Average Error: 37.3 → 0.4
Time: 25.4s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1625623720612472 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.143319438209034 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.6
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.1625623720612472e-08

    1. Initial program 30.5

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

    3. Applied sin-sum0.5

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

    if -1.1625623720612472e-08 < eps < 8.143319438209034e-09

    1. Initial program 44.7

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

    3. Applied diff-sin44.7

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

    4. Simplified0.3

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

    if 8.143319438209034e-09 < eps

    1. Initial program 30.5

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1625623720612472 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.143319438209034 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(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: 22.9s)Debug log

sample371.0ms

Algorithm
intervals
Results
112×202.0ms(body real 1280)
54×56.0ms(body real 640)
21×55.0ms(body real 2560)
30×22.0ms(body real 320)
256×11.0ms(pre true 80)
31×7.0ms(body real 80)
8×4.0ms(body real 160)

simplify8.0ms

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

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.7b

localize38.0ms

Local error

Found 2 expressions with local error:

3.4b
(sin (+ x eps))
0.9b
(- (sin (+ x eps)) (sin x))

rewrite19.0ms

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
14.0ms
(- (sin (+ x eps)) (sin x))
4.0ms
(sin (+ x eps))

series116.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
84.0ms
(- (sin (+ x eps)) (sin x))
32.0ms
(sin (+ x eps))

simplify586.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
388.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
74.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
42.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
27.0ms
(/ (exp (sin (+ x eps))) (exp (sin x)))
9.0ms
(- (sqrt (sin (+ x eps))) (sqrt (sin x)))

prune486.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize62.0ms

Local error

Found 2 expressions with local error:

3.2b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite26.0ms

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
15.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
11.0ms
(cos (/ (+ (+ x x) eps) 2))

series145.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
107.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
38.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.0s

Counts
14 → 30
Calls
14 calls:
Slowest
434.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
271.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
120.0ms
(* (* (* (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)))
27.0ms
(* (cos (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))

prune583.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.4b

localize35.0ms

Local error

Found 4 expressions with local error:

3.2b
(cos (/ (+ (+ x x) eps) 2))
0.4b
(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)))

rewrite79.0ms

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
34.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
20.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))
12.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
11.0ms
(cos (/ (+ (+ x x) eps) 2))

series254.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
109.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
78.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
34.0ms
(cos (/ (+ (+ x x) eps) 2))
33.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))

simplify2.0s

Counts
34 → 66
Calls
34 calls:
Slowest
448.0ms
(- E (+ (* 1/2 (* x (* E eps))) (* 1/2 (* (pow x 2) E))))
442.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
276.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
270.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
112.0ms
(* (* (* (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))))

prune1.4s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.4b

localize38.0ms

Local error

Found 4 expressions with local error:

3.2b
(cos (/ (+ (+ x x) eps) 2))
3.2b
(cos (/ (+ (+ x x) eps) 2))
3.2b
(cos (/ (+ (+ x x) eps) 2))
0.4b
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))

rewrite55.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
1×rem-cbrt-cube
1×cos-mult
1×associate-*l/
1×pow3
1×pow1/3
1×cbrt-div
1×cbrt-prod
Counts
4 → 32
Calls
4 calls:
Slowest
18.0ms
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))
15.0ms
(cos (/ (+ (+ x x) eps) 2))
11.0ms
(cos (/ (+ (+ x x) eps) 2))
11.0ms
(cos (/ (+ (+ x x) eps) 2))

series136.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
35.0ms
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))
34.0ms
(cos (/ (+ (+ x x) eps) 2))
34.0ms
(cos (/ (+ (+ x x) eps) 2))
33.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.4s

Counts
15 → 44
Calls
15 calls:
Slowest
294.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
277.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
273.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
270.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
97.0ms
(cbrt (* (+ (cos (+ (/ (+ (+ x x) eps) 2) (/ (+ (+ x x) eps) 2))) (cos (- (/ (+ (+ x x) eps) 2) (/ (+ (+ x x) eps) 2)))) (cos (/ (+ (+ x x) eps) 2))))

prune986.0ms

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0.4b

regimes358.0ms

Accuracy

99.5% (0.1b remaining)

Error of 0.4b against oracle of 0.4b and baseline of 15.6b

bsearch367.0ms

end0.0ms

sample12.3s

Algorithm
intervals
Results
3271×6.8s(body real 1280)
1842×2.2s(body real 640)
635×1.7s(body real 2560)
819×637.0ms(body real 320)
8000×264.0ms(pre true 80)
1073×248.0ms(body real 80)
360×175.0ms(body real 160)