Average Error: 17.7 → 0.4
Time: 35.2s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + J \cdot \left(\left(\ell \cdot 2 + \left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]

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 17.7

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

  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. Using strategy rm

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019007 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 34.0s)Debug log

sample396.0ms

Algorithm
intervals

simplify168.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

168.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 17.2b

localize40.0ms

Local error

Found 3 expressions with local error:

7.5b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

18.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
7.0ms
(* J (- (exp l) (exp (- l))))
4.0ms
(- (exp l) (exp (- l)))

series548.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

428.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
72.0ms
(* J (- (exp l) (exp (- l))))
48.0ms
(- (exp l) (exp (- l)))

simplify1.6s

Counts
32 → 59
Calls

32 calls. Slowest were:

387.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
155.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))
148.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune815.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize33.0ms

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)))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

27.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
6.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
2.0ms
(* 1/3 (pow l 3))

series268.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

159.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
70.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
24.0ms
(* 1/60 (pow l 5))
16.0ms
(* 1/3 (pow l 3))

simplify2.0s

Counts
28 → 64
Calls

28 calls. Slowest were:

612.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
303.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
274.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.1s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

27.0ms
(* (+ (* 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))))
3.0ms
(* 1/60 (pow l 5))

series298.0ms

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))))
69.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
35.0ms
(* 1/60 (pow l 5))
28.0ms
(* 1/3 (pow l 3))

simplify2.5s

Counts
32 → 66
Calls

32 calls. Slowest were:

415.0ms
(* (- (* (* 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)))
323.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
286.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune965.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

14.4b
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
14.4b
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
14.4b
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
0.5b
(* (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))

rewrite95.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

43.0ms
(* (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
25.0ms
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
13.0ms
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

340.0ms
(* (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
303.0ms
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
301.0ms
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
300.0ms
(cbrt (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))

simplify8.8s

Counts
49 → 77
Calls

49 calls. Slowest were:

630.0ms
(cbrt (* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2))))
578.0ms
(cbrt (* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2))))
545.0ms
(* (cbrt (* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))) (cbrt (* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))))

prune1.8s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes600.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample10.6s

Algorithm
intervals