Average Error: 36.9 → 0.3
Time: 19.1s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot (\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos x\right) + \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-\sin x\right)\right))_*\right)\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target14.9
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied diff-sin37.2

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

  4. Simplified14.9

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

  5. Taylor expanded around -inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  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 fma-neg0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019002 +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: 17.8s)Debug log

sample265.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

8.0ms
(- (sin (+ x eps)) (sin x))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.6b

localize28.0ms

Local error

Found 2 expressions with local error:

3.5b
(sin (+ x eps))
1.2b
(- (sin (+ x eps)) (sin x))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

12.0ms
(- (sin (+ x eps)) (sin x))
3.0ms
(sin (+ x eps))

series87.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(- (sin (+ x eps)) (sin x))
24.0ms
(sin (+ x eps))

simplify983.0ms

Counts
33 → 46
Calls

33 calls. Slowest were:

464.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
75.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
41.0ms
(fma (sqrt (sin (+ x eps))) (sqrt (sin (+ x eps))) (- (* (sqrt (sin x)) (sqrt (sin x)))))

prune444.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.3b

localize25.0ms

Local error

Found 2 expressions with local error:

3.5b
(cos (/ (+ (+ x x) eps) 2))
0.3b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

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

series68.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

51.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
17.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify643.0ms

Counts
14 → 34
Calls

14 calls. Slowest were:

330.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
116.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
89.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune369.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.3b

localize34.0ms

Local error

Found 2 expressions with local error:

3.5b
(cos (fma 1/2 eps x))
0.3b
(* (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)))
3.0ms
(cos (fma 1/2 eps x))

series86.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

57.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
29.0ms
(cos (fma 1/2 eps x))

simplify699.0ms

Counts
16 → 36
Calls

16 calls. Slowest were:

338.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
185.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/8 (pow eps 2))))
92.0ms
(* (* (* (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))))

prune429.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.1b

localize52.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (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))

rewrite112.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

65.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
22.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
12.0ms
(* (sin (* 1/2 eps)) (sin x))

series348.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
103.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
78.0ms
(* (sin (* 1/2 eps)) (sin x))
40.0ms
(* (cos (* 1/2 eps)) (cos x))

simplify3.5s

Counts
54 → 99
Calls

54 calls. Slowest were:

674.0ms
(* (* (* (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)))))
420.0ms
(- (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x))))
331.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))

prune1.5s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

regimes230.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch17.0ms

end0.0ms

sample7.9s

Algorithm
intervals