Average Error: 17.7 → 0.4
Time: 45.9s
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 2019022 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 44.8s)Debug log

sample278.0ms

Algorithm
intervals
Results
109.0ms104×body1280valid
54.0ms77×body640valid
44.0ms41×body320valid
38.0ms229×body80valid
7.0ms23×body160valid

simplify190.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
190.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.3b

localize44.0ms

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))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
9×add-sqr-sqrt
8×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×add-exp-log
5×add-cbrt-cube
5×pow1
4×associate-*r/
3×flip--
3×flip3--
3×sub-neg
2×difference-of-squares
2×distribute-lft-out--
2×*-commutative
2×associate-*l/
2×distribute-lft-in
2×distribute-rgt-in
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×diff-log
Counts
3 → 50
Calls
3 calls:
Slowest
13.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
6.0ms
(* J (- (exp l) (exp (- l))))
4.0ms
(- (exp l) (exp (- l)))

series224.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
94.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
73.0ms
(* J (- (exp l) (exp (- l))))
57.0ms
(- (exp l) (exp (- l)))

simplify1.7s

Counts
32 → 59
Calls
32 calls:
Slowest
372.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
165.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))
157.0ms
(* (* J (- (* (exp l) (exp l)) (* (exp (- l)) (exp (- l))))) (cos (/ K 2)))
148.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))
140.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune901.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize29.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* 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
(* 1/60 (pow l 5))
0.0b
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
11×associate-*r*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×pow1
3×*-commutative
2×flip-+
2×associate-*r/
2×associate-*l/
2×flip3-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×unpow3
1×sum-log
1×+-commutative
1×cube-mult
Counts
4 → 52
Calls
4 calls:
Slowest
17.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
4.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
1.0ms
(* 1/3 (pow l 3))
1.0ms
(* 1/60 (pow l 5))

series283.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
162.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
82.0ms
(+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))
20.0ms
(* 1/60 (pow l 5))
18.0ms
(* 1/3 (pow l 3))

simplify1.8s

Counts
28 → 64
Calls
28 calls:
Slowest
465.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
272.0ms
(* (* (* (* 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))))
270.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
250.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
168.0ms
(+ (log (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (log (cos (/ K 2))))

prune925.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

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))))
0.1b
(* 1/60 (pow l 5))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*r*
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×*-commutative
4×associate-*l/
3×associate-*l*
2×flip-+
2×associate-*r/
2×flip3-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×unpow3
1×cube-mult
Counts
4 → 54
Calls
4 calls:
Slowest
11.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
8.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
2.0ms
(* 1/60 (pow l 5))
1.0ms
(* 1/3 (pow l 3))

series273.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
162.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
64.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
31.0ms
(* 1/60 (pow l 5))
16.0ms
(* 1/3 (pow l 3))

simplify2.2s

Counts
32 → 66
Calls
32 calls:
Slowest
342.0ms
(* (- (* (* 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)))
266.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
251.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
245.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
202.0ms
(* (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3)) (cos (/ K 2)))

prune1.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

28.0b
(cbrt (* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
0.2b
(* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
0.2b
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
0.2b
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))

rewrite478.0ms

Algorithm
rewrite-expression-head
Rules
86×associate-*l/
80×associate-*r/
35×flip-+
35×flip3-+
35×pow1
32×frac-times
26×cbrt-div
20×add-exp-log
20×add-cbrt-cube
15×pow-prod-down
10×cbrt-unprod
10×prod-exp
7×associate-*r*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×pow-prod-up
3×*-commutative
3×associate-*l*
2×pow-plus
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×pow2
1×cbrt-prod
Counts
4 → 107
Calls
4 calls:
Slowest
316.0ms
(cbrt (* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
126.0ms
(* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
16.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
15.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))

series629.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
172.0ms
(* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))))
163.0ms
(cbrt (* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
155.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
139.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))

simplify20.3s

Counts
92 → 119
Calls
92 calls:
Slowest
793.0ms
(cbrt (* (* (* (* 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))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))))
666.0ms
(* (* (* (* (* 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)))) (* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))))
624.0ms
(* (* (* (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))) (* (* (* (* 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)))))
604.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
540.0ms
(* (- (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (- (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))

prune2.3s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes686.0ms

Accuracy

0% (0.3b remaining)

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

bsearch5.0ms

end0.0ms

sample10.3s

Algorithm
intervals
Results
3.8s2736×body1280valid
2.6s8371×body80valid
2.2s2553×body640valid
908.0ms1413×body320valid
271.0ms727×body160valid