Average Error: 39.1 → 0.8
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -25.400968012366302 \lor \neg \left(\varepsilon \le 4.162454335163805 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

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. Split input into 2 regimes

  2. if eps < -25.400968012366302 or 4.162454335163805e-05 < eps

    1. Initial program 29.7

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

    3. Applied cos-sum0.8

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

    if -25.400968012366302 < eps < 4.162454335163805e-05

    1. Initial program 49.0

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

    3. Applied diff-cos37.5

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

    4. Simplified0.8

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -25.400968012366302 \lor \neg \left(\varepsilon \le 4.162454335163805 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 16.5s)Debug log

sample180.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

7.0ms
(- (cos (+ x eps)) (cos x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.8b

localize65.0ms

Local error

Found 2 expressions with local error:

3.8b
(cos (+ x eps))
0.7b
(- (cos (+ x eps)) (cos x))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

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

series89.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

69.0ms
(- (cos (+ x eps)) (cos x))
20.0ms
(cos (+ x eps))

simplify485.0ms

Counts
13 → 30
Calls

13 calls. Slowest were:

249.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
126.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
41.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))

prune316.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize38.0ms

Local error

Found 2 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

10.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
4.0ms
(sin (/ (+ (+ x x) eps) 2))

series114.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

88.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
26.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify551.0ms

Counts
15 → 31
Calls

15 calls. Slowest were:

221.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
88.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
79.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune449.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.3b

localize25.0ms

Local error

Found 4 expressions with local error:

4.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
3.9b
(sin (/ (+ (+ x x) eps) 2))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 37
Calls

4 calls. Slowest were:

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

series468.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

156.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
150.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
128.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
35.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.1s

Counts
18 → 49
Calls

18 calls. Slowest were:

297.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))
295.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))
251.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))

prune751.0ms

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 0.3b

localize51.0ms

Local error

Found 4 expressions with local error:

4.9b
(cbrt (* (* (* (cbrt (sin (/ (+ (+ x x) eps) 2))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2))))
4.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
3.9b
(sin (/ (+ (+ x x) eps) 2))

rewrite106.0ms

Algorithm
rewrite-expression-head
Counts
4 → 36
Calls

4 calls. Slowest were:

93.0ms
(cbrt (* (* (* (cbrt (sin (/ (+ (+ x x) eps) 2))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2))))
5.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.0ms
(sin (/ (+ (+ x x) eps) 2))

series411.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

134.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
126.0ms
(cbrt (* (* (* (cbrt (sin (/ (+ (+ x x) eps) 2))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2))))
117.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
33.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.2s

Counts
17 → 48
Calls

17 calls. Slowest were:

318.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))
305.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))
296.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))) (* 1/3 (* x (exp (* 1/3 (- (log 2) (log eps))))))) (* 13/162 (* (pow x 2) (exp (* 1/3 (+ (* 2 (log eps)) (log 1/4)))))))

prune859.0ms

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 0.3b

regimes364.0ms

Accuracy

97% (0.4b remaining)

Error of 0.8b against oracle of 0.4b and baseline of 15.2b

bsearch176.0ms

end0.0ms

sample8.6s

Algorithm
intervals