Average Error: 17.4 → 0.6
Time: 33.8s
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 17.4

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019016 +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: 32.7s)Debug log

sample340.0ms

Algorithm
intervals
Results
529×(pre true 80)
276×(body real 80)
93×(body real 640)
89×(body real 1280)
50×(body real 320)
21×(body real 160)

simplify256.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
256.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.7b

localize35.0ms

Local error

Found 3 expressions with local error:

4.8b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
28×*-un-lft-identity
28×add-sqr-sqrt
27×prod-diff
24×add-cube-cbrt
11×distribute-lft-in
11×distribute-rgt-in
8×associate-*r*
5×add-log-exp
5×add-exp-log
5×add-cbrt-cube
5×pow1
4×associate-*r/
3×flip--
3×log1p-expm1-u
3×fma-neg
3×flip3--
3×sub-neg
3×expm1-log1p-u
2×difference-of-squares
2×distribute-lft-out--
2×*-commutative
2×associate-*l/
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×diff-log
Counts
3 → 86
Calls
3 calls:
Slowest
15.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
8.0ms
(* J (- (exp l) (exp (- l))))
7.0ms
(- (exp l) (exp (- l)))

series235.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
91.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
84.0ms
(* J (- (exp l) (exp (- l))))
61.0ms
(- (exp l) (exp (- l)))

simplify4.0s

Counts
70 → 95
Calls
70 calls:
Slowest
431.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
221.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))
160.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
156.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))
146.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune1.4s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 3 expressions with local error:

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
5×add-exp-log
5×add-cbrt-cube
5×pow1
4×add-cube-cbrt
4×associate-*r*
4×*-un-lft-identity
4×add-sqr-sqrt
3×add-log-exp
3×log1p-expm1-u
3×expm1-log1p-u
2×*-commutative
1×flip-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×flip3-+
1×fma-def
1×+-commutative
Counts
3 → 41
Calls
3 calls:
Slowest
11.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
6.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
1.0ms
(* 2 (* J l))

series186.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
111.0ms
(+ (* (* 2 (* J l)) (cos (/ K 2))) U)
63.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
12.0ms
(* 2 (* J l))

simplify1.1s

Counts
16 → 50
Calls
16 calls:
Slowest
815.0ms
(* (* (* (* 2 (* J l)) (* 2 (* J l))) (* 2 (* J l))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
54.0ms
(* (* 2 (* J l)) (cos (/ K 2)))
52.0ms
(+ (log (* 2 (* J l))) (log (cos (/ K 2))))
50.0ms
(* 2 (* (cos (* 1/2 K)) (* J l)))
43.0ms
(* 2 (* (cos (* 1/2 K)) (* J l)))

prune564.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

11.7b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
11.7b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
11.7b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
0.5b
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
7×pow1/3
6×add-cube-cbrt
6×add-exp-log
6×*-un-lft-identity
6×add-sqr-sqrt
5×cbrt-prod
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 59
Calls
4 calls:
Slowest
30.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
6.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
6.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
5.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

series732.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
228.0ms
(* (cbrt (* (* 2 (* J l)) (cos (/ K 2)))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
173.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
166.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
165.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

simplify2.7s

Counts
31 → 71
Calls
31 calls:
Slowest
488.0ms
(* (pow (cbrt 2) 2) (exp (* 1/3 (- (log (pow (cos (* 1/2 K)) 2)) (+ (* 2 (log (/ -1 l))) (* 2 (log (/ -1 J))))))))
216.0ms
(* (exp (* 1/3 (- (log (cos (* 1/2 K))) (+ (log (/ 1 l)) (log (/ 1 J)))))) (cbrt 2))
215.0ms
(* (exp (* 1/3 (+ (* 2 (log J)) (* 2 (log l))))) (pow (cbrt 2) 2))
188.0ms
(* (exp (* 1/3 (- (log (cos (* 1/2 K))) (+ (log (/ 1 l)) (log (/ 1 J)))))) (cbrt 2))
180.0ms
(* (pow (cbrt 2) 2) (exp (* 1/3 (- (log (pow (cos (* 1/2 K)) 2)) (+ (* 2 (log (/ 1 l))) (* 2 (log (/ 1 J))))))))

prune1.0s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 4 expressions with local error:

26.2b
(cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2)))))
11.7b
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
0.3b
(* (cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
0.2b
(* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))

rewrite89.0ms

Algorithm
rewrite-expression-head
Rules
33×pow1
18×add-exp-log
16×add-cbrt-cube
15×pow-prod-down
9×cbrt-unprod
9×prod-exp
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×associate-*l*
5×associate-*r*
5×cbrt-prod
4×add-log-exp
4×log1p-expm1-u
4×pow-prod-up
4×pow1/3
4×expm1-log1p-u
2×*-commutative
2×pow-plus
1×pow3
1×pow2
Counts
4 → 76
Calls
4 calls:
Slowest
33.0ms
(* (cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
27.0ms
(cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2)))))
23.0ms
(* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))
5.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))

series674.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
215.0ms
(* (cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))) (cbrt (* (* 2 (* J l)) (cos (/ K 2)))))
214.0ms
(cbrt (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2)))))
174.0ms
(cbrt (* (* 2 (* J l)) (cos (/ K 2))))
71.0ms
(* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2))))

simplify6.5s

Counts
50 → 88
Calls
50 calls:
Slowest
465.0ms
(* (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2)))) (* (* 2 (* J l)) (cos (/ K 2))))
424.0ms
(* (* (* (* 2 (* J l)) (cos (/ K 2))) (* (* 2 (* J l)) (cos (/ K 2)))) (* (* 2 (* J l)) (cos (/ K 2))))
379.0ms
(+ (+ (log (* 2 (* J l))) (log (cos (/ K 2)))) (log (* (* 2 (* J l)) (cos (/ K 2)))))
321.0ms
(* (cos (* 1/2 K)) (* J (* (cbrt 2) (* l (cbrt 4)))))
309.0ms
(+ (+ (log (* 2 (* J l))) (log (cos (/ K 2)))) (+ (log (* 2 (* J l))) (log (cos (/ K 2)))))

prune1.5s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes581.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch3.0ms

end0.0ms

sample10.7s

Algorithm
intervals
Results
15941×(pre true 80)
8459×(body real 80)
2689×(body real 1280)
2636×(body real 640)
1410×(body real 320)
747×(body real 160)