Average Error: 37.6 → 0.5
Time: 21.0s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \frac{\left({\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}{(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left((\left(\sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right))_*\right) + \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right))_*}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.6
Target15.0
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied diff-sin37.9

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  9. Applied cos-sum0.3

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

  11. Applied flip3--0.4

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}^{3} - {\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}^{3}}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)}}\right)\]

  12. Applied associate-*r/0.5

    \[\leadsto 2 \cdot \color{blue}{\frac{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left({\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}^{3} - {\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}^{3}\right)}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)}}\]

  13. Simplified0.5

    \[\leadsto 2 \cdot \frac{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left({\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}^{3} - {\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}^{3}\right)}{\color{blue}{(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left((\left(\sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right))_*\right) + \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right))_*}}\]

  14. Final simplification0.5

    \[\leadsto 2 \cdot \frac{\left({\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}^{3}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}{(\left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left((\left(\sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right))_*\right) + \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right))_*}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(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: 19.3s)Debug log

sample207.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.2b

localize64.0ms

Local error

Found 2 expressions with local error:

3.8b
(sin (+ x eps))
0.8b
(- (sin (+ x eps)) (sin x))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

13.0ms
(- (sin (+ x eps)) (sin x))
3.0ms
(sin (+ x eps))

series62.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(- (sin (+ x eps)) (sin x))
27.0ms
(sin (+ x eps))

simplify1.3s

Counts
33 → 46
Calls

33 calls. Slowest were:

762.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
46.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
41.0ms
(fma (sqrt (sin (+ x eps))) (sqrt (sin (+ x eps))) (- (* (sqrt (sin x)) (sqrt (sin x)))))

prune437.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.5b

localize25.0ms

Local error

Found 3 expressions with local error:

3.6b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.2b
(/ (+ (+ x x) eps) 2)

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
3 → 45
Calls

3 calls. Slowest were:

12.0ms
(/ (+ (+ x x) eps) 2)
6.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
5.0ms
(cos (/ (+ (+ x x) eps) 2))

series94.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

60.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
17.0ms
(/ (+ (+ x x) eps) 2)
17.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.1s

Counts
21 → 54
Calls

21 calls. Slowest were:

369.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
259.0ms
(- (* (+ x x) (+ x x)) (* eps eps))
128.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))

prune575.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.5b

localize38.0ms

Local error

Found 2 expressions with local error:

3.6b
(cos (fma 1/2 eps x))
0.2b
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

14.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
4.0ms
(cos (fma 1/2 eps x))

series74.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

50.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
24.0ms
(cos (fma 1/2 eps x))

simplify758.0ms

Counts
16 → 36
Calls

16 calls. Slowest were:

353.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
226.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/8 (pow eps 2))))
91.0ms
(* (* (* (sin (* eps 1/2)) (sin (* eps 1/2))) (sin (* eps 1/2))) (* (* (cos (fma 1/2 eps x)) (cos (fma 1/2 eps x))) (cos (fma 1/2 eps x))))

prune453.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.1b

localize49.0ms

Local error

Found 4 expressions with local error:

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

rewrite99.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

61.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
18.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
11.0ms
(* (sin (* 1/2 eps)) (sin x))

series315.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

132.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
83.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
55.0ms
(* (sin (* 1/2 eps)) (sin x))
46.0ms
(* (cos (* 1/2 eps)) (cos x))

simplify3.3s

Counts
54 → 99
Calls

54 calls. Slowest were:

624.0ms
(* (* (* (sin (* eps 1/2)) (sin (* eps 1/2))) (sin (* eps 1/2))) (* (* (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))))
401.0ms
(- (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x))))
380.0ms
(- (* (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) 2) (* 2 (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x)))))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

regimes216.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch5.0ms

end0.0ms

sample8.7s

Algorithm
intervals