Average Error: 39.6 → 0.7
Time: 18.8s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.345664659215098 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.0015573934673695188:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

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 < -9.345664659215098e-07

    1. Initial program 30.1

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

    3. Applied cos-sum1.0

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

    if -9.345664659215098e-07 < eps < 0.0015573934673695188

    1. Initial program 49.6

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

    3. Applied diff-cos38.0

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

    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

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

    if 0.0015573934673695188 < eps

    1. Initial program 29.8

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

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.345664659215098 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.0015573934673695188:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))

Details

Time bar (total: 17.1s)Debug log

sample263.0ms

Algorithm
intervals
Results
256×(pre true 80)
145×(body real 1280)
57×(body real 640)
27×(body real 2560)
16×(body real 320)
8×(body real 160)
3×(body real 80)

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (cos (+ x eps)) (cos x))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 37.5b

localize14.0ms

Local error

Found 2 expressions with local error:

4.0b
(cos (+ x eps))
0.7b
(- (cos (+ x eps)) (cos x))

rewrite8.0ms

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
5.0ms
(- (cos (+ x eps)) (cos x))
2.0ms
(cos (+ x eps))

series101.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
83.0ms
(- (cos (+ x eps)) (cos x))
18.0ms
(cos (+ x eps))

simplify810.0ms

Counts
33 → 46
Calls
33 calls:
Slowest
258.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
134.0ms
(- 1 (+ (* x eps) (* 1/2 (pow x 2))))
41.0ms
(fma (sqrt (cos (+ x eps))) (sqrt (cos (+ x eps))) (- (* (sqrt (cos x)) (sqrt (cos x)))))
37.0ms
(fma 1 (cos (+ x eps)) (- (* (cos x) 1)))
30.0ms
(fma (- (cbrt (cos x))) (* (cbrt (cos x)) (cbrt (cos x))) (* (cbrt (cos x)) (* (cbrt (cos x)) (cbrt (cos x)))))

prune426.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize22.0ms

Local error

Found 3 expressions with local error:

4.0b
(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))))

rewrite31.0ms

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×log1p-expm1-u
3×associate-*l*
3×expm1-log1p-u
2×sin-mult
2×*-commutative
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
Counts
3 → 41
Calls
3 calls:
Slowest
13.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
13.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
5.0ms
(sin (/ (+ (+ x x) eps) 2))

series191.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
80.0ms
(* -2 (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
77.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
34.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify931.0ms

Counts
19 → 50
Calls
19 calls:
Slowest
345.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
200.0ms
(- (* 1/6 (* (pow x 3) eps)) (+ (* 1/2 (pow eps 2)) (* x eps)))
113.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
91.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
32.0ms
(* -2 (* (sin (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps))))

prune410.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.3b

localize9.0ms

Local error

Found 4 expressions with local error:

4.0b
(sin (/ (+ (+ x x) eps) 2))
0.3b
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.2b
(expm1 (log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))))
0.1b
(log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite15.0ms

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)))
4.0ms
(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)))))

series399.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
156.0ms
(log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
146.0ms
(expm1 (log1p (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))))
77.0ms
(* (sin (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
20.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify1.5s

Counts
23 → 63
Calls
23 calls:
Slowest
356.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
344.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
322.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
120.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
60.0ms
(* (* (* (sin (/ (+ (+ x x) eps) 2)) (sin (/ (+ (+ x x) eps) 2))) (sin (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune955.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.3b

localize25.0ms

Local error

Found 4 expressions with local error:

4.0b
(sin (/ (+ (+ x x) eps) 2))
0.6b
(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))))

rewrite21.0ms

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)))
9.0ms
(sin (/ (+ (+ x x) eps) 2))
1.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
0.0ms
(expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))

series282.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
124.0ms
(* (expm1 (log1p (sin (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
71.0ms
(expm1 (log1p (sin (/ (+ (+ x x) eps) 2))))
52.0ms
(log1p (sin (/ (+ (+ x x) eps) 2)))
34.0ms
(sin (/ (+ (+ x x) eps) 2))

simplify984.0ms

Counts
22 → 62
Calls
22 calls:
Slowest
329.0ms
(- (+ (* 1/4 (pow eps 2)) (* 1/2 (* x eps))) (* 1/12 (* (pow x 3) eps)))
136.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
111.0ms
(- (+ x (* 1/2 eps)) (* 1/6 (pow x 3)))
97.0ms
(- (+ x (* 1/2 eps)) (* 1/2 (pow x 2)))
78.0ms
(* (* (* (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))))

prune722.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

regimes232.0ms

Accuracy

97.5% (0.4b remaining)

Error of 0.7b against oracle of 0.4b and baseline of 15.0b

bsearch319.0ms

end0.0ms

sample8.4s

Algorithm
intervals
Results
8000×(pre true 80)
4433×(body real 1280)
1827×(body real 640)
709×(body real 320)
664×(body real 2560)
277×(body real 160)
90×(body real 80)