Average Error: 39.2 → 0.8
Time: 22.2s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.7184081031776776 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.41130289542898757:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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 3 regimes

  2. if eps < -4.7184081031776776e-05

    1. Initial program 30.0

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

    3. Applied cos-sum0.9

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

    if -4.7184081031776776e-05 < eps < 0.41130289542898757

    1. Initial program 49.2

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

    3. Applied diff-cos38.3

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

    4. Simplified0.6

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

    if 0.41130289542898757 < eps

    1. Initial program 29.0

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.7184081031776776 \cdot 10^{-05}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.41130289542898757:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 323378853 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 19.7s)Debug log

sample324.0ms

Algorithm
intervals
Results
202.0ms151×body1280valid
46.0ms51×body640valid
44.0ms19×body2560valid
15.0ms24×body320valid
3.0ms8×body160valid
1.0ms3×body80valid

simplify8.0ms

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

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.1b

localize35.0ms

Local error

Found 2 expressions with local error:

3.6b
(cos (+ x eps))
0.7b
(- (cos (+ x eps)) (cos x))

rewrite42.0ms

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
11.0ms
(- (cos (+ x eps)) (cos x))
4.0ms
(cos (+ x eps))

series113.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
85.0ms
(- (cos (+ x eps)) (cos x))
28.0ms
(cos (+ x eps))

simplify629.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
354.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
145.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
42.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
28.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
10.0ms
(- (sqrt (cos (+ x eps))) (sqrt (cos x)))

prune470.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.4b

localize62.0ms

Local error

Found 2 expressions with local error:

3.6b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite34.0ms

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
23.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
11.0ms
(sin (/ (+ (+ x x) eps) 2))

series176.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
134.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
42.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify784.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
356.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
121.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
104.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
27.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
27.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))

prune592.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.3b

localize38.0ms

Local error

Found 4 expressions with local error:

3.6b
(sin (/ (+ (+ x x) eps) 2))
3.6b
(sin (/ (+ (+ x x) eps) 2))
3.6b
(sin (/ (+ (+ x x) eps) 2))
0.6b
(cbrt (sin (/ (+ (+ x x) eps) 2)))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×cbrt-prod
1×pow1/3
Counts
4 → 32
Calls
4 calls:
Slowest
14.0ms
(sin (/ (+ (+ x x) eps) 2))
10.0ms
(sin (/ (+ (+ x x) eps) 2))
10.0ms
(sin (/ (+ (+ x x) eps) 2))
5.0ms
(cbrt (sin (/ (+ (+ x x) eps) 2)))

series280.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
154.0ms
(cbrt (sin (/ (+ (+ x x) eps) 2)))
42.0ms
(sin (/ (+ (+ x x) eps) 2))
42.0ms
(sin (/ (+ (+ x x) eps) 2))
41.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify991.0ms

Counts
15 → 44
Calls
15 calls:
Slowest
438.0ms
(- (+ (* 1/3 (* x (exp (* 1/3 (- (log 4) (* 2 (log eps))))))) (exp (* 1/3 (+ (log 1/2) (log eps))))) (* 7/81 (* (pow x 2) (exp (* 1/3 (+ (log 1/2) (log eps)))))))
105.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
103.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
103.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
30.0ms
(pow (sin (* 1/2 (+ (* 2 x) eps))) 1/3)

prune1.0s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize42.0ms

Local error

Found 4 expressions with local error:

3.6b
(sin (/ (+ (+ x x) eps) 2))
3.6b
(sin (/ (+ (+ x x) eps) 2))
3.6b
(sin (/ (+ (+ x x) eps) 2))
3.6b
(sin (/ (+ (+ x x) eps) 2))

rewrite47.0ms

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
Counts
4 → 28
Calls
4 calls:
Slowest
15.0ms
(sin (/ (+ (+ x x) eps) 2))
11.0ms
(sin (/ (+ (+ x x) eps) 2))
10.0ms
(sin (/ (+ (+ x x) eps) 2))
10.0ms
(sin (/ (+ (+ x x) eps) 2))

series171.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
44.0ms
(sin (/ (+ (+ x x) eps) 2))
43.0ms
(sin (/ (+ (+ x x) eps) 2))
42.0ms
(sin (/ (+ (+ x x) eps) 2))
41.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify617.0ms

Counts
12 → 40
Calls
12 calls:
Slowest
107.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
106.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
105.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
102.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
24.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune986.0ms

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.3b

regimes350.0ms

Accuracy

97.5% (0.4b remaining)

Error of 0.8b against oracle of 0.4b and baseline of 14.9b

bsearch341.0ms

end0.0ms

sample11.4s

Algorithm
intervals
Results
6.2s4449×body1280valid
2.3s1885×body640valid
2.2s670×body2560valid
493.0ms647×body320valid
117.0ms277×body160valid
15.0ms72×body80valid