Average Error: 17.8 → 0.4
Time: 31.7s
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 17.8

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

  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. Using strategy rm

  7. Applied fma-def0.4

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

  8. 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 2019004 +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: 30.3s)Debug log

sample358.0ms

Algorithm
intervals

simplify235.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

235.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.5b

localize37.0ms

Local error

Found 4 expressions with local error:

6.3b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)
0.0b
(* J (- (exp l) (exp (- l))))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 99
Calls

4 calls. Slowest were:

18.0ms
(* J (- (exp l) (exp (- l))))
18.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
14.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

series469.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

239.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)
111.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
67.0ms
(* J (- (exp l) (exp (- l))))
51.0ms
(- (exp l) (exp (- l)))

simplify4.6s

Counts
74 → 111
Calls

74 calls. Slowest were:

437.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
328.0ms
(- (+ U (* (exp l) (* (cos (* 1/2 K)) J))) (* (cos (* 1/2 K)) (* J (exp (- l)))))
268.0ms
(- (+ U (* (cos (* 1/2 K)) (* (exp l) J))) (* (cos (* 1/2 K)) (* J (exp (* -1 l)))))

prune1.7s

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize57.0ms

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

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

12.0ms
(+ (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))) U)
11.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
4.0ms
(* (pow l 5) 1/60)

series468.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

229.0ms
(+ (* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2))) U)
167.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
38.0ms
(fma 1/3 (* l l) 2)
34.0ms
(* (pow l 5) 1/60)

simplify2.1s

Counts
21 → 65
Calls

21 calls. Slowest were:

606.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
279.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
275.0ms
(+ (* 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 (7 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

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 (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)) U)

series372.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

236.0ms
(fma (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)) U)
83.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
27.0ms
(fma 1/3 (* l l) 2)
25.0ms
(* (pow l 5) 1/60)

simplify1.6s

Counts
19 → 60
Calls

19 calls. Slowest were:

364.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
363.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
356.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune995.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

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

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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

series474.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

198.0ms
(+ (* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))) U)
156.0ms
(* J (* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2))))
84.0ms
(* (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)) (cos (/ K 2)))
36.0ms
(* (pow l 5) 1/60)

simplify2.2s

Counts
23 → 68
Calls

23 calls. Slowest were:

344.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
336.0ms
(* (* (* (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))))
283.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.3s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

regimes678.0ms

Accuracy

0% (0.4b remaining)

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

bsearch3.0ms

end0.0ms

sample11.4s

Algorithm
intervals