Average Error: 40.1 → 0.8
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \left(\sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

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 40.1

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.3

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

  5. Taylor expanded around inf 15.3

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

  6. Simplified15.3

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

  10. Applied add-cube-cbrt0.8

    \[\leadsto \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \color{blue}{\left(\sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  11. Final simplification0.8

    \[\leadsto \left(\sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \left(\sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sqrt[3]{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 8.0s)Debug log

start271.0ms

Algorithm
intervals

setup11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 39.6b

localize19.0ms

Local error

Found 2 expressions with local error:

3.1b
(cos (+ x eps))
2.0b
(- (cos (+ x eps)) (cos x))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

9.0ms
(- (cos (+ x eps)) (cos x))
3.0ms
(cos (+ x eps))

series85.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

64.0ms
(- (cos (+ x eps)) (cos x))
21.0ms
(cos (+ x eps))

simplify393.0ms

Counts
13 → 30
Calls

13 calls. Slowest were:

225.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
98.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
22.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))

prune276.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.2b

localize35.0ms

Local error

Found 3 expressions with local error:

3.1b
(sin (/ (+ (+ x x) eps) 2))
0.2b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.0b
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

9.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
8.0ms
(sin (/ (+ (+ x x) eps) 2))
7.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

series206.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

93.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
88.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
25.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify853.0ms

Counts
19 → 44
Calls

19 calls. Slowest were:

244.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
239.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
75.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune593.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize32.0ms

Local error

Found 2 expressions with local error:

3.1b
(sin (+ x (* eps 1/2)))
0.2b
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

10.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
2.0ms
(sin (+ x (* eps 1/2)))

series155.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

135.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
19.0ms
(sin (+ x (* eps 1/2)))

simplify633.0ms

Counts
13 → 29
Calls

13 calls. Slowest were:

247.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
239.0ms
(* (* (* (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)))))
66.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))

prune385.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.1b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite80.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

41.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
18.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
10.0ms
(* (cos x) (sin (* eps 1/2)))

series421.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

177.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
123.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
60.0ms
(* (cos x) (sin (* eps 1/2)))
60.0ms
(* (sin x) (cos (* eps 1/2)))

simplify2.1s

Counts
38 → 75
Calls

38 calls. Slowest were:

428.0ms
(- x (+ (* 1/8 (* x (pow eps 2))) (* 1/6 (pow x 3))))
317.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
278.0ms
(- (* 1/2 eps) (+ (* 1/4 (* (pow x 2) eps)) (* 1/48 (pow eps 3))))

prune1.2s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

regimes194.0ms

Accuracy

0% (0.8b remaining)

Error of 0.8b against oracle of 0.1b and baseline of 0.8b

bsearch4.0ms