Average Error: 39.6 → 0.9
Time: 17.8s
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 39.6

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

  3. Applied diff-cos34.0

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

  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. Using strategy rm

  10. Applied add-cube-cbrt0.9

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

  11. Final simplification0.9

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

Reproduce

herbie shell --seed 2019016 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 16.0s)Debug log

sample280.0ms

Algorithm
intervals
Results
256×(pre true 80)
145×(body real 1280)
57×(body real 640)
27×(body real 2560)
16×(body real 320)
8×(body real 160)
3×(body real 80)

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (cos (+ x eps)) (cos x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 37.5b

localize72.0ms

Local error

Found 2 expressions with local error:

4.0b
(cos (+ x eps))
0.7b
(- (cos (+ x eps)) (cos x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×cos-sum
2×pow1
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×diff-cos
1×diff-log
1×flip3--
1×sub-neg
1×associate--l-
Counts
2 → 24
Calls
2 calls:
Slowest
5.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series85.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
63.0ms
(- (cos (+ x eps)) (cos x))
22.0ms
(cos (+ x eps))

simplify438.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
270.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
92.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
31.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
18.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
4.0ms
(- (sqrt (cos (+ x eps))) (sqrt (cos x)))

prune274.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize43.0ms

Local error

Found 3 expressions with local error:

4.0b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.0b
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×add-exp-log
5×add-cbrt-cube
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×associate-*r*
3×add-log-exp
3×associate-*l*
2×sin-mult
2×*-commutative
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
Counts
3 → 35
Calls
3 calls:
Slowest
10.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
9.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
4.0ms
(sin (/ (+ (+ x x) eps) 2))

series261.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
118.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
107.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
36.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify854.0ms

Counts
19 → 44
Calls
19 calls:
Slowest
279.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
258.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
95.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
48.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
27.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))

prune520.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.3b

localize35.0ms

Local error

Found 2 expressions with local error:

4.0b
(sin (+ x (* eps 1/2)))
0.3b
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×sin-sum
Counts
2 → 23
Calls
2 calls:
Slowest
18.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
4.0ms
(sin (+ x (* eps 1/2)))

series122.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
83.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
38.0ms
(sin (+ x (* eps 1/2)))

simplify725.0ms

Counts
13 → 29
Calls
13 calls:
Slowest
323.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
220.0ms
(* (* (* (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)))))
72.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
27.0ms
(+ (log (sin (+ x (* eps 1/2)))) (log (* -2 (sin (* eps 1/2)))))
22.0ms
(* -2 (* (sin (+ x (* 1/2 eps))) (sin (* 1/2 eps))))

prune373.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize39.0ms

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

rewrite85.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
10×pow1
9×add-cube-cbrt
9×associate-*l*
9×*-un-lft-identity
9×add-sqr-sqrt
7×associate-*r*
6×add-log-exp
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
2×flip-+
2×associate-*l/
2×flip3-+
1×sin-cos-mult
1×sum-log
1×+-commutative
Counts
4 → 63
Calls
4 calls:
Slowest
45.0ms
(* (+ (* (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))))
11.0ms
(* (cos x) (sin (* eps 1/2)))
10.0ms
(* (sin x) (cos (* eps 1/2)))

series460.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
196.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
117.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
74.0ms
(* (cos x) (sin (* eps 1/2)))
72.0ms
(* (sin x) (cos (* eps 1/2)))

simplify2.0s

Counts
38 → 75
Calls
38 calls:
Slowest
342.0ms
(- x (+ (* 1/8 (* x (pow eps 2))) (* 1/6 (pow x 3))))
339.0ms
(- (* 1/2 eps) (+ (* 1/4 (* (pow x 2) eps)) (* 1/48 (pow eps 3))))
248.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
185.0ms
(* (- (* (* (sin x) (cos (* eps 1/2))) (* (sin x) (cos (* eps 1/2)))) (* (* (cos x) (sin (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))) (* -2 (sin (* eps 1/2))))
144.0ms
(* -2 (* (sin (* 1/2 eps)) (+ (* (sin x) (cos (* 1/2 eps))) (* (cos x) (sin (* 1/2 eps))))))

prune908.0ms

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 0.0b

regimes234.0ms

Accuracy

0% (0.8b remaining)

Error of 0.9b against oracle of 0.0b and baseline of 0.9b

bsearch5.0ms

end0.0ms

sample8.1s

Algorithm
intervals
Results
8000×(pre true 80)
4433×(body real 1280)
1827×(body real 640)
709×(body real 320)
664×(body real 2560)
277×(body real 160)
90×(body real 80)