Average Error: 39.5 → 0.7
Time: 17.9s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001097902846600216:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 5.359431187098088 \cdot 10^{-05}:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^* \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

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

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

    6. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \color{blue}{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right))} - 1)^*}\]

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001097902846600216:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 5.359431187098088 \cdot 10^{-05}:\\ \;\;\;\;(e^{\log_* (1 + \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^* \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

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

Details

Time bar (total: 16.5s)Debug log

sample212.0ms

Algorithm
intervals
Results
120.0ms136×body1280valid
34.0ms59×body640valid
32.0ms21×body2560valid
9.0ms24×body320valid
5.0ms16×body160valid

simplify8.0ms

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.5b

localize53.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
11×*-un-lft-identity
11×add-sqr-sqrt
9×add-cube-cbrt
9×prod-diff
4×add-log-exp
3×fma-neg
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×cos-sum
2×pow1
2×expm1-log1p-u
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 → 40
Calls
2 calls:
Slowest
7.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series75.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
62.0ms
(- (cos (+ x eps)) (cos x))
13.0ms
(cos (+ x eps))

simplify836.0ms

Counts
33 → 46
Calls
33 calls:
Slowest
226.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
178.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
40.0ms
(fma (sqrt (cos (+ x eps))) (sqrt (cos (+ x eps))) (- (* (sqrt (cos x)) (sqrt (cos x)))))
40.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
31.0ms
(fma (- (cbrt (cos x))) (* (cbrt (cos x)) (cbrt (cos x))) (* (cbrt (cos x)) (* (cbrt (cos x)) (cbrt (cos x)))))

prune407.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.4b

localize26.0ms

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)))

rewrite15.0ms

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
2×log1p-expm1-u
2×expm1-log1p-u
1×sin-mult
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
2 → 29
Calls
2 calls:
Slowest
10.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
4.0ms
(sin (/ (+ (+ x x) eps) 2))

series104.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
76.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
28.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify599.0ms

Counts
15 → 35
Calls
15 calls:
Slowest
310.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
82.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
62.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
26.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
26.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune427.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.3b

localize9.0ms

Local error

Found 4 expressions with local error:

3.8b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.1b
(expm1 (log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))))
0.1b
(log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×expm1-log1p-u
4×add-log-exp
4×log1p-expm1-u
3×associate-*l*
3×associate-*r*
1×expm1-udef
1×sin-mult
1×cbrt-unprod
1×*-commutative
1×log1p-expm1
1×prod-exp
1×pow-prod-down
1×expm1-log1p
1×log1p-udef
Counts
4 → 51
Calls
4 calls:
Slowest
10.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
7.0ms
(sin (/ (+ (+ x x) eps) 2))
0.0ms
(log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
0.0ms
(expm1 (log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))))

series454.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
171.0ms
(expm1 (log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))))
160.0ms
(log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
100.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
23.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.9s

Counts
23 → 63
Calls
23 calls:
Slowest
800.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
354.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
323.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
105.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
67.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune730.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize28.0ms

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)))

rewrite19.0ms

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×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×pow1/3
Counts
4 → 48
Calls
4 calls:
Slowest
14.0ms
(sin (/ (+ (+ x x) eps) 2))
1.0ms
(cbrt (sin (/ eps 2)))
1.0ms
(cbrt (sin (/ eps 2)))
1.0ms
(cbrt (sin (/ eps 2)))

series499.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
179.0ms
(cbrt (sin (/ eps 2)))
152.0ms
(cbrt (sin (/ eps 2)))
142.0ms
(cbrt (sin (/ eps 2)))
27.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify569.0ms

Counts
21 → 60
Calls
21 calls:
Slowest
144.0ms
(- (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)))))
131.0ms
(- (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)))))
120.0ms
(- (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)))))
90.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
26.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune823.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

regimes180.0ms

Accuracy

97.7% (0.3b remaining)

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

bsearch214.0ms

end0.0ms

sample8.2s

Algorithm
intervals
Results
4.7s4405×body1280valid
1.6s1856×body640valid
1.4s651×body2560valid
317.0ms704×body320valid
79.0ms300×body160valid
11.0ms84×body80valid