Average Error: 17.6 → 0.4
Time: 29.2s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left((\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

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. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{(\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied associate-*l*0.4

    \[\leadsto \color{blue}{J \cdot \left((\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* \cdot \cos \left(\frac{K}{2}\right)\right)} + U\]

  6. Final simplification0.4

    \[\leadsto J \cdot \left((\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U\]

Reproduce

herbie shell --seed 2019010 +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: 27.5s)Debug log

sample362.0ms

Algorithm
intervals

simplify280.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
280.0ms
(+ (* (* 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

localize25.0ms

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))))

rewrite48.0ms

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
28.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
10.0ms
(- (exp l) (exp (- l)))
8.0ms
(* J (- (exp l) (exp (- l))))

series247.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
130.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
76.0ms
(* J (- (exp l) (exp (- l))))
40.0ms
(- (exp l) (exp (- l)))

simplify4.1s

Counts
70 → 95
Calls
70 calls:
Slowest
537.0ms
(* J (fma (- (cbrt (exp (- l)))) (* (cbrt (exp (- l))) (cbrt (exp (- l)))) (* (cbrt (exp (- l))) (* (cbrt (exp (- l))) (cbrt (exp (- l)))))))
405.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
203.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
183.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
170.0ms
(* (* J (- (* (exp l) (exp l)) (* (exp (- l)) (exp (- l))))) (cos (/ K 2)))

prune1.4s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize30.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
0.0b
(* (pow l 5) 1/60)
0.0b
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
0.0b
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×expm1-log1p-u
3×fma-udef
3×*-commutative
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×distribute-lft-in
1×distribute-rgt-in
Counts
4 → 55
Calls
4 calls:
Slowest
6.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
2.0ms
(* (pow l 5) 1/60)
1.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
0.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

series298.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
160.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
79.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
35.0ms
(* (pow l 5) 1/60)
23.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

simplify2.3s

Counts
25 → 67
Calls
25 calls:
Slowest
389.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
375.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
343.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
339.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
206.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))

prune1.2s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×associate-*l*
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*r*
4×expm1-log1p-u
3×*-commutative
1×fma-udef
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
Counts
4 → 53
Calls
4 calls:
Slowest
10.0ms
(* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))
9.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
2.0ms
(* (pow l 5) 1/60)
0.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

series252.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
129.0ms
(* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))
64.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
36.0ms
(* (pow l 5) 1/60)
23.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

simplify2.0s

Counts
23 → 65
Calls
23 calls:
Slowest
321.0ms
(* (* (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
300.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
292.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
251.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
209.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))

prune1.1s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

localize16.0ms

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))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×log1p-expm1-u
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 → 69
Calls
4 calls:
Slowest
11.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))
1.0ms
(cbrt (cos (/ K 2)))

series746.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
284.0ms
(* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))
163.0ms
(cbrt (cos (/ K 2)))
163.0ms
(cbrt (cos (/ K 2)))
136.0ms
(cbrt (cos (/ K 2)))

simplify492.0ms

Counts
41 → 81
Calls
41 calls:
Slowest
125.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
103.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
93.0ms
(- 1 (+ (* 1/25920 (pow K 6)) (* 1/12 (pow K 2))))
77.0ms
(- 1 (+ (* 1/1152 (pow K 4)) (* 1/24 (pow K 2))))
6.0ms
(pow (pow (cos (* 1/2 K)) 2) 1/3)

prune1.5s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.0b

regimes569.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample10.5s

Algorithm
intervals