Average Error: 27.7 → 8.2
Time: 21.0s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(\left(sin \cdot x\right) \cdot cos\right) \cdot sin\right) \cdot x\right)}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 27.7

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
  2. Using strategy rm

  3. Applied unpow227.7

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot \color{blue}{\left(sin \cdot sin\right)}\right) \cdot x\right)}\]

  4. Applied associate-*r*21.3

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\color{blue}{\left(\left(x \cdot sin\right) \cdot sin\right)} \cdot x\right)}\]
  5. Using strategy rm

  6. Applied unpow221.3

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(cos \cdot cos\right)} \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)}\]

  7. Applied associate-*l*15.9

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{cos \cdot \left(cos \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)\right)}}\]
  8. Using strategy rm

  9. Applied associate-*r*13.2

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \color{blue}{\left(\left(cos \cdot \left(\left(x \cdot sin\right) \cdot sin\right)\right) \cdot x\right)}}\]
  10. Using strategy rm

  11. Applied associate-*r*8.2

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\color{blue}{\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right)} \cdot x\right)}\]

  12. Final simplification8.2

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(\left(sin \cdot x\right) \cdot cos\right) \cdot sin\right) \cdot x\right)}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))

Details

Time bar (total: 19.6s)Debug log

sample178.0ms

Algorithm
intervals

simplify181.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

181.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.2b

localize24.0ms

Local error

Found 4 expressions with local error:

18.1b
(* (pow cos 2) (* (* x (pow sin 2)) x))
8.5b
(* x (pow sin 2))
4.1b
(* (* x (pow sin 2)) x)
0.7b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

12.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
10.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
3.0ms
(* (* x (pow sin 2)) x)

series138.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
34.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
21.0ms
(* (* x (pow sin 2)) x)
16.0ms
(* x (pow sin 2))

simplify2.6s

Counts
35 → 72
Calls

35 calls. Slowest were:

486.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
222.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
217.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

prune773.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 9.8b

localize13.0ms

Local error

Found 4 expressions with local error:

18.1b
(* (pow cos 2) (* (* (* x sin) sin) x))
4.1b
(* (* (* x sin) sin) x)
0.7b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.2b
(cos (* 2 x))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

26.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
12.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
9.0ms
(* (* (* x sin) sin) x)

series120.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
45.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
16.0ms
(* (* (* x sin) sin) x)
8.0ms
(cos (* 2 x))

simplify2.2s

Counts
34 → 68
Calls

34 calls. Slowest were:

257.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
241.0ms
(- (log (cos (* 2 x))) (+ (* (log cos) 2) (log (* (* (* x sin) sin) x))))
204.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))

prune758.0ms

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 6.9b

localize17.0ms

Local error

Found 4 expressions with local error:

6.5b
(* cos (* (* (* x sin) sin) x))
5.3b
(* cos (* cos (* (* (* x sin) sin) x)))
4.1b
(* (* (* x sin) sin) x)
0.7b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite41.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

13.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
12.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
10.0ms
(* cos (* (* (* x sin) sin) x))

series141.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
35.0ms
(* cos (* (* (* x sin) sin) x))
34.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
22.0ms
(* (* (* x sin) sin) x)

simplify2.0s

Counts
26 → 59
Calls

26 calls. Slowest were:

268.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
220.0ms
(/ (* (* (cos (* 2 x)) (cos (* 2 x))) (cos (* 2 x))) (* (* (* cos (* cos (* (* (* x sin) sin) x))) (* cos (* cos (* (* (* x sin) sin) x)))) (* cos (* cos (* (* (* x sin) sin) x)))))
213.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))

prune787.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 4.7b

localize21.0ms

Local error

Found 4 expressions with local error:

6.2b
(* cos (* (* x sin) sin))
5.3b
(* cos (* (* cos (* (* x sin) sin)) x))
3.9b
(* (* cos (* (* x sin) sin)) x)
0.7b
(/ (cos (* 2 x)) (* cos (* (* cos (* (* x sin) sin)) x)))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

20.0ms
(/ (cos (* 2 x)) (* cos (* (* cos (* (* x sin) sin)) x)))
11.0ms
(* cos (* (* cos (* (* x sin) sin)) x))
7.0ms
(* (* cos (* (* x sin) sin)) x)

series178.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(/ (cos (* 2 x)) (* cos (* (* cos (* (* x sin) sin)) x)))
48.0ms
(* (* cos (* (* x sin) sin)) x)
38.0ms
(* cos (* (* cos (* (* x sin) sin)) x))
37.0ms
(* cos (* (* x sin) sin))

simplify2.4s

Counts
24 → 59
Calls

24 calls. Slowest were:

361.0ms
(/ (* (cos x) (cos x)) (* cos (* (* cos (* (* x sin) sin)) x)))
306.0ms
(- (+ (/ 1 (* (pow x 2) (* (pow sin 2) (pow cos 2)))) (* 2/3 (/ (pow x 2) (* (pow sin 2) (pow cos 2))))) (* 2 (/ 1 (* (pow sin 2) (pow cos 2)))))
246.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune579.0ms

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 3.4b

regimes731.0ms

Accuracy

0% (4.4b remaining)

Error of 8.2b against oracle of 3.8b and baseline of 8.2b

bsearch2.0ms

end0.0ms

sample5.6s

Algorithm
intervals