Average Error: 39.5 → 0.8
Time: 16.7s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.452743699206042 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00191500921306136:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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 < -7.452743699206042e-09

    1. Initial program 30.1

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

    3. Applied cos-sum1.2

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

    if -7.452743699206042e-09 < eps < 0.00191500921306136

    1. Initial program 49.0

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

    3. Applied diff-cos37.7

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

    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 0.00191500921306136 < eps

    1. Initial program 30.0

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.452743699206042 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00191500921306136:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 14.9s)Debug log

sample215.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(- (cos (+ x eps)) (cos x))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 39.6b

localize13.0ms

Local error

Found 2 expressions with local error:

2.9b
(cos (+ x eps))
0.7b
(- (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
5.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series73.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
44.0ms
(- (cos (+ x eps)) (cos x))
29.0ms
(cos (+ x eps))

simplify423.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
282.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
73.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
20.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
13.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
8.0ms
(- (cos (+ x eps)) (cos x))

prune277.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.6b

localize44.0ms

Local error

Found 2 expressions with local error:

2.8b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite27.0ms

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
18.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
9.0ms
(sin (/ (+ (+ x x) eps) 2))

series122.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
97.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
25.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify603.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
255.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
101.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
83.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
24.0ms
(- (cos (- (/ (+ (+ x x) eps) 2) (/ eps 2))) (cos (+ (/ (+ (+ x x) eps) 2) (/ eps 2))))
23.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune322.0ms

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.5b

localize19.0ms

Local error

Found 4 expressions with local error:

2.8b
(sin (/ (+ (+ x x) eps) 2))
2.8b
(sin (/ (+ (+ x x) eps) 2))
2.8b
(sin (/ (+ (+ x x) eps) 2))
0.6b
(cbrt (sin (/ (+ (+ x x) eps) 2)))

rewrite25.0ms

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
8.0ms
(sin (/ (+ (+ x x) eps) 2))
8.0ms
(sin (/ (+ (+ x x) eps) 2))
6.0ms
(sin (/ (+ (+ x x) eps) 2))
2.0ms
(cbrt (sin (/ (+ (+ x x) eps) 2)))

series164.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
95.0ms
(cbrt (sin (/ (+ (+ x x) eps) 2)))
24.0ms
(sin (/ (+ (+ x x) eps) 2))
24.0ms
(sin (/ (+ (+ x x) eps) 2))
21.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify776.0ms

Counts
15 → 44
Calls
15 calls:
Slowest
312.0ms
(- (+ (* 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)))))))
96.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
96.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
84.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
26.0ms
(pow (sin (* 1/2 (+ (* 2 x) eps))) 1/3)

prune667.0ms

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.5b

localize15.0ms

Local error

Found 4 expressions with local error:

4.8b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.8b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
4.8b
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
2.8b
(sin (/ (+ (+ x x) eps) 2))

rewrite21.0ms

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
7.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
6.0ms
(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))))

series439.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
148.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
134.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
130.0ms
(cbrt (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
27.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.1s

Counts
18 → 49
Calls
18 calls:
Slowest
309.0ms
(- (+ (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)))))))
302.0ms
(- (+ (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)))))))
270.0ms
(- (+ (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)))))))
46.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
27.0ms
(cbrt (- (cos (- (/ (+ (+ x x) eps) 2) (/ eps 2))) (cos (+ (/ (+ (+ x x) eps) 2) (/ eps 2)))))

prune787.0ms

Pruning

14 alts after pruning (12 fresh and 2 done)

Merged error: 0.5b

regimes333.0ms

Accuracy

97.5% (0.4b remaining)

Error of 0.8b against oracle of 0.4b and baseline of 14.7b

bsearch208.0ms

end0.0ms

sample8.1s

Algorithm
intervals