Average Error: 37.0 → 0.5
Time: 17.2s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2868657370910494 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2175984351199754 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target14.6
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.2868657370910494e-08

    1. Initial program 29.3

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

    3. Applied sin-sum0.6

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

    if -1.2868657370910494e-08 < eps < 1.2175984351199754e-08

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    4. Simplified0.4

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

    if 1.2175984351199754e-08 < eps

    1. Initial program 29.4

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2868657370910494 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2175984351199754 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +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: 15.9s)Debug log

sample210.0ms

Algorithm
intervals

simplify8.0ms

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

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.2b

localize48.0ms

Local error

Found 2 expressions with local error:

2.8b
(sin (+ x eps))
1.8b
(- (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
6.0ms
(- (sin (+ x eps)) (sin x))
2.0ms
(sin (+ x eps))

series51.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(- (sin (+ x eps)) (sin x))
18.0ms
(sin (+ x eps))

simplify1.3s

Counts
33 → 46
Calls
33 calls:
Slowest
436.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
345.0ms
(fma (- (cbrt (sin x))) (* (cbrt (sin x)) (cbrt (sin x))) (* (cbrt (sin x)) (* (cbrt (sin x)) (cbrt (sin x)))))
59.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
42.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
33.0ms
(fma 1 (sin (+ x eps)) (- (* (sin x) 1)))

prune435.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize47.0ms

Local error

Found 2 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
0.2b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite24.0ms

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
13.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
10.0ms
(cos (/ (+ (+ x x) eps) 2))

series108.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
30.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify775.0ms

Counts
14 → 34
Calls
14 calls:
Slowest
387.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
141.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
85.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
28.0ms
(cos (* 1/2 (+ (* 2 x) eps)))
24.0ms
(cos (* 1/2 (+ (* 2 x) eps)))

prune385.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 4 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
0.4b
(expm1 (cos (/ (+ (+ x x) eps) 2)))
0.3b
(log1p (expm1 (cos (/ (+ (+ x x) eps) 2))))
0.2b
(* (log1p (expm1 (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×log1p-expm1-u
4×add-log-exp
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
1×expm1-udef
1×cbrt-unprod
1×*-commutative
1×log1p-expm1
1×prod-exp
1×pow-prod-down
1×expm1-log1p
1×log1p-udef
Counts
4 → 50
Calls
4 calls:
Slowest
5.0ms
(* (log1p (expm1 (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
5.0ms
(cos (/ (+ (+ x x) eps) 2))
0.0ms
(expm1 (cos (/ (+ (+ x x) eps) 2)))
0.0ms
(log1p (expm1 (cos (/ (+ (+ x x) eps) 2))))

series289.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
137.0ms
(* (log1p (expm1 (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
73.0ms
(log1p (expm1 (cos (/ (+ (+ x x) eps) 2))))
55.0ms
(expm1 (cos (/ (+ (+ x x) eps) 2)))
24.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.3s

Counts
22 → 62
Calls
22 calls:
Slowest
323.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
322.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
157.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
149.0ms
(- E (+ (* 1/2 (* x (* E eps))) (+ (* 1/2 (* (pow x 2) E)) 1)))
87.0ms
(* (* (* (log1p (expm1 (cos (/ (+ (+ x x) eps) 2)))) (log1p (expm1 (cos (/ (+ (+ x x) eps) 2))))) (log1p (expm1 (cos (/ (+ (+ x x) eps) 2))))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune717.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.5b

localize20.0ms

Local error

Found 4 expressions with local error:

3.0b
(cos (/ (+ (+ x x) eps) 2))
3.0b
(cos (/ (+ (+ x x) eps) 2))
3.0b
(cos (/ (+ (+ x x) eps) 2))
0.5b
(cbrt (cos (/ (+ (+ x x) eps) 2)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×cbrt-prod
1×pow1/3
Counts
4 → 40
Calls
4 calls:
Slowest
9.0ms
(cos (/ (+ (+ x x) eps) 2))
9.0ms
(cos (/ (+ (+ x x) eps) 2))
9.0ms
(cos (/ (+ (+ x x) eps) 2))
1.0ms
(cbrt (cos (/ (+ (+ x x) eps) 2)))

series172.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
109.0ms
(cbrt (cos (/ (+ (+ x x) eps) 2)))
27.0ms
(cos (/ (+ (+ x x) eps) 2))
20.0ms
(cos (/ (+ (+ x x) eps) 2))
16.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.4s

Counts
15 → 52
Calls
15 calls:
Slowest
374.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
302.0ms
(- 1 (+ (* 1/6 (* x eps)) (* 1/6 (pow x 2))))
291.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
261.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
28.0ms
(cos (* 1/2 (+ (* 2 x) eps)))

prune645.0ms

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.4b

regimes169.0ms

Accuracy

99.6% (0.1b remaining)

Error of 0.5b against oracle of 0.4b and baseline of 14.6b

bsearch201.0ms

end0.0ms

sample7.5s

Algorithm
intervals