Average Error: 37.0 → 0.5
Time: 17.2s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2868657370910494 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2175984351199754 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target14.6
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.2868657370910494e-08

    1. Initial program 29.3

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

    3. Applied sin-sum0.6

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

    if -1.2868657370910494e-08 < eps < 1.2175984351199754e-08

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    4. Simplified0.4

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

    if 1.2175984351199754e-08 < eps

    1. Initial program 29.4

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2868657370910494 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2175984351199754 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))

Details

Time bar (total: 16.2s)Debug log

sample276.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (sin (+ x eps)) (sin x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.2b

localize28.0ms

Local error

Found 2 expressions with local error:

2.8b
(sin (+ x eps))
1.8b
(- (sin (+ x eps)) (sin x))

rewrite15.0ms

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×sin-sum
2×add-cbrt-cube
2×pow1
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×associate--l+
1×diff-sin
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 24
Calls
2 calls:
Slowest
12.0ms
(- (sin (+ x eps)) (sin x))
3.0ms
(sin (+ x eps))

series87.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
58.0ms
(- (sin (+ x eps)) (sin x))
28.0ms
(sin (+ x eps))

simplify417.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
266.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
68.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
25.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
16.0ms
(/ (exp (sin (+ x eps))) (exp (sin x)))
9.0ms
(- (sqrt (sin (+ x eps))) (sqrt (sin x)))

prune270.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize37.0ms

Local error

Found 2 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite14.0ms

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×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
2 → 24
Calls
2 calls:
Slowest
9.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
5.0ms
(cos (/ (+ (+ x x) eps) 2))

series80.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
62.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
18.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify728.0ms

Counts
14 → 30
Calls
14 calls:
Slowest
307.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
191.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
82.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
27.0ms
(* (cos (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
20.0ms
(+ (log (cos (/ (+ (+ x x) eps) 2))) (log (sin (/ eps 2))))

prune361.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 4 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
0.5b
(log (exp (cos (/ (+ (+ x x) eps) 2))))
0.2b
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
0.1b
(exp (cos (/ (+ (+ x x) eps) 2)))

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×exp-prod
8×log-pow
8×pow1
7×associate-*l*
6×add-exp-log
6×add-cbrt-cube
5×add-log-exp
3×associate-*r*
3×log-prod
1×rem-exp-log
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×rem-log-exp
Counts
4 → 54
Calls
4 calls:
Slowest
25.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
14.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))
10.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
4.0ms
(cos (/ (+ (+ x x) eps) 2))

series168.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
70.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
43.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
28.0ms
(cos (/ (+ (+ x x) eps) 2))
27.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))

simplify1.6s

Counts
34 → 66
Calls
34 calls:
Slowest
373.0ms
(- E (+ (* 1/2 (* x (* E eps))) (* 1/2 (* (pow x 2) E))))
361.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
239.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
172.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
88.0ms
(* (* (* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (log (exp (cos (/ (+ (+ x x) eps) 2))))) (log (exp (cos (/ (+ (+ x x) eps) 2))))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune881.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.5b

localize17.0ms

Local error

Found 4 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
0.5b
(log (log (exp (exp (cos (/ (+ (+ x x) eps) 2))))))
0.5b
(exp (exp (cos (/ (+ (+ x x) eps) 2))))
0.2b
(log (exp (exp (cos (/ (+ (+ x x) eps) 2)))))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
10×log-prod
9×add-cube-cbrt
9×log-pow
9×*-un-lft-identity
9×exp-prod
9×add-sqr-sqrt
7×pow1
5×add-log-exp
5×add-exp-log
4×add-cbrt-cube
2×rem-log-exp
1×rem-exp-log
Counts
4 → 49
Calls
4 calls:
Slowest
18.0ms
(exp (exp (cos (/ (+ (+ x x) eps) 2))))
8.0ms
(log (exp (exp (cos (/ (+ (+ x x) eps) 2)))))
5.0ms
(cos (/ (+ (+ x x) eps) 2))
3.0ms
(log (log (exp (exp (cos (/ (+ (+ x x) eps) 2))))))

series203.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
98.0ms
(exp (exp (cos (/ (+ (+ x x) eps) 2))))
48.0ms
(log (exp (exp (cos (/ (+ (+ x x) eps) 2)))))
28.0ms
(log (log (exp (exp (cos (/ (+ (+ x x) eps) 2))))))
28.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.5s

Counts
31 → 61
Calls
31 calls:
Slowest
386.0ms
(- E (+ (* 1/2 (* x (* E eps))) (* 1/2 (* (pow x 2) E))))
283.0ms
(- (exp E) (+ (* 1/2 (* (pow x 2) (* E (exp E)))) (* 1/2 (* x (* E (* (exp E) eps))))))
245.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
195.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
26.0ms
(exp (cos (* 1/2 (+ (* 2 x) eps))))

prune917.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.5b

regimes167.0ms

Accuracy

99.6% (0.1b remaining)

Error of 0.5b against oracle of 0.4b and baseline of 14.6b

bsearch254.0ms

end0.0ms

sample8.0s

Algorithm
intervals