Average Error: 36.7 → 0.5
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \left(\sqrt[3]{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x} \cdot \sqrt[3]{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right) \cdot \sqrt[3]{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.8
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.7

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

  3. Applied diff-sin37.0

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

  4. Simplified14.8

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

  5. Taylor expanded around -inf 14.8

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

  6. Simplified14.8

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

  8. Applied fma-udef14.8

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

  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 add-cube-cbrt0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 +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: 16.7s)Debug log

sample172.0ms

Algorithm
intervals
Results
78.0ms97×body1280valid
35.0ms21×body2560valid
34.0ms66×body640valid
8.0ms19×body320valid
5.0ms16×body160valid
4.0ms37×body80valid

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (sin (+ x eps)) (sin x))

prune34.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.3b

localize19.0ms

Local error

Found 2 expressions with local error:

3.9b
(sin (+ x eps))
1.4b
(- (sin (+ x eps)) (sin x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
11×*-un-lft-identity
11×add-sqr-sqrt
9×add-cube-cbrt
9×prod-diff
4×add-log-exp
3×fma-neg
2×log1p-expm1-u
2×add-exp-log
2×sin-sum
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
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 → 40
Calls
2 calls:
Slowest
7.0ms
(- (sin (+ x eps)) (sin x))
2.0ms
(sin (+ x eps))

series49.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(- (sin (+ x eps)) (sin x))
17.0ms
(sin (+ x eps))

simplify819.0ms

Counts
33 → 46
Calls
33 calls:
Slowest
401.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
43.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
38.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
28.0ms
(fma (sqrt (sin (+ x eps))) (sqrt (sin (+ x eps))) (- (* (sqrt (sin x)) (sqrt (sin x)))))
25.0ms
(/ (exp (sin (+ x eps))) (exp (sin x)))

prune427.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.2b

localize31.0ms

Local error

Found 2 expressions with local error:

4.0b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite11.0ms

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
2×log1p-expm1-u
2×expm1-log1p-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
2 → 28
Calls
2 calls:
Slowest
6.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
5.0ms
(cos (/ (+ (+ x x) eps) 2))

series71.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
52.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
19.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify649.0ms

Counts
14 → 34
Calls
14 calls:
Slowest
281.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
163.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
71.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
32.0ms
(* (cos (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
28.0ms
(* (cos (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))

prune385.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize34.0ms

Local error

Found 2 expressions with local error:

4.0b
(cos (fma 1/2 eps x))
0.2b
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))

rewrite18.0ms

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
2×log1p-expm1-u
2×expm1-log1p-u
1×sin-cos-mult
1×fma-udef
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×cos-sum
Counts
2 → 30
Calls
2 calls:
Slowest
14.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
4.0ms
(cos (fma 1/2 eps x))

series83.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
58.0ms
(* (sin (* eps 1/2)) (cos (fma 1/2 eps x)))
25.0ms
(cos (fma 1/2 eps x))

simplify813.0ms

Counts
16 → 36
Calls
16 calls:
Slowest
378.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
252.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/8 (pow eps 2))))
74.0ms
(* (* (* (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))))
34.0ms
(+ (sin (- (* eps 1/2) (fma 1/2 eps x))) (sin (+ (* eps 1/2) (fma 1/2 eps x))))
10.0ms
(+ (log (sin (* eps 1/2))) (log (cos (fma 1/2 eps x))))

prune484.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize30.0ms

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))

rewrite74.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×add-exp-log
10×add-cbrt-cube
10×*-un-lft-identity
10×pow1
10×add-sqr-sqrt
9×associate-*l*
9×associate-*r*
6×add-log-exp
5×sin-mult
5×cos-mult
4×log1p-expm1-u
4×associate-*r/
4×expm1-log1p-u
3×cbrt-unprod
3×*-commutative
3×prod-exp
3×pow-prod-down
3×prod-diff
3×distribute-lft-in
3×sub-neg
3×distribute-rgt-in
2×sub-div
2×flip--
2×frac-sub
2×flip3--
1×diff-log
1×fma-neg
Counts
4 → 87
Calls
4 calls:
Slowest
40.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
17.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
11.0ms
(* (sin (* 1/2 eps)) (sin x))
5.0ms
(* (cos (* 1/2 eps)) (cos x))

series337.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
139.0ms
(* (sin (* eps 1/2)) (- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x))))
86.0ms
(- (* (cos (* 1/2 eps)) (cos x)) (* (sin (* 1/2 eps)) (sin x)))
63.0ms
(* (sin (* 1/2 eps)) (sin x))
49.0ms
(* (cos (* 1/2 eps)) (cos x))

simplify3.5s

Counts
54 → 99
Calls
54 calls:
Slowest
660.0ms
(* (* (* (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)))))
628.0ms
(- (* 1/2 eps) (+ (* 1/12 (pow eps 3)) (* 1/4 (* x (pow eps 2)))))
322.0ms
(- (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x))))
291.0ms
(- (* (+ (cos (+ (* 1/2 eps) x)) (cos (- (* 1/2 eps) x))) 2) (* 2 (- (cos (- (* 1/2 eps) x)) (cos (+ (* 1/2 eps) x)))))
235.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/8 (pow eps 2))))

prune1.4s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

regimes184.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample7.0s

Algorithm
intervals
Results
3.6s3259×body1280valid
1.5s651×body2560valid
1.1s1822×body640valid
325.0ms785×body320valid
136.0ms1122×body80valid
91.0ms361×body160valid