Average Error: 17.8 → 0.4
Time: 35.8s
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 17.8

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

  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 2019004 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 34.4s)Debug log

sample368.0ms

Algorithm
intervals

simplify191.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

190.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.5b

localize26.0ms

Local error

Found 4 expressions with local error:

6.3b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)
0.0b
(* J (- (exp l) (exp (- l))))

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

19.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)
15.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
7.0ms
(* J (- (exp l) (exp (- l))))

series482.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

253.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)
119.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
57.0ms
(* J (- (exp l) (exp (- l))))
52.0ms
(- (exp l) (exp (- l)))

simplify2.9s

Counts
35 → 72
Calls

35 calls. Slowest were:

712.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
444.0ms
(- (+ U (* (cos (* 1/2 K)) (* (exp l) J))) (* (cos (* 1/2 K)) (* J (exp (* -1 l)))))
304.0ms
(- (+ U (* (exp l) (* (cos (* 1/2 K)) J))) (* (cos (* 1/2 K)) (* J (exp (- l)))))

prune955.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize36.0ms

Local error

Found 4 expressions with local error:

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

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

21.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
17.0ms
(+ (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) U)
1.0ms
(* 1/3 (pow l 3))

series518.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

321.0ms
(+ (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) U)
148.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
34.0ms
(* 1/60 (pow l 5))
15.0ms
(* 1/3 (pow l 3))

simplify2.2s

Counts
27 → 63
Calls

27 calls. Slowest were:

571.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
252.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
242.0ms
(* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))

prune1.1s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

13.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
12.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
2.0ms
(* 1/3 (pow l 3))

series271.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

136.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
86.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
34.0ms
(* 1/60 (pow l 5))
15.0ms
(* 1/3 (pow l 3))

simplify2.1s

Counts
32 → 66
Calls

32 calls. Slowest were:

336.0ms
(* (- (* (* 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)))
239.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
225.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.2s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

15.6b
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))
15.6b
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))
15.6b
(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))))))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

44.0ms
(* (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))))))
4.0ms
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))
4.0ms
(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:

323.0ms
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))
318.0ms
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))
302.0ms
(* (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))))))
286.0ms
(cbrt (* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))))

simplify7.6s

Counts
49 → 77
Calls

49 calls. Slowest were:

390.0ms
(+ (* 20/3 (/ (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))) (pow l 2))) (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))))
390.0ms
(cbrt (* J (* (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3)) (cos (/ K 2)))))
381.0ms
(cbrt (* J (* (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3)) (cos (/ K 2)))))

prune1.7s

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 0.0b

regimes567.0ms

Accuracy

0% (0.4b remaining)

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

bsearch5.0ms

end0.0ms

sample10.5s

Algorithm
intervals