Average Error: 17.6 → 0.4
Time: 35.8s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left((\ell \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U\]

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

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2018360 +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: 21.9s)Debug log

start361.0ms

Algorithm
intervals

setup310.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 19.0b

localize28.0ms

Local error

Found 3 expressions with local error:

7.0b
(- (exp l) (exp (- l)))
0.0b
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
0.0b
(* J (- (exp l) (exp (- l))))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
3 → 86
Calls

3 calls. Slowest were:

19.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
13.0ms
(* J (- (exp l) (exp (- l))))
5.0ms
(- (exp l) (exp (- l)))

series214.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

99.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))
58.0ms
(* J (- (exp l) (exp (- l))))
56.0ms
(- (exp l) (exp (- l)))

simplify3.3s

Counts
70 → 95
Calls

70 calls. Slowest were:

323.0ms
(* (* J (- (pow (exp l) 3) (pow (exp (- l)) 3))) (cos (/ K 2)))
181.0ms
(* (cos (* 1/2 K)) (* J (- (exp l) (exp (- l)))))
168.0ms
(* (* J (- (exp l) (exp (- l)))) (cos (/ K 2)))

prune1.5s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize25.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

6.0ms
(* (* J (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))) (cos (/ K 2)))
2.0ms
(* (pow l 5) 1/60)
1.0ms
(* J (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)))

series317.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(* (* J (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))) (cos (/ K 2)))
112.0ms
(* J (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)))
38.0ms
(fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))
36.0ms
(* (pow l 5) 1/60)

simplify2.0s

Counts
25 → 67
Calls

25 calls. Slowest were:

327.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
318.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))
274.0ms
(* (* J (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))) (cos (/ K 2)))

prune1.4s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

11.0ms
(* J (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
9.0ms
(* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))
4.0ms
(* (pow l 5) 1/60)

series257.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

138.0ms
(* J (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
77.0ms
(* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))
23.0ms
(fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))
20.0ms
(* (pow l 5) 1/60)

simplify2.3s

Counts
23 → 65
Calls

23 calls. Slowest were:

711.0ms
(* (* (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))) (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60))) (* (* (cos (/ K 2)) (cos (/ K 2))) (cos (/ K 2))))
378.0ms
(- (+ (* 1/3 (pow l 3)) (* 2 l)) (* 1/4 (* (pow K 2) l)))
280.0ms
(+ (* 1/60 (* (cos (* 1/2 K)) (* J (pow l 5)))) (* 1/3 (* (cos (* 1/2 K)) (* J (pow l 3)))))

prune1.3s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
0.6b
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
0.6b
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
0.5b
(* (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))) (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

11.0ms
(* (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))) (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))))
4.0ms
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
2.0ms
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))

series742.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

225.0ms
(* (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))) (cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2)))))
183.0ms
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
169.0ms
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
164.0ms
(cbrt (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))

simplify4.7s

Counts
31 → 71
Calls

31 calls. Slowest were:

283.0ms
(+ (* 40/3 (/ (exp (* 1/3 (- (log (* 1/3600 (pow (cos (* 1/2 K)) 2))) (* 10 (log (/ -1 l)))))) (pow l 2))) (exp (* 1/3 (- (log (* 1/3600 (pow (cos (* 1/2 K)) 2))) (* 10 (log (/ -1 l)))))))
243.0ms
(* (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))) (* (fma l (fma 1/3 (* l l) 2) (* (pow l 5) 1/60)) (cos (/ K 2))))
235.0ms
(+ (exp (* 1/3 (- (log (* 1/3600 (pow (cos (* 1/2 K)) 2))) (* 10 (log (/ 1 l)))))) (* 40/3 (/ (exp (* 1/3 (- (log (* 1/3600 (pow (cos (* 1/2 K)) 2))) (* 10 (log (/ 1 l)))))) (pow l 2))))

prune1.9s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

regimes910.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms