Average Error: 26.7 → 5.9
Time: 18.1s
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 26.7

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

  3. Applied unpow226.7

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

  4. Applied associate-*r*20.9

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

  6. Applied unpow220.9

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

  7. Applied associate-*l*15.6

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

  9. Applied associate-*l*12.8

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(cos \cdot \color{blue}{\left(\left(x \cdot sin\right) \cdot \left(sin \cdot x\right)\right)}\right)}\]
  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 2019016 
(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: 16.6s)Debug log

sample106.0ms

Algorithm
intervals
Results
363×(pre true 80)
282×(body real 80)
29×(body real 640)
24×(body real 1280)
19×(body real 320)
9×(body real 160)

simplify182.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
182.0ms
(/ (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: 26.5b

localize28.0ms

Local error

Found 4 expressions with local error:

16.8b
(* (pow cos 2) (* (* x (pow sin 2)) x))
7.9b
(* x (pow sin 2))
5.2b
(* (* x (pow sin 2)) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
11×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
5×associate-*l*
5×associate-*r*
4×add-log-exp
4×prod-exp
4×pow1
3×associate-/l*
3×*-commutative
3×div-exp
3×times-frac
2×unpow2
2×cbrt-unprod
2×cbrt-undiv
2×pow-to-exp
1×div-inv
1×cos-2
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 60
Calls
4 calls:
Slowest
10.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
8.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
3.0ms
(* (* x (pow sin 2)) x)
1.0ms
(* x (pow sin 2))

series386.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
286.0ms
(* (* x (pow sin 2)) x)
50.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
33.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
16.0ms
(* x (pow sin 2))

simplify2.2s

Counts
35 → 72
Calls
35 calls:
Slowest
391.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
197.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
176.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
176.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* x (pow sin 2)) x))))
169.0ms
(- (+ (/ 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)))))

prune624.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 9.9b

localize13.0ms

Local error

Found 4 expressions with local error:

16.8b
(* (pow cos 2) (* (* (* x sin) sin) x))
5.2b
(* (* (* x sin) sin) x)
0.5b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.1b
(* (* x sin) sin)

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
14×add-exp-log
11×add-cbrt-cube
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*l*
4×add-log-exp
4×prod-exp
4×pow1
3×associate-/l*
3×*-commutative
3×div-exp
3×times-frac
2×cbrt-unprod
2×cbrt-undiv
2×pow-to-exp
1×unpow2
1×div-inv
1×cos-2
1×associate-*r*
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 57
Calls
4 calls:
Slowest
15.0ms
(/ (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)
2.0ms
(* (* x sin) sin)

series114.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
49.0ms
(/ (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)
16.0ms
(* (* x sin) sin)

simplify2.0s

Counts
33 → 69
Calls
33 calls:
Slowest
256.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
203.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
173.0ms
(- (+ (/ 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)))))
171.0ms
(- (log (cos (* 2 x))) (+ (* (log cos) 2) (log (* (* (* x sin) sin) x))))
149.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune549.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 7.4b

localize8.0ms

Local error

Found 4 expressions with local error:

6.3b
(* cos (* cos (* (* (* x sin) sin) x)))
6.2b
(* cos (* (* (* x sin) sin) x))
5.2b
(* (* (* x sin) sin) x)
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×pow1
3×associate-/l*
3×*-commutative
3×times-frac
2×associate-*r*
1×div-inv
1×cos-2
1×associate-*l*
1×div-exp
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 47
Calls
4 calls:
Slowest
13.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
12.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
7.0ms
(* (* (* x sin) sin) x)
6.0ms
(* cos (* (* (* x sin) sin) x))

series132.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
48.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
34.0ms
(* cos (* (* (* x sin) sin) x))
33.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
17.0ms
(* (* (* x sin) sin) x)

simplify1.5s

Counts
26 → 59
Calls
26 calls:
Slowest
172.0ms
(- (+ (/ 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)))))
169.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
149.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
149.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
140.0ms
(/ (* (* (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)))))

prune506.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 5.2b

localize12.0ms

Local error

Found 4 expressions with local error:

6.3b
(* cos (* cos (* (* x sin) (* sin x))))
6.2b
(* cos (* (* x sin) (* sin x)))
0.5b
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
0.2b
(* (* x sin) (* sin x))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
8×add-exp-log
8×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×add-log-exp
3×associate-/l*
3×*-commutative
3×associate-*r*
3×times-frac
1×div-inv
1×cos-2
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×div-exp
1×div-sub
1×frac-2neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
4 → 51
Calls
4 calls:
Slowest
8.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
8.0ms
(* cos (* cos (* (* x sin) (* sin x))))
5.0ms
(* (* x sin) (* sin x))
4.0ms
(* cos (* (* x sin) (* sin x)))

series131.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
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))))
16.0ms
(* (* x sin) (* sin x))

simplify1.8s

Counts
29 → 63
Calls
29 calls:
Slowest
235.0ms
(* (* (* (* x sin) (* x sin)) (* x sin)) (* (* (* sin x) (* sin x)) (* sin x)))
172.0ms
(- (+ (/ 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)))))
169.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))
153.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
151.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune533.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 3.3b

regimes520.0ms

Accuracy

0% (2.2b remaining)

Error of 5.9b against oracle of 3.6b and baseline of 5.9b

bsearch2.0ms

end0.0ms

sample5.1s

Algorithm
intervals
Results
11760×(pre true 80)
9181×(body real 80)
898×(body real 640)
820×(body real 1280)
569×(body real 320)
292×(body real 160)