Average Error: 39.9 → 0.4
Time: 26.4s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x + \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)\]

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 39.9

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

  3. Applied diff-cos34.4

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

  4. Simplified15.5

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

  5. Taylor expanded around inf 15.5

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

  6. Simplified15.4

    \[\leadsto \color{blue}{\sin \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
  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 *-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 1042149663 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 23.7s)Debug log

sample344.0ms

Algorithm
intervals
Results
256×(pre true 80)
139×(body real 1280)
61×(body real 640)
25×(body real 320)
19×(body real 2560)
10×(body real 160)
2×(body real 80)

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
9.0ms
(- (cos (+ x eps)) (cos x))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.4b

localize36.0ms

Local error

Found 2 expressions with local error:

3.1b
(cos (+ x eps))
0.9b
(- (cos (+ x eps)) (cos x))

rewrite14.0ms

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
10.0ms
(- (cos (+ x eps)) (cos x))
4.0ms
(cos (+ x eps))

series134.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
110.0ms
(- (cos (+ x eps)) (cos x))
25.0ms
(cos (+ x eps))

simplify1.0s

Counts
13 → 30
Calls
13 calls:
Slowest
527.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
373.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
46.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
30.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
10.0ms
(- (sqrt (cos (+ x eps))) (sqrt (cos x)))

prune467.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize64.0ms

Local error

Found 3 expressions with local error:

2.9b
(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))))

rewrite50.0ms

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
22.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
17.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
10.0ms
(sin (/ (+ (+ x x) eps) 2))

series319.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
140.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
132.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
46.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.3s

Counts
19 → 44
Calls
19 calls:
Slowest
380.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
376.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
126.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
109.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
39.0ms
(* -2 (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))))

prune782.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.3b

localize49.0ms

Local error

Found 2 expressions with local error:

2.9b
(sin (+ x (* eps 1/2)))
0.3b
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

rewrite28.0ms

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
23.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
5.0ms
(sin (+ x (* eps 1/2)))

series180.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
140.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
39.0ms
(sin (+ x (* eps 1/2)))

simplify899.0ms

Counts
13 → 29
Calls
13 calls:
Slowest
369.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
281.0ms
(* (* (* (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)))))
113.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
30.0ms
(+ (log (sin (+ x (* eps 1/2)))) (log (* -2 (sin (* eps 1/2)))))
23.0ms
(* -2 (* (sin (+ x (* 1/2 eps))) (sin (* 1/2 eps))))

prune536.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize47.0ms

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))))
0.2b
(* (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))))

rewrite103.0ms

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
53.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
24.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
13.0ms
(* (cos x) (sin (* eps 1/2)))
12.0ms
(* (sin x) (cos (* eps 1/2)))

series622.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
285.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
164.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
88.0ms
(* (cos x) (sin (* eps 1/2)))
83.0ms
(* (sin x) (cos (* eps 1/2)))

simplify3.2s

Counts
38 → 75
Calls
38 calls:
Slowest
531.0ms
(- x (+ (* 1/8 (* x (pow eps 2))) (* 1/6 (pow x 3))))
471.0ms
(- (* 1/2 eps) (+ (* 1/4 (* (pow x 2) eps)) (* 1/48 (pow eps 3))))
382.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
266.0ms
(* (- (* (* (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))))
247.0ms
(* (+ (pow (* (sin x) (cos (* eps 1/2))) 3) (pow (* (cos x) (sin (* eps 1/2))) 3)) (* -2 (sin (* eps 1/2))))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

regimes313.0ms

Accuracy

0% (0.3b remaining)

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

bsearch4.0ms

end0.0ms

sample11.7s

Algorithm
intervals
Results
8000×(pre true 80)
4436×(body real 1280)
1870×(body real 640)
690×(body real 320)
644×(body real 2560)
292×(body real 160)
68×(body real 80)