Average Error: 31.5 → 0.2
Time: 26.0s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.5

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

  3. Applied flip--31.7

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

  4. Applied associate-/l/31.7

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

  5. Simplified15.7

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

  7. Applied times-frac16.1

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

  8. Simplified16.0

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

  10. Applied *-un-lft-identity16.0

    \[\leadsto \frac{\color{blue}{1 \cdot \sin x}}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)\]

  11. Applied times-frac0.2

    \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \frac{\sin x}{x}\right)} \cdot \tan \left(\frac{x}{2}\right)\]

  12. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right)}\]

  13. Final simplification0.2

    \[\leadsto \frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 17.7s)Debug log

start202.0ms

Algorithm
intervals

setup59.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.8b

localize20.0ms

Local error

Found 2 expressions with local error:

16.5b
(/ (- 1 (cos x)) (* x x))
1.3b
(- 1 (cos x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

4.0ms
(/ (- 1 (cos x)) (* x x))
1.0ms
(- 1 (cos x))

series69.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

39.0ms
(/ (- 1 (cos x)) (* x x))
31.0ms
(- 1 (cos x))

simplify692.0ms

Counts
19 → 43
Calls

19 calls. Slowest were:

343.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
108.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
57.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))

prune354.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

16.5b
(/ (* (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)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

8.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
4.0ms
(* (* x x) (+ 1 (cos x)))
3.0ms
(* (sin x) (sin x))

series118.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

44.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
34.0ms
(* (* x x) (+ 1 (cos x)))
27.0ms
(* (sin x) (sin x))
13.0ms
(+ 1 (cos x))

simplify4.4s

Counts
51 → 96
Calls

51 calls. Slowest were:

567.0ms
(/ (* (* (* (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)))))
464.0ms
(/ (* (* (* (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)))))
362.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))

prune1.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

16.5b
(/ (* (sin x) (sin x)) (* (* x x) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))))
0.8b
(* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))
0.4b
(+ 1 (cos x))
0.4b
(+ 1 (cos x))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

14.0ms
(/ (* (sin x) (sin x)) (* (* x x) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))))
11.0ms
(* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))
1.0ms
(+ 1 (cos x))

series92.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (* (sin x) (sin x)) (* (* x x) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))))
14.0ms
(* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))
14.0ms
(+ 1 (cos x))
13.0ms
(+ 1 (cos x))

simplify6.5s

Counts
72 → 113
Calls

72 calls. Slowest were:

415.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (* (* (* (* x x) (* x x)) (* x x)) (* (* (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x)))) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x)))))))
405.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (+ (log (sqrt (+ 1 (cos x)))) (log (sqrt (+ 1 (cos x)))))))
402.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* (* x x) (* x x)) (* x x)) (* (* (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x)))) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x))))) (* (sqrt (+ 1 (cos x))) (sqrt (+ 1 (cos x)))))))

prune1.6s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 2 expressions with local error:

16.0b
(/ (sin x) (* x x))
0.2b
(* (/ (sin x) (* x x)) (tan (/ x 2)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
2 → 49
Calls

2 calls. Slowest were:

15.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
6.0ms
(/ (sin x) (* x x))

series135.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

101.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
34.0ms
(/ (sin x) (* x x))

simplify1.8s

Counts
34 → 55
Calls

34 calls. Slowest were:

485.0ms
(* (* (* (/ (sin x) (* x x)) (/ (sin x) (* x x))) (/ (sin x) (* x x))) (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))))
248.0ms
(/ (* (* (sin x) (sin x)) (sin x)) (* (* (* x x) (* x x)) (* x x)))
192.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))

prune545.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes44.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms