Average Error: 17.7 → 0.4
Time: 35.9s
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 2019007 +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: 34.3s)Debug log

sample326.0ms

Algorithm
intervals

simplify266.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

265.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 17.2b

localize50.0ms

Local error

Found 3 expressions with local error:

7.5b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

33.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
10.0ms
(- (exp l) (exp (- l)))
9.0ms
(* J (- (exp l) (exp (- l))))

series233.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

107.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
69.0ms
(* J (- (exp l) (exp (- l))))
57.0ms
(- (exp l) (exp (- l)))

simplify3.9s

Counts
70 → 95
Calls

70 calls. Slowest were:

470.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
190.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
180.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))

prune1.6s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

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)))

series317.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

181.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
71.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
33.0ms
(* (pow l 5) 1/60)
32.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))

simplify2.2s

Counts
25 → 67
Calls

25 calls. Slowest were:

343.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
312.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
305.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.1s

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
(* (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))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

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)

series279.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

154.0ms
(* 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)))
23.0ms
(fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))
23.0ms
(* (pow l 5) 1/60)

simplify1.9s

Counts
23 → 65
Calls

23 calls. Slowest were:

328.0ms
(+ (* 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))))
256.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))

prune981.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

14.4b
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
14.4b
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
14.4b
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
0.5b
(* (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))) (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))))

rewrite48.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

27.0ms
(* (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))) (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))))
6.0ms
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
6.0ms
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

360.0ms
(* (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))) (cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))))
332.0ms
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
317.0ms
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))
308.0ms
(cbrt (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))))

simplify6.8s

Counts
31 → 71
Calls

31 calls. Slowest were:

389.0ms
(+ (* 20/3 (/ (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))) (pow l 2))) (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))))
377.0ms
(+ (* 20/3 (/ (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))) (pow l 2))) (exp (* 1/3 (- (log (* 1/60 (cos (* 1/2 K)))) (+ (* 5 (log (/ 1 l))) (log (/ 1 J)))))))
374.0ms
(* (* (* 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))))

prune1.5s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes813.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample10.4s

Algorithm
intervals