Average Error: 17.7 → 0.3
Time: 30.5s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot (\left(\ell \cdot J\right) \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) + \left(\left(J \cdot \frac{1}{60}\right) \cdot {\ell}^{5}\right))_*\]

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 0.3

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

  3. Simplified0.3

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

  4. Taylor expanded around inf 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019002 +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: 29.4s)Debug log

sample389.0ms

Algorithm
intervals

simplify311.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

311.0ms
(+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U)

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.4b

localize24.0ms

Local error

Found 3 expressions with local error:

5.9b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

15.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
8.0ms
(* J (- (exp l) (exp (- l))))
5.0ms
(- (exp l) (exp (- l)))

series190.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

95.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
56.0ms
(* J (- (exp l) (exp (- l))))
40.0ms
(- (exp l) (exp (- l)))

simplify4.2s

Counts
70 → 95
Calls

70 calls. Slowest were:

417.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
228.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (* -1 l)))))
211.0ms
(* (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

localize52.0ms

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
(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)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

12.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
4.0ms
(* (pow l 5) 1/60)
2.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))

series313.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

155.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))
102.0ms
(* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60)))
32.0ms
(* (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 were:

389.0ms
(+ (* 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)))))
335.0ms
(* (* J (fma (fma 1/3 (* l l) 2) l (* (pow l 5) 1/60))) (cos (/ K 2)))

prune1.2s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

2.6b
(* (* 1/60 J) (pow l 5))
0.1b
(* (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5))) (cos (/ K 2)))
0.0b
(* 1/60 J)
0.0b
(fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

5.0ms
(* (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5))) (cos (/ K 2)))
4.0ms
(* (* 1/60 J) (pow l 5))
0.0ms
(* 1/60 J)

series212.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(* (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5))) (cos (/ K 2)))
54.0ms
(* (* 1/60 J) (pow l 5))
45.0ms
(fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5)))
7.0ms
(* 1/60 J)

simplify3.3s

Counts
26 → 68
Calls

26 calls. Slowest were:

665.0ms
(* (* (* (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5))) (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5)))) (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5)))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
372.0ms
(+ (* 1/3 (* J (pow l 3))) (+ (* 2 (* J l)) (* 1/60 (* J (pow l 5)))))
370.0ms
(sqrt (fma (* l J) (fma 1/3 (* l l) 2) (* (* 1/60 J) (pow l 5))))

prune1.2s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

8.0ms
(* (* (cos (/ K 2)) J) (fma (fma 1/3 (* l l) 2) l (* 1/60 (pow l 5))))
3.0ms
(* (cos (/ K 2)) J)
1.0ms
(* 1/60 (pow l 5))

series345.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

232.0ms
(* (* (cos (/ K 2)) J) (fma (fma 1/3 (* l l) 2) l (* 1/60 (pow l 5))))
40.0ms
(fma (fma 1/3 (* l l) 2) l (* 1/60 (pow l 5)))
38.0ms
(* (cos (/ K 2)) J)
34.0ms
(* 1/60 (pow l 5))

simplify1.6s

Counts
26 → 67
Calls

26 calls. Slowest were:

315.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) (+ (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))) (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5))))))
294.0ms
(+ (* 2 (* (cos (* 1/2 K)) (* J l))) (+ (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))) (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5))))))
246.0ms
(* (* (cos (/ K 2)) J) (fma (fma 1/3 (* l l) 2) l (* 1/60 (pow l 5))))

prune1.2s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

regimes579.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample10.5s

Algorithm
intervals