Average Error: 27.7 → 8.2
Time: 22.2s
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 +o rules:numerics
(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: 20.2s)Debug log

sample186.0ms

Algorithm
intervals

simplify172.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

172.0ms
(/ (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)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

11.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
9.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
4.0ms
(* (* x (pow sin 2)) x)

series117.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
36.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
16.0ms
(* (* x (pow sin 2)) x)
15.0ms
(* x (pow sin 2))

simplify2.4s

Counts
35 → 80
Calls

35 calls. Slowest were:

388.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
249.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
204.0ms
(- (+ (/ 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)))))

prune755.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 8.7b

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))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

14.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
13.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
7.0ms
(* (* (* x sin) sin) x)

series105.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
32.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
16.0ms
(* (* (* x sin) sin) x)
7.0ms
(cos (* 2 x))

simplify2.5s

Counts
34 → 76
Calls

34 calls. Slowest were:

280.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
267.0ms
(- (+ (/ 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)))))
257.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))

prune867.0ms

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 6.1b

localize13.0ms

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))))

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

17.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
15.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
7.0ms
(* cos (* (* (* x sin) sin) x))

series221.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
60.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
59.0ms
(* cos (* (* (* x sin) sin) x))
29.0ms
(* (* (* x sin) sin) x)

simplify2.1s

Counts
26 → 67
Calls

26 calls. Slowest were:

285.0ms
(- (+ (/ 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)))))
250.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
227.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune921.0ms

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 4.3b

localize14.0ms

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)))

rewrite58.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

24.0ms
(* cos (* (* cos (* (* x sin) sin)) x))
14.0ms
(* (* cos (* (* x sin) sin)) x)
13.0ms
(/ (cos (* 2 x)) (* cos (* (* cos (* (* x sin) sin)) x)))

series191.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

69.0ms
(/ (cos (* 2 x)) (* cos (* (* cos (* (* x sin) sin)) x)))
49.0ms
(* cos (* (* cos (* (* x sin) sin)) x))
37.0ms
(* (* cos (* (* x sin) sin)) x)
36.0ms
(* cos (* (* x sin) sin))

simplify2.4s

Counts
24 → 67
Calls

24 calls. Slowest were:

409.0ms
(/ (* (cos x) (cos x)) (* cos (* (* cos (* (* x sin) sin)) x)))
283.0ms
(- (+ (/ 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)))))
227.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune743.0ms

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 3.2b

regimes775.0ms

Accuracy

0% (4.6b remaining)

Error of 8.2b against oracle of 3.6b and baseline of 8.2b

bsearch2.0ms

end0.0ms

sample5.5s

Algorithm
intervals