Average Error: 40.0 → 0.4
Time: 17.2s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 40.0

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  2. Using strategy rm

  3. Applied diff-cos34.5

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

    \[\leadsto \color{blue}{\sin \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
  7. Using strategy rm

  8. Applied sin-sum0.4

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  9. Final simplification0.4

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 16.1s)Debug log

sample208.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

8.0ms
(- (cos (+ x eps)) (cos x))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.5b

localize13.0ms

Local error

Found 2 expressions with local error:

3.8b
(cos (+ x eps))
1.2b
(- (cos (+ x eps)) (cos x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

5.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series109.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

70.0ms
(- (cos (+ x eps)) (cos x))
39.0ms
(cos (+ x eps))

simplify376.0ms

Counts
13 → 30
Calls

13 calls. Slowest were:

220.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
77.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
19.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))

prune274.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.6b

localize48.0ms

Local error

Found 4 expressions with local error:

3.9b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.2b
(/ (+ (+ x x) eps) 2)
0.0b
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite57.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

18.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
15.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
14.0ms
(/ (+ (+ x x) eps) 2)

series235.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
80.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
35.0ms
(sin (/ (+ (+ x x) eps) 2))
27.0ms
(/ (+ (+ x x) eps) 2)

simplify1.1s

Counts
26 → 62
Calls

26 calls. Slowest were:

270.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
216.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
186.0ms
(- (* (+ x x) (+ x x)) (* eps eps))

prune746.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.5b

localize26.0ms

Local error

Found 2 expressions with local error:

3.9b
(sin (+ x (* eps 1/2)))
0.3b
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

19.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
4.0ms
(sin (+ x (* eps 1/2)))

series123.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

100.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
23.0ms
(sin (+ x (* eps 1/2)))

simplify577.0ms

Counts
13 → 29
Calls

13 calls. Slowest were:

212.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
205.0ms
(* (* (* (sin (+ x (* eps 1/2))) (sin (+ x (* eps 1/2)))) (sin (+ x (* eps 1/2)))) (* (* (* -2 (sin (* eps 1/2))) (* -2 (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2)))))
60.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))

prune332.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.1b

localize35.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
0.2b
(* (sin x) (cos (* eps 1/2)))
0.2b
(* (cos x) (sin (* eps 1/2)))
0.1b
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))

rewrite92.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

51.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
19.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
11.0ms
(* (cos x) (sin (* eps 1/2)))

series380.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

184.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
98.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
54.0ms
(* (cos x) (sin (* eps 1/2)))
44.0ms
(* (sin x) (cos (* eps 1/2)))

simplify2.1s

Counts
38 → 75
Calls

38 calls. Slowest were:

364.0ms
(- x (+ (* 1/8 (* x (pow eps 2))) (* 1/6 (pow x 3))))
294.0ms
(- (* 1/2 eps) (+ (* 1/4 (* (pow x 2) eps)) (* 1/48 (pow eps 3))))
229.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))

prune1.1s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

regimes213.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample8.0s

Algorithm
intervals