Average Error: 16.9 → 0.5
Time: 25.7s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\cos \left(\frac{K}{2}\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right) + U\]

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 16.9

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019005 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 24.5s)Debug log

sample347.0ms

Algorithm
intervals

simplify212.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

212.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.9b

localize55.0ms

Local error

Found 3 expressions with local error:

6.1b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

22.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
9.0ms
(- (exp l) (exp (- l)))
8.0ms
(* J (- (exp l) (exp (- l))))

series234.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

114.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
77.0ms
(* J (- (exp l) (exp (- l))))
43.0ms
(- (exp l) (exp (- l)))

simplify1.8s

Counts
32 → 59
Calls

32 calls. Slowest were:

400.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
189.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
184.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))

prune938.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 2 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

15.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
12.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)

series195.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

119.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
75.0ms
(* (* 2 (* J l)) (cos (/ K 2)))

simplify941.0ms

Counts
12 → 31
Calls

12 calls. Slowest were:

667.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
65.0ms
(+ (log (* 2 (* J l))) (log (cos (/ K 2))))
40.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) U)

prune393.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
0.5b
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

26.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
5.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
4.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

series777.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

239.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
192.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
186.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
159.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

simplify2.5s

Counts
31 → 63
Calls

31 calls. Slowest were:

256.0ms
(* (exp (* 1/3 (+ (* 2 (log J)) (* 2 (log l))))) (pow (cbrt 2) 2))
215.0ms
(* (exp (* 1/3 (- (log (cos (* 1/2 K))) (+ (log (/ 1 l)) (log (/ 1 J)))))) (cbrt 2))
206.0ms
(* (exp (* 1/3 (- (log (cos (* 1/2 K))) (+ (log (/ 1 l)) (log (/ 1 J)))))) (cbrt 2))

prune995.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
12.8b
(cbrt (* 2 (* J l)))
0.5b
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (* (cbrt (* 2 (* J l))) (cbrt (cos (/ K 2)))))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

27.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (* (cbrt (* 2 (* J l))) (cbrt (cos (/ K 2)))))
3.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
3.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

series763.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

252.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (* (cbrt (* 2 (* J l))) (cbrt (cos (/ K 2)))))
199.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
189.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
122.0ms
(cbrt (* 2 (* J l)))

simplify2.4s

Counts
28 → 60
Calls

28 calls. Slowest were:

255.0ms
(* (exp (* 1/3 (+ (* 2 (log J)) (* 2 (log l))))) (pow (cbrt 2) 2))
205.0ms
(* (pow (cbrt 2) 2) (exp (* 1/3 (- (log (pow (cos (* 1/2 K)) 2)) (+ (* 2 (log (/ 1 l))) (* 2 (log (/ 1 J))))))))
187.0ms
(* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))

prune1.0s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes481.0ms

Accuracy

0% (0.3b remaining)

Error of 0.5b against oracle of 0.2b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample10.2s

Algorithm
intervals