Average Error: 39.5 → 0.8
Time: 19.2s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.452743699206042 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00191500921306136:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} - 1)^*)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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

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

    6. Applied expm1-log1p-u0.5

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

    8. Applied log1p-expm1-u0.6

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.452743699206042 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00191500921306136:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log_* (1 + (e^{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} - 1)^*)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

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

Details

Time bar (total: 17.7s)Debug log

sample266.0ms

Algorithm
intervals

simplify8.0ms

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

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 39.6b

localize27.0ms

Local error

Found 2 expressions with local error:

2.9b
(cos (+ x eps))
0.7b
(- (cos (+ x eps)) (cos x))

rewrite14.0ms

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
10.0ms
(- (cos (+ x eps)) (cos x))
3.0ms
(cos (+ x eps))

series114.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
93.0ms
(- (cos (+ x eps)) (cos x))
21.0ms
(cos (+ x eps))

simplify958.0ms

Counts
33 → 46
Calls
33 calls:
Slowest
315.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
128.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
39.0ms
(fma 1 (cos (+ x eps)) (- (* (cos x) 1)))
39.0ms
(* (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)))))

prune502.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.6b

localize53.0ms

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)))

rewrite21.0ms

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
12.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
9.0ms
(sin (/ (+ (+ x x) eps) 2))

series135.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
112.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
23.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify785.0ms

Counts
15 → 35
Calls
15 calls:
Slowest
386.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
112.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
105.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
28.0ms
(* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
26.0ms
(sin (* 1/2 (+ (* 2 x) eps)))

prune388.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.5b

localize25.0ms

Local error

Found 4 expressions with local error:

2.8b
(sin (/ (+ (+ x x) eps) 2))
0.4b
(log1p (sin (/ (+ (+ x x) eps) 2)))
0.3b
(* (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
0.2b
(expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))

rewrite22.0ms

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×cbrt-unprod
1×*-commutative
1×log1p-expm1
1×prod-exp
1×pow-prod-down
1×expm1-log1p
1×log1p-udef
Counts
4 → 50
Calls
4 calls:
Slowest
10.0ms
(* (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
10.0ms
(sin (/ (+ (+ x x) eps) 2))
1.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
0.0ms
(expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))

series273.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
131.0ms
(* (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
65.0ms
(expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))
54.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
22.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.0s

Counts
22 → 62
Calls
22 calls:
Slowest
354.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
116.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
104.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
87.0ms
(- (+ x (* 1/2 eps)) (* 1/2 (pow x 2)))
72.0ms
(* (* (* (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))) (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))) (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune878.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 4 expressions with local error:

2.8b
(sin (/ (+ (+ x x) eps) 2))
0.4b
(log1p (sin (/ (+ (+ x x) eps) 2)))
0.3b
(* (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2)))
0.3b
(log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))))

rewrite10.0ms

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*
2×log1p-expm1
2×log1p-udef
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
4 → 50
Calls
4 calls:
Slowest
5.0ms
(* (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2)))
5.0ms
(sin (/ (+ (+ x x) eps) 2))
0.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
0.0ms
(log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))))

series399.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
216.0ms
(* (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))))) (sin (/ eps 2)))
120.0ms
(log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))))
41.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
21.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.1s

Counts
22 → 62
Calls
22 calls:
Slowest
401.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
139.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
120.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
91.0ms
(* (* (* (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))))) (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))))) (log1p (expm1 (expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
68.0ms
(- (+ x (* 1/2 eps)) (* 1/2 (pow x 2)))

prune932.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.5b

regimes346.0ms

Accuracy

97.2% (0.4b remaining)

Error of 0.8b against oracle of 0.4b and baseline of 14.8b

bsearch238.0ms

end0.0ms

sample9.1s

Algorithm
intervals