Average Error: 39.8 → 0.4
Time: 31.4s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) - \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)}{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x} \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right)\]

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. Initial program 39.8

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

  3. Applied diff-cos34.4

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

  4. Simplified15.6

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

  5. Taylor expanded around -inf 15.6

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

  6. Simplified15.6

    \[\leadsto \color{blue}{\sin \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}\]
  7. Using strategy rm

  8. Applied sin-sum0.4

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) + \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
  9. Using strategy rm

  10. Applied flip-+0.4

    \[\leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)\right) - \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}{\sin x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

  11. Final simplification0.4

    \[\leadsto \frac{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) - \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x\right)}{\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x} \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot -2\right)\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 26.9s)Debug log

sample468.0ms

Algorithm
intervals
Results
146×276.0ms(body real 1280)
57×68.0ms(body real 640)
21×67.0ms(body real 2560)
23×21.0ms(body real 320)
256×10.0ms(pre true 80)
8×4.0ms(body real 160)
1×0.0ms(body real 80)

simplify10.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
10.0ms
(- (cos (+ x eps)) (cos x))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 41.1b

localize40.0ms

Local error

Found 2 expressions with local error:

3.4b
(cos (+ x eps))
1.0b
(- (cos (+ x eps)) (cos x))

rewrite17.0ms

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
12.0ms
(- (cos (+ x eps)) (cos x))
4.0ms
(cos (+ x eps))

series178.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
145.0ms
(- (cos (+ x eps)) (cos x))
32.0ms
(cos (+ x eps))

simplify991.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
578.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
212.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
62.0ms
(* (sin (/ (- (+ x eps) x) 2)) (sin (/ (+ (+ x eps) x) 2)))
38.0ms
(/ (exp (cos (+ x eps))) (exp (cos x)))
22.0ms
(- (cos (+ x eps)) (cos x))

prune718.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.5b

localize78.0ms

Local error

Found 3 expressions with local error:

3.3b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.0b
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×add-exp-log
5×add-cbrt-cube
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×associate-*r*
3×add-log-exp
3×associate-*l*
2×sin-mult
2×*-commutative
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
Counts
3 → 35
Calls
3 calls:
Slowest
27.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
22.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
13.0ms
(sin (/ (+ (+ x x) eps) 2))

series373.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
164.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
153.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
56.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.3s

Counts
19 → 44
Calls
19 calls:
Slowest
405.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
384.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
125.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
107.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
42.0ms
(* -2 (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))))

prune912.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.4b

localize49.0ms

Local error

Found 2 expressions with local error:

3.3b
(sin (+ x (* eps 1/2)))
0.3b
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×sin-sum
Counts
2 → 23
Calls
2 calls:
Slowest
25.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
5.0ms
(sin (+ x (* eps 1/2)))

series614.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
570.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))
43.0ms
(sin (+ x (* eps 1/2)))

simplify930.0ms

Counts
13 → 29
Calls
13 calls:
Slowest
394.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
290.0ms
(* (* (* (sin (+ x (* eps 1/2))) (sin (+ x (* eps 1/2)))) (sin (+ x (* eps 1/2)))) (* (* (* -2 (sin (* eps 1/2))) (* -2 (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2)))))
111.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
29.0ms
(+ (log (sin (+ x (* eps 1/2)))) (log (* -2 (sin (* eps 1/2)))))
25.0ms
(* (sin (+ x (* eps 1/2))) (* -2 (sin (* eps 1/2))))

prune638.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.1b

localize50.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
0.2b
(* (cos x) (sin (* eps 1/2)))
0.1b
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
0.1b
(* (sin x) (cos (* eps 1/2)))

rewrite122.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
10×pow1
9×add-cube-cbrt
9×associate-*l*
9×*-un-lft-identity
9×add-sqr-sqrt
7×associate-*r*
6×add-log-exp
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
2×flip-+
2×associate-*l/
2×flip3-+
1×sin-cos-mult
1×sum-log
1×+-commutative
Counts
4 → 63
Calls
4 calls:
Slowest
67.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
24.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
15.0ms
(* (cos x) (sin (* eps 1/2)))
14.0ms
(* (sin x) (cos (* eps 1/2)))

series721.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
334.0ms
(* (+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2)))) (* -2 (sin (* eps 1/2))))
196.0ms
(+ (* (sin x) (cos (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))
100.0ms
(* (cos x) (sin (* eps 1/2)))
90.0ms
(* (sin x) (cos (* eps 1/2)))

simplify3.2s

Counts
38 → 75
Calls
38 calls:
Slowest
529.0ms
(- x (+ (* 1/8 (* x (pow eps 2))) (* 1/6 (pow x 3))))
468.0ms
(- (* 1/2 eps) (+ (* 1/4 (* (pow x 2) eps)) (* 1/48 (pow eps 3))))
367.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
262.0ms
(* (- (* (* (sin x) (cos (* eps 1/2))) (* (sin x) (cos (* eps 1/2)))) (* (* (cos x) (sin (* eps 1/2))) (* (cos x) (sin (* eps 1/2))))) (* -2 (sin (* eps 1/2))))
258.0ms
(* (+ (pow (* (sin x) (cos (* eps 1/2))) 3) (pow (* (cos x) (sin (* eps 1/2))) 3)) (* -2 (sin (* eps 1/2))))

prune1.7s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 0.0b

regimes769.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch5.0ms

end0.0ms

sample12.9s

Algorithm
intervals
Results
4400×7.5s(body real 1280)
1887×2.5s(body real 640)
635×1.7s(body real 2560)
713×546.0ms(body real 320)
296×249.0ms(body real 160)
8000×242.0ms(pre true 80)
69×16.0ms(body real 80)