Average Error: 17.6 → 0.4
Time: 36.2s
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 17.6

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

  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 2019010 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Details

Time bar (total: 34.8s)Debug log

sample399.0ms

Algorithm
intervals

simplify204.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
204.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.9b

localize42.0ms

Local error

Found 3 expressions with local error:

4.9b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite32.0ms

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
15.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
8.0ms
(- (exp l) (exp (- l)))
7.0ms
(* J (- (exp l) (exp (- l))))

series256.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
101.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
98.0ms
(* J (- (exp l) (exp (- l))))
57.0ms
(- (exp l) (exp (- l)))

simplify1.7s

Counts
32 → 59
Calls
32 calls:
Slowest
401.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
175.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))
163.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
132.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))
116.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))

prune896.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize55.0ms

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

rewrite54.0ms

Algorithm
rewrite-expression-head
Rules
14×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-*r/
2×flip-+
2×associate-*l/
2×flip3-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×unpow3
1×distribute-lft-in
1×cube-mult
1×distribute-rgt-in
Counts
4 → 56
Calls
4 calls:
Slowest
32.0ms
(* (* 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)))))
2.0ms
(* 1/3 (pow l 3))
2.0ms
(* 1/60 (pow l 5))

series272.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
146.0ms
(* (* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cos (/ K 2)))
79.0ms
(* J (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
30.0ms
(* 1/60 (pow l 5))
17.0ms
(* 1/3 (pow l 3))

simplify3.4s

Counts
34 → 68
Calls
34 calls:
Slowest
557.0ms
(* (* J (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3))) (cos (/ K 2)))
433.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
419.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
312.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
273.0ms
(* (* (* (* 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
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
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))))
0.0b
(* 1/60 (pow l 5))

rewrite25.0ms

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
15.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
7.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
1.0ms
(* 1/3 (pow l 3))
1.0ms
(* 1/60 (pow l 5))

series256.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
143.0ms
(* J (* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2))))
67.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (cos (/ K 2)))
26.0ms
(* 1/3 (pow l 3))
20.0ms
(* 1/60 (pow l 5))

simplify2.6s

Counts
32 → 66
Calls
32 calls:
Slowest
403.0ms
(* (- (* (* 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)))
300.0ms
(* (+ (pow (* 2 l) 3) (pow (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) 3)) (cos (/ K 2)))
296.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
287.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
272.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.2s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
0.6b
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
0.6b
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
0.5b
(* (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))))

rewrite61.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-div
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×flip-+
9×flip3-+
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
4×frac-times
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 75
Calls
4 calls:
Slowest
36.0ms
(* (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))))
8.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
8.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
8.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))

series665.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
172.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
168.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
163.0ms
(* (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))) (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))))
161.0ms
(cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))

simplify8.2s

Counts
59 → 87
Calls
59 calls:
Slowest
592.0ms
(* (cbrt (- (* (* 2 l) (* 2 l)) (* (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (cbrt (- (* (* 2 l) (* 2 l)) (* (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))))
441.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
370.0ms
(* (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))) (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))
362.0ms
(* (cbrt (- (* (* 2 l) (* 2 l)) (* (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5)))))) (cbrt (+ (* 2 l) (+ (* 1/3 (pow l 3)) (* 1/60 (pow l 5))))))
350.0ms
(+ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 l)))))) (+ (* 320/9 (/ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 l)))))) (pow l 4))) (* 40/3 (/ (exp (* 1/3 (- (log 1/3600) (* 10 (log (/ 1 l)))))) (pow l 2)))))

prune2.1s

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 0.0b

regimes704.0ms

Accuracy

0% (0.3b remaining)

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

bsearch5.0ms

end0.0ms

sample10.6s

Algorithm
intervals