Average Error: 39.1 → 0.8
Time: 18.6s
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 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 17.0s)Debug log

sample159.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.8b

localize13.0ms

Local error

Found 2 expressions with local error:

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

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

37.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series94.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

70.0ms
(- (cos (+ x eps)) (cos x))
24.0ms
(cos (+ x eps))

simplify846.0ms

Counts
33 → 46
Calls

33 calls. Slowest were:

282.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
162.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
30.0ms
(fma (- (cbrt (cos x))) (* (cbrt (cos x)) (cbrt (cos x))) (* (cbrt (cos x)) (* (cbrt (cos x)) (cbrt (cos x)))))

prune448.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.4b

localize32.0ms

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)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

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

series130.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

101.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
29.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify572.0ms

Counts
15 → 35
Calls

15 calls. Slowest were:

300.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
96.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
52.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune537.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize24.0ms

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))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

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

series502.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

182.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
164.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
128.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
28.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify869.0ms

Counts
18 → 57
Calls

18 calls. Slowest were:

163.0ms
(- (+ (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)))))))
162.0ms
(- (+ (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)))))))
136.0ms
(- (+ (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)))))))

prune870.0ms

Pruning

14 alts after pruning (12 fresh and 2 done)

Merged error: 0.3b

localize49.0ms

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))

rewrite80.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

67.0ms
(cbrt (* (* (* (cbrt (sin (/ (+ (+ x x) eps) 2))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2))))
4.0ms
(sin (/ (+ (+ x x) eps) 2))
4.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

series437.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
136.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
117.0ms
(cbrt (* (* (* (cbrt (sin (/ (+ (+ x x) eps) 2))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (cbrt (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2))))
32.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify982.0ms

Counts
17 → 56
Calls

17 calls. Slowest were:

187.0ms
(- (+ (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)))))))
165.0ms
(- (+ (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)))))))
154.0ms
(- (+ (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)))))))

prune1.1s

Pruning

14 alts after pruning (11 fresh and 3 done)

Merged error: 0.3b

regimes295.0ms

Accuracy

97% (0.4b remaining)

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

bsearch203.0ms

end1.0ms

sample8.6s

Algorithm
intervals