Average Error: 27.1 → 5.9
Time: 15.7s
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(sin \cdot x\right) \cdot cos\right) \cdot \left(sin \cdot x\right)\right)}\]

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 27.1

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

  3. Applied unpow227.1

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

  4. Applied associate-*r*21.1

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

  6. Applied unpow221.1

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

  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-*l*13.2

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

  11. Applied associate-*r*5.9

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

  12. Final simplification5.9

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

Reproduce

herbie shell --seed 2019007 
(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: 14.7s)Debug log

sample182.0ms

Algorithm
intervals

simplify188.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

187.0ms
(/ (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: 25.9b

localize32.0ms

Local error

Found 4 expressions with local error:

16.6b
(* (pow cos 2) (* (* x (pow sin 2)) x))
9.0b
(* x (pow sin 2))
4.6b
(* (* x (pow sin 2)) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

19.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
18.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
6.0ms
(* (* x (pow sin 2)) x)

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
53.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
27.0ms
(* (* x (pow sin 2)) x)
25.0ms
(* x (pow sin 2))

simplify2.4s

Counts
35 → 72
Calls

35 calls. Slowest were:

578.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
212.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
207.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* x (pow sin 2)) x))))

prune632.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 8.5b

localize14.0ms

Local error

Found 4 expressions with local error:

16.6b
(* (pow cos 2) (* (* (* x sin) sin) x))
4.6b
(* (* (* x sin) sin) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.2b
(cos (* 2 x))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

13.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
12.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
5.0ms
(* (* (* x sin) sin) x)

series110.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
36.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
16.0ms
(* (* (* x sin) sin) x)
8.0ms
(cos (* 2 x))

simplify2.1s

Counts
34 → 68
Calls

34 calls. Slowest were:

253.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
214.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
198.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))

prune551.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 6.8b

localize9.0ms

Local error

Found 4 expressions with local error:

6.2b
(* cos (* (* (* x sin) sin) x))
4.6b
(* (* (* x sin) sin) x)
4.3b
(* cos (* cos (* (* (* x sin) sin) x)))
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

15.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
12.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
6.0ms
(* cos (* (* (* x sin) sin) x))

series174.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

52.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
49.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
48.0ms
(* cos (* (* (* x sin) sin) x))
25.0ms
(* (* (* x sin) sin) x)

simplify1.4s

Counts
26 → 59
Calls

26 calls. Slowest were:

167.0ms
(- (+ (/ 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)))))
162.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
145.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune466.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 3.7b

localize13.0ms

Local error

Found 4 expressions with local error:

6.2b
(* cos (* (* x sin) (* sin x)))
4.3b
(* cos (* cos (* (* x sin) (* sin x))))
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
0.2b
(cos (* 2 x))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

11.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
8.0ms
(* cos (* cos (* (* x sin) (* sin x))))
4.0ms
(* cos (* (* x sin) (* sin x)))

series123.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
34.0ms
(* cos (* (* x sin) (* sin x)))
32.0ms
(* cos (* cos (* (* x sin) (* sin x))))
7.0ms
(cos (* 2 x))

simplify1.4s

Counts
27 → 58
Calls

27 calls. Slowest were:

170.0ms
(- (+ (/ 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)))))
167.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
164.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))

prune406.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 2.8b

regimes359.0ms

Accuracy

0% (2.2b remaining)

Error of 5.9b against oracle of 3.7b and baseline of 5.9b

bsearch1.0ms

end0.0ms

sample3.8s

Algorithm
intervals