Average Error: 17.7 → 0.4
Time: 31.2s
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 17.7

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

  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. 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 2019022 +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: 29.9s)Debug log

sample315.0ms

Algorithm
intervals
Results
124.0ms104×body1280valid
83.0ms229×body80valid
55.0ms77×body640valid
23.0ms41×body320valid
8.0ms23×body160valid

simplify221.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
220.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.3b

localize47.0ms

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))))

rewrite55.0ms

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)))
15.0ms
(* J (- (exp l) (exp (- l))))
9.0ms
(- (exp l) (exp (- l)))

series242.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
101.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
76.0ms
(* J (- (exp l) (exp (- l))))
65.0ms
(- (exp l) (exp (- l)))

simplify4.0s

Counts
70 → 95
Calls
70 calls:
Slowest
352.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
212.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
204.0ms
(+ (* (exp l) (exp l)) (+ (* (exp (- l)) (exp (- l))) (* (exp l) (exp (- l)))))
193.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
179.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))

prune1.3s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize38.0ms

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

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×expm1-log1p-u
3×associate-*r*
2×fma-udef
2×*-commutative
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
Counts
4 → 50
Calls
4 calls:
Slowest
16.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
3.0ms
(* (pow l 5) 1/60)
0.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))
0.0ms
(fma 1/3 (* l l) 2)

series238.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
159.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
35.0ms
(fma 1/3 (* l l) 2)
24.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))
20.0ms
(* (pow l 5) 1/60)

simplify1.4s

Counts
20 → 62
Calls
20 calls:
Slowest
329.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
276.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
248.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
160.0ms
(+ (log (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))) (log (cos (/ K 2))))
90.0ms
(* (* (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))) (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))

prune965.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

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

rewrite14.0ms

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
6.0ms
(* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))
5.0ms
(* (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))

series245.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
114.0ms
(* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))
78.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
32.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))
20.0ms
(* (pow l 5) 1/60)

simplify1.9s

Counts
23 → 65
Calls
23 calls:
Slowest
341.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
315.0ms
(* (* (* (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))))
235.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
207.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
176.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))

prune920.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

28.0b
(cbrt (* (* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))))
0.2b
(* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))))
0.1b
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
0.1b
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))

rewrite83.0ms

Algorithm
rewrite-expression-head
Rules
13×pow1
10×add-exp-log
10×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*l*
7×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
1×rem-cbrt-cube
1×pow-prod-up
1×pow3
1×pow1/3
1×pow-plus
1×pow2
1×cbrt-prod
Counts
4 → 68
Calls
4 calls:
Slowest
51.0ms
(cbrt (* (* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))))
23.0ms
(* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))))
4.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
4.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))

series451.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
167.0ms
(cbrt (* (* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))))
153.0ms
(* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))))
78.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
52.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))

simplify5.4s

Counts
36 → 80
Calls
36 calls:
Slowest
428.0ms
(* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))))
414.0ms
(+ (log (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))) (log (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))))
413.0ms
(cbrt (* (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))))
348.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
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))))

prune1.4s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes584.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms

end0.0ms

sample10.1s

Algorithm
intervals
Results
4.5s2736×body1280valid
2.1s2553×body640valid
2.0s8371×body80valid
849.0ms1413×body320valid
290.0ms727×body160valid