Average Error: 31.2 → 0.2
Time: 21.2s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x}}{x} \cdot \tan \left(\frac{x}{2}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

    \[\frac{1 - \cos x}{x \cdot x}\]
  2. Using strategy rm

  3. Applied flip--31.3

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

  4. Applied associate-/l/31.3

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}}\]

  5. Simplified15.4

    \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
  6. Using strategy rm

  7. Applied times-frac15.8

    \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}}\]

  8. Simplified15.6

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

  10. Applied associate-/r*0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 20.4s)Debug log

sample147.0ms

Algorithm
intervals

simplify15.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

15.0ms
(/ (- 1 (cos x)) (* x x))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.4b

localize22.0ms

Local error

Found 2 expressions with local error:

17.9b
(/ (- 1 (cos x)) (* x x))
1.8b
(- 1 (cos x))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 33
Calls

2 calls. Slowest were:

8.0ms
(/ (- 1 (cos x)) (* x x))
2.0ms
(- 1 (cos x))

series122.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

92.0ms
(/ (- 1 (cos x)) (* x x))
29.0ms
(- 1 (cos x))

simplify612.0ms

Counts
19 → 39
Calls

19 calls. Slowest were:

389.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
53.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
39.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))

prune260.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

17.6b
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
0.4b
(+ 1 (cos x))
0.2b
(* (sin x) (sin x))
0.1b
(* (* x x) (+ 1 (cos x)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

16.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
7.0ms
(* (sin x) (sin x))
6.0ms
(* (* x x) (+ 1 (cos x)))

series107.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
22.0ms
(* (* x x) (+ 1 (cos x)))
20.0ms
(* (sin x) (sin x))
8.0ms
(+ 1 (cos x))

simplify5.4s

Counts
51 → 88
Calls

51 calls. Slowest were:

1.1s
(* (* (* (* x x) (* x x)) (* x x)) (* (* (+ 1 (cos x)) (+ 1 (cos x))) (+ 1 (cos x))))
460.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (* (* (* (* x x) (+ 1 (cos x))) (* (* x x) (+ 1 (cos x)))) (* (* x x) (+ 1 (cos x)))))
440.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* (* x x) (+ 1 (cos x))) (* (* x x) (+ 1 (cos x)))) (* (* x x) (+ 1 (cos x)))))

prune823.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize49.0ms

Local error

Found 4 expressions with local error:

17.6b
(/ (* (sin x) (sin x)) (* (* x x) (+ (pow 1 3) (pow (cos x) 3))))
1.8b
(+ (cos x) -1)
0.3b
(pow (cos x) 3)
0.3b
(+ (pow 1 3) (pow (cos x) 3))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

14.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ (pow 1 3) (pow (cos x) 3))))
4.0ms
(+ (pow 1 3) (pow (cos x) 3))
3.0ms
(+ (cos x) -1)

series150.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ (pow 1 3) (pow (cos x) 3))))
32.0ms
(+ (pow 1 3) (pow (cos x) 3))
26.0ms
(pow (cos x) 3)
17.0ms
(+ (cos x) -1)

simplify4.2s

Counts
43 → 81
Calls

43 calls. Slowest were:

487.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (* (* (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (* (* x x) (+ (pow 1 3) (pow (cos x) 3)))) (* (* x x) (+ (pow 1 3) (pow (cos x) 3)))))
401.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ (pow 1 3) (pow (cos x) 3)))))
389.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (* (* x x) (+ (pow 1 3) (pow (cos x) 3)))) (* (* x x) (+ (pow 1 3) (pow (cos x) 3)))))

prune947.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

17.1b
(/ (sin x) (* x x))
0.2b
(* (/ (sin x) (* x x)) (tan (/ x 2)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 45
Calls

2 calls. Slowest were:

10.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
3.0ms
(/ (sin x) (* x x))

series105.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

69.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
35.0ms
(/ (sin x) (* x x))

simplify1.6s

Counts
34 → 51
Calls

34 calls. Slowest were:

475.0ms
(* (* (* (/ (sin x) (* x x)) (/ (sin x) (* x x))) (/ (sin x) (* x x))) (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))))
190.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))
189.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))

prune480.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0.0b

regimes48.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch1.0ms

end0.0ms

sample5.2s

Algorithm
intervals