Average Error: 36.6 → 0.3
Time: 21.2s
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

Original36.6
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.6

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

  3. Applied diff-sin36.9

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

  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 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 2018365 +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: 12.0s)Debug log

start173.0ms

Algorithm
intervals

setup20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.8b

localize61.0ms

Local error

Found 2 expressions with local error:

3.1b
(sin (+ x eps))
1.5b
(- (sin (+ x eps)) (sin x))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

13.0ms
(- (sin (+ x eps)) (sin x))
4.0ms
(sin (+ x eps))

series77.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

51.0ms
(- (sin (+ x eps)) (sin x))
25.0ms
(sin (+ x eps))

simplify1.0s

Counts
33 → 46
Calls

33 calls. Slowest were:

449.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
42.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
41.0ms
(fma (sqrt (sin (+ x eps))) (sqrt (sin (+ x eps))) (- (* (sqrt (sin x)) (sqrt (sin x)))))

prune467.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.4b

localize23.0ms

Local error

Found 3 expressions with local error:

3.2b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
0.0b
(* 2 (* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
3 → 39
Calls

3 calls. Slowest were:

8.0ms
(* 2 (* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
7.0ms
(cos (/ (+ (+ x x) eps) 2))
6.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

series196.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

100.0ms
(* 2 (* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2))))
79.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
17.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.3s

Counts
18 → 48
Calls

18 calls. Slowest were:

520.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
378.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
135.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))

prune589.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.4b

localize22.0ms

Local error

Found 3 expressions with local error:

3.2b
(cos (fma 1/2 eps x))
0.2b
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
0.0b
(* 2 (* (sin (* eps 1/2)) (cos (fma 1/2 eps x))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
3 → 42
Calls

3 calls. Slowest were:

19.0ms
(* 2 (* (sin (* eps 1/2)) (cos (fma 1/2 eps x))))
9.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
2.0ms
(cos (fma 1/2 eps x))

series99.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

41.0ms
(* 2 (* (sin (* eps 1/2)) (cos (fma 1/2 eps x))))
35.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
23.0ms
(cos (fma 1/2 eps x))

simplify1.0s

Counts
20 → 51
Calls

20 calls. Slowest were:

312.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
273.0ms
(- eps (+ (* 1/6 (pow eps 3)) (* 1/2 (* x (pow eps 2)))))
249.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/8 (pow eps 2))))

prune842.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize54.0ms

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))

rewrite87.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

64.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
11.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
6.0ms
(* (sin (* 1/2 eps)) (sin x))

series286.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
87.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
49.0ms
(* (cos (* 1/2 eps)) (cos x))
47.0ms
(* (sin (* 1/2 eps)) (sin x))

simplify3.6s

Counts
54 → 99
Calls

54 calls. Slowest were:

844.0ms
(* (* (* (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)))))
355.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
346.0ms
(- (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x))))

prune1.7s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

regimes192.0ms

Accuracy

0% (0.2b remaining)

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

bsearch19.0ms