 (pow l 3))

series310.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

166.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
78.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
36.0ms
(* 1/60 (pow l 5))
30.0ms
(* 1/3 (pow l 3))

simplify2.5s

Counts
28 → 64
Calls

28 calls. Slowest were:

554.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
511.0ms
(* (* J (- (* (* 2 l) (* 2 l)) (* (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (cos (/ K 2)))
374.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.5s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.1b
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
0.1b
(* 1/3 (pow l 3))
0.1b
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
0.0b
(* 1/60 (pow l 5))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

23.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
14.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
2.0ms
(* 1/3 (pow l 3))

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

149.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
62.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
27.0ms
(* 1/3 (pow l 3))
20.0ms
(* 1/60 (pow l 5))

simplify2.4s

Counts
32 → 66
Calls

32 calls. Slowest were:

453.0ms
(* (- (* (* 2 l) (* 2 l)) (* (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
373.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
308.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.7s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

localize35.0ms

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))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series755.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

210.0ms
(cbrt (cos (/ K 2)))
200.0ms
(cbrt (cos (/ K 2)))
195.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
150.0ms
(cbrt (cos (/ K 2)))

simplify341.0ms

Counts
41 → 73
Calls

41 calls. Slowest were:

68.0ms
(- 1 (+ (* 1/25920 (pow K 6)) (* 1/12 (pow K 2))))
68.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
65.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))

prune2.0s

Pruning

9 alts after pruning (6 fresh and 3 done)

Merged error: 0.0b

regimes1.4s

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch5.0ms