Average Error: 39.5 → 0.7
Time: 16.2s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.025970436598878203 \lor \neg \left(\varepsilon \le 8.149891340493173 \cdot 10^{-05}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

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 < -0.025970436598878203 or 8.149891340493173e-05 < eps

    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 -0.025970436598878203 < eps < 8.149891340493173e-05

    1. Initial program 48.9

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

    3. Applied diff-cos36.9

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

    4. Simplified0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.025970436598878203 \lor \neg \left(\varepsilon \le 8.149891340493173 \cdot 10^{-05}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019018 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 14.8s)Debug log

sample258.0ms

Algorithm
intervals
Results
148.0ms129×body1280valid
44.0ms65×body640valid
35.0ms17×body2560valid
19.0ms36×body320valid
2.0ms8×body160valid
0.0ms1×body80valid

simplify7.0ms

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: 40.5b

localize27.0ms

Local error

Found 2 expressions with local error:

3.8b
(cos (+ x eps))
0.8b
(- (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
9.0ms
(- (cos (+ x eps)) (cos x))
3.0ms
(cos (+ x eps))

series111.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
64.0ms
(- (cos (+ x eps)) (cos x))
47.0ms
(cos (+ x eps))

simplify370.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
225.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
76.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
18.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
16.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
9.0ms
(+ (sqrt (cos (+ x eps))) (sqrt (cos x)))

prune304.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.4b

localize22.0ms

Local error

Found 2 expressions with local error:

3.7b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite13.0ms

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
9.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
4.0ms
(sin (/ (+ (+ x x) eps) 2))

series98.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
70.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
27.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify470.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
222.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
78.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
54.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
18.0ms
(- (cos (- (/ (+ (+ x x) eps) 2) (/ eps 2))) (cos (+ (/ (+ (+ x x) eps) 2) (/ eps 2))))
16.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

prune399.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize23.0ms

Local error

Found 4 expressions with local error:

5.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
5.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
5.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
3.7b
(sin (/ (+ (+ x x) eps) 2))

rewrite33.0ms

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

series499.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
169.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
156.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
145.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
28.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.4s

Counts
18 → 49
Calls
18 calls:
Slowest
378.0ms
(- (+ (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)))))))
369.0ms
(- (+ (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)))))))
306.0ms
(- (+ (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)))))))
88.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
29.0ms
(pow (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))) 1/3)

prune738.0ms

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 0.3b

localize25.0ms

Local error

Found 4 expressions with local error:

5.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
5.9b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
3.7b
(sin (/ (+ (+ x x) eps) 2))
3.7b
(sin (/ (+ (+ x x) eps) 2))

rewrite29.0ms

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
2×sin-mult
2×pow1/3
2×cbrt-div
2×cbrt-prod
Counts
4 → 34
Calls
4 calls:
Slowest
8.0ms
(sin (/ (+ (+ x x) eps) 2))
7.0ms
(sin (/ (+ (+ x x) eps) 2))
7.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
7.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

series258.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
113.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
104.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
21.0ms
(sin (/ (+ (+ x x) eps) 2))
20.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify843.0ms

Counts
16 → 46
Calls
16 calls:
Slowest
248.0ms
(- (+ (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)))))))
247.0ms
(- (+ (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)))))))
90.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
86.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
23.0ms
(pow (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))) 1/3)

prune550.0ms

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 0.3b

regimes184.0ms

Accuracy

97.6% (0.3b remaining)

Error of 0.7b against oracle of 0.4b and baseline of 15.1b

bsearch187.0ms

end0.0ms

sample7.9s

Algorithm
intervals
Results
4.0s4478×body1280valid
1.6s1863×body640valid
1.5s618×body2560valid
515.0ms297×body160valid
243.0ms671×body320valid
8.0ms73×body80valid