Average Error: 17.0 → 0.5
Time: 25.1s
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 17.0

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

  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 2019008 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 24.3s)Debug log

sample309.0ms

Algorithm
intervals

simplify571.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

570.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.2b

localize48.0ms

Local error

Found 3 expressions with local error:

6.4b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

37.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
10.0ms
(- (exp l) (exp (- l)))
9.0ms
(* J (- (exp l) (exp (- l))))

series206.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
75.0ms
(* J (- (exp l) (exp (- l))))
38.0ms
(- (exp l) (exp (- l)))

simplify4.0s

Counts
70 → 95
Calls

70 calls. Slowest were:

473.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
188.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))
179.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune1.5s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize26.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (* 2 (* J l)) (cos (/ K 2)))
0.0b
(* 2 (* J l))
0.0b
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

11.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
6.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
1.0ms
(* 2 (* J l))

series147.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

91.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
45.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
11.0ms
(* 2 (* J l))

simplify1.1s

Counts
16 → 50
Calls

16 calls. Slowest were:

696.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
80.0ms
(+ (log (* 2 (* J l))) (log (cos (/ K 2))))
71.0ms
(* (* 2 (* J l)) (cos (/ K 2)))

prune576.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize29.0ms

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))) (cbrt (cos (/ K 2))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

11.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series733.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

233.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
181.0ms
(cbrt (cos (/ K 2)))
180.0ms
(cbrt (cos (/ K 2)))
139.0ms
(cbrt (cos (/ K 2)))

simplify557.0ms

Counts
41 → 81
Calls

41 calls. Slowest were:

125.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
123.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
114.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))

prune1.1s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

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

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

17.0ms
(* (* 2 (* J l)) (cbrt (* (cos (/ K 2)) (cos (/ K 2)))))
11.0ms
(* (cos (/ K 2)) (cos (/ K 2)))
5.0ms
(cbrt (* (cos (/ K 2)) (cos (/ K 2))))

series572.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

229.0ms
(cbrt (* (cos (/ K 2)) (cos (/ K 2))))
165.0ms
(cbrt (cos (/ K 2)))
163.0ms
(* (* 2 (* J l)) (cbrt (* (cos (/ K 2)) (cos (/ K 2)))))
16.0ms
(* (cos (/ K 2)) (cos (/ K 2)))

simplify1.2s

Counts
38 → 80
Calls

38 calls. Slowest were:

605.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (cos (/ K 2)) (cos (/ K 2))))
146.0ms
(- 1 (+ (* 1/25920 (pow K 6)) (* 1/12 (pow K 2))))
116.0ms
(- 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: 0.0b

regimes540.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.8s

Algorithm
intervals