Average Error: 39.5 → 0.7
Time: 16.8s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001097902846600216:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 5.359431187098088 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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 3 regimes

  2. if eps < -0.0001097902846600216

    1. Initial program 30.3

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

    if -0.0001097902846600216 < eps < 5.359431187098088e-05

    1. Initial program 49.3

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

    3. Applied diff-cos37.9

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

    4. Simplified0.5

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

    if 5.359431187098088e-05 < eps

    1. Initial program 29.6

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

    3. Applied cos-sum0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001097902846600216:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 5.359431187098088 \cdot 10^{-05}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 15.7s)Debug log

sample214.0ms

Algorithm
intervals
Results
122.0ms136×body1280valid
36.0ms21×body2560valid
32.0ms59×body640valid
11.0ms24×body320valid
4.0ms16×body160valid

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (cos (+ x eps)) (cos x))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.5b

localize12.0ms

Local error

Found 2 expressions with local error:

3.7b
(cos (+ x eps))
2.0b
(- (cos (+ x eps)) (cos x))

rewrite6.0ms

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
4.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series83.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
71.0ms
(- (cos (+ x eps)) (cos x))
12.0ms
(cos (+ x eps))

simplify345.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
200.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
72.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
25.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
14.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
7.0ms
(- (cos (+ x eps)) (cos x))

prune286.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize36.0ms

Local error

Found 2 expressions with local error:

3.8b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×associate-*l*
3×associate-*r*
2×add-log-exp
1×sin-mult
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
2 → 25
Calls
2 calls:
Slowest
12.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
8.0ms
(sin (/ (+ (+ x x) eps) 2))

series100.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
79.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
20.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify504.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
230.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
77.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
55.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
22.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
21.0ms
(- (cos (- (/ (+ (+ x x) eps) 2) (/ eps 2))) (cos (+ (/ (+ (+ x x) eps) 2) (/ eps 2))))

prune395.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize14.0ms

Local error

Found 4 expressions with local error:

3.8b
(sin (/ (+ (+ x x) eps) 2))
0.6b
(cbrt (sin (/ eps 2)))
0.6b
(cbrt (sin (/ eps 2)))
0.6b
(cbrt (sin (/ eps 2)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
9×cbrt-prod
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×pow1/3
Counts
4 → 40
Calls
4 calls:
Slowest
8.0ms
(sin (/ (+ (+ x x) eps) 2))
2.0ms
(cbrt (sin (/ eps 2)))
1.0ms
(cbrt (sin (/ eps 2)))
1.0ms
(cbrt (sin (/ eps 2)))

series478.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
160.0ms
(cbrt (sin (/ eps 2)))
153.0ms
(cbrt (sin (/ eps 2)))
136.0ms
(cbrt (sin (/ eps 2)))
30.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.2s

Counts
21 → 52
Calls
21 calls:
Slowest
375.0ms
(- (exp (* 1/3 (+ (log 1/2) (log eps)))) (+ (* 1/51840 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 4))) (* 1/72 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 2)))))
346.0ms
(- (exp (* 1/3 (+ (log 1/2) (log eps)))) (+ (* 1/51840 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 4))) (* 1/72 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 2)))))
291.0ms
(- (exp (* 1/3 (+ (log 1/2) (log eps)))) (+ (* 1/51840 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 4))) (* 1/72 (* (exp (* 1/3 (+ (log 1/2) (log eps)))) (pow eps 2)))))
72.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
18.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune685.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize14.0ms

Local error

Found 4 expressions with local error:

7.2b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
7.2b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
7.2b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
3.8b
(sin (/ (+ (+ x x) eps) 2))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×sin-mult
3×pow1/3
3×cbrt-div
3×cbrt-prod
Counts
4 → 37
Calls
4 calls:
Slowest
4.0ms
(sin (/ (+ (+ x x) eps) 2))
4.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

series394.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
132.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
127.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
108.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
27.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.2s

Counts
18 → 49
Calls
18 calls:
Slowest
331.0ms
(- (+ (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)))))))
311.0ms
(- (+ (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)))))))
275.0ms
(- (+ (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)))))))
43.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
33.0ms
(pow (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))) 1/3)

prune733.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.3b

regimes189.0ms

Accuracy

97.7% (0.3b remaining)

Error of 0.7b against oracle of 0.4b and baseline of 14.9b

bsearch267.0ms

end0.0ms

sample8.5s

Algorithm
intervals
Results
5.1s4405×body1280valid
1.5s651×body2560valid
1.2s1856×body640valid
328.0ms704×body320valid
165.0ms300×body160valid
11.0ms84×body80valid