Average Error: 27.5 → 5.9
Time: 21.8s
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 27.5

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

  3. Applied unpow227.5

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

  4. Applied associate-*r*21.4

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

  6. Applied unpow221.4

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

  7. Applied associate-*l*16.3

    \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{cos \cdot \left(cos \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)\right)}}\]
  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 2019022 
(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: 20.7s)Debug log

sample99.0ms

Algorithm
intervals
Results
45.0ms293×body80valid
17.0ms25×body1280valid
15.0ms29×body640valid
6.0ms18×body320valid
3.0ms12×body160valid

simplify230.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
230.0ms
(/ (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: 25.2b

localize24.0ms

Local error

Found 4 expressions with local error:

16.7b
(* (pow cos 2) (* (* x (pow sin 2)) x))
7.8b
(* x (pow sin 2))
4.3b
(* (* x (pow sin 2)) x)
0.7b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))

rewrite33.0ms

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
14.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
10.0ms
(* (* x (pow sin 2)) x)
8.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
1.0ms
(* x (pow sin 2))

series150.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
59.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x)))
47.0ms
(* (pow cos 2) (* (* x (pow sin 2)) x))
27.0ms
(* x (pow sin 2))
17.0ms
(* (* x (pow sin 2)) x)

simplify2.8s

Counts
35 → 72
Calls
35 calls:
Slowest
449.0ms
(* (* (* (pow cos 2) (pow cos 2)) (pow cos 2)) (* (* (* (* x (pow sin 2)) x) (* (* x (pow sin 2)) x)) (* (* x (pow sin 2)) x)))
230.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
219.0ms
(- (log (cos (* 2 x))) (log (* (pow cos 2) (* (* x (pow sin 2)) x))))
217.0ms
(- (+ (/ 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)))))
206.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* x (pow sin 2)) x))))

prune877.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 11.4b

localize18.0ms

Local error

Found 4 expressions with local error:

16.7b
(* (pow cos 2) (* (* (* x sin) sin) x))
4.3b
(* (* (* x sin) sin) x)
0.7b
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
0.2b
(* (* x sin) sin)

rewrite47.0ms

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
25.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
11.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
5.0ms
(* (* (* x sin) sin) x)
4.0ms
(* (* x sin) sin)

series173.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
81.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
52.0ms
(* (pow cos 2) (* (* (* x sin) sin) x))
23.0ms
(* (* x sin) sin)
16.0ms
(* (* (* x sin) sin) x)

simplify3.0s

Counts
33 → 69
Calls
33 calls:
Slowest
411.0ms
(- (log (cos (* 2 x))) (+ (log (pow cos 2)) (log (* (* (* x sin) sin) x))))
282.0ms
(/ (* (cos x) (cos x)) (* (pow cos 2) (* (* (* x sin) sin) x)))
280.0ms
(- (log (cos (* 2 x))) (+ (* (log cos) 2) (log (* (* (* x sin) sin) x))))
252.0ms
(- (+ (/ 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)))))
212.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune913.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 7.4b

localize16.0ms

Local error

Found 4 expressions with local error:

5.5b
(* cos (* (* (* x sin) sin) x))
4.7b
(* cos (* cos (* (* (* x sin) sin) x)))
4.3b
(* (* (* x sin) sin) x)
0.7b
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))

rewrite50.0ms

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
24.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
12.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
7.0ms
(* cos (* (* (* x sin) sin) x))
5.0ms
(* (* (* x sin) sin) x)

series204.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
71.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* (* x sin) sin) x))))
58.0ms
(* cos (* (* (* x sin) sin) x))
53.0ms
(* cos (* cos (* (* (* x sin) sin) x)))
21.0ms
(* (* (* x sin) sin) x)

simplify2.2s

Counts
26 → 59
Calls
26 calls:
Slowest
271.0ms
(- (+ (/ 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)))))
245.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* (* x sin) sin) x))))
239.0ms
(/ (* (* (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)))))
208.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
206.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune680.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 5.3b

localize12.0ms

Local error

Found 4 expressions with local error:

5.5b
(* cos (* (* x sin) (* sin x)))
4.7b
(* cos (* cos (* (* x sin) (* sin x))))
0.7b
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
0.3b
(* (* x sin) (* sin x))

rewrite44.0ms

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
16.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
16.0ms
(* cos (* cos (* (* x sin) (* sin x))))
6.0ms
(* (* x sin) (* sin x))
5.0ms
(* cos (* (* x sin) (* sin x)))

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
75.0ms
(/ (cos (* 2 x)) (* cos (* cos (* (* x sin) (* sin x)))))
66.0ms
(* cos (* (* x sin) (* sin x)))
48.0ms
(* cos (* cos (* (* x sin) (* sin x))))
26.0ms
(* (* x sin) (* sin x))

simplify2.5s

Counts
29 → 63
Calls
29 calls:
Slowest
292.0ms
(/ (* (cos x) (cos x)) (* cos (* cos (* (* x sin) (* sin x)))))
258.0ms
(- (+ (/ 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
(* (* (* (* x sin) (* x sin)) (* x sin)) (* (* (* sin x) (* sin x)) (* sin x)))
232.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))
219.0ms
(/ (cos (* 2 x)) (* (pow cos 2) (* (pow x 2) (pow sin 2))))

prune727.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 2.6b

regimes390.0ms

Accuracy

0% (2.3b remaining)

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

bsearch2.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
1.9s9085×body80valid
1.5s835×body1280valid
1.2s942×body640valid
352.0ms562×body320valid
163.0ms337×body160valid