Average Error: 37.2 → 0.4
Time: 16.0s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.225340949031534 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.0250181228359064 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.225340949031534e-08

    1. Initial program 29.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -1.225340949031534e-08 < eps < 2.0250181228359064e-08

    1. Initial program 45.1

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

    3. Applied diff-sin45.1

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

    4. Simplified0.3

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

    if 2.0250181228359064e-08 < eps

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.225340949031534 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.0250181228359064 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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: 14.4s)Debug log

sample226.0ms

Algorithm
intervals
Results
122.0ms107×body1280valid
40.0ms23×body2560valid
39.0ms63×body640valid
9.0ms21×body320valid
4.0ms16×body160valid
4.0ms26×body80valid

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (sin (+ x eps)) (sin x))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.3b

localize43.0ms

Local error

Found 2 expressions with local error:

3.7b
(sin (+ x eps))
1.6b
(- (sin (+ x eps)) (sin x))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-cube-cbrt
2×add-exp-log
2×sin-sum
2×add-cbrt-cube
2×pow1
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 → 24
Calls
2 calls:
Slowest
15.0ms
(- (sin (+ x eps)) (sin x))
3.0ms
(sin (+ x eps))

series61.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
35.0ms
(- (sin (+ x eps)) (sin x))
26.0ms
(sin (+ x eps))

simplify395.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
252.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
54.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
37.0ms
(* (sin (/ (- (+ x eps) x) 2)) (cos (/ (+ (+ x eps) x) 2)))
22.0ms
(/ (exp (sin (+ x eps))) (exp (sin x)))
4.0ms
(- (sqrt (sin (+ x eps))) (sqrt (sin x)))

prune257.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.5b

localize41.0ms

Local error

Found 2 expressions with local error:

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

rewrite10.0ms

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
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
Counts
2 → 24
Calls
2 calls:
Slowest
6.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
4.0ms
(cos (/ (+ (+ x x) eps) 2))

series113.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
88.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
25.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify745.0ms

Counts
14 → 30
Calls
14 calls:
Slowest
326.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
190.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
102.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
20.0ms
(cos (* 1/2 (+ (* 2 x) eps)))
18.0ms
(+ (log (cos (/ (+ (+ x x) eps) 2))) (log (sin (/ eps 2))))

prune407.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.4b

localize20.0ms

Local error

Found 4 expressions with local error:

3.6b
(cos (/ (+ (+ x x) eps) 2))
3.6b
(cos (/ (+ (+ x x) eps) 2))
3.6b
(cos (/ (+ (+ x x) eps) 2))
0.4b
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
1×rem-cbrt-cube
1×cos-mult
1×associate-*l/
1×pow3
1×pow1/3
1×cbrt-div
1×cbrt-prod
Counts
4 → 32
Calls
4 calls:
Slowest
8.0ms
(cos (/ (+ (+ x x) eps) 2))
7.0ms
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))
6.0ms
(cos (/ (+ (+ x x) eps) 2))
4.0ms
(cos (/ (+ (+ x x) eps) 2))

series87.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
27.0ms
(cos (/ (+ (+ x x) eps) 2))
25.0ms
(cos (/ (+ (+ x x) eps) 2))
17.0ms
(cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))
17.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify889.0ms

Counts
15 → 44
Calls
15 calls:
Slowest
199.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
195.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
150.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
141.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
56.0ms
(cbrt (* (+ (cos (+ (/ (+ (+ x x) eps) 2) (/ (+ (+ x x) eps) 2))) (cos (- (/ (+ (+ x x) eps) 2) (/ (+ (+ x x) eps) 2)))) (cos (/ (+ (+ x x) eps) 2))))

prune544.0ms

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0.4b

localize30.0ms

Local error

Found 4 expressions with local error:

3.6b
(cos (/ (+ (+ x x) eps) 2))
3.6b
(cos (/ (+ (+ x x) eps) 2))
3.6b
(cos (/ (+ (+ x x) eps) 2))
0.5b
(log (exp (cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×log-pow
5×pow1
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×exp-prod
3×log-prod
1×rem-log-exp
1×cbrt-prod
Counts
4 → 37
Calls
4 calls:
Slowest
9.0ms
(cos (/ (+ (+ x x) eps) 2))
9.0ms
(cos (/ (+ (+ x x) eps) 2))
8.0ms
(cos (/ (+ (+ x x) eps) 2))
3.0ms
(log (exp (cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))))

series82.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
27.0ms
(cos (/ (+ (+ x x) eps) 2))
20.0ms
(log (exp (cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))))))
18.0ms
(cos (/ (+ (+ x x) eps) 2))
16.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify965.0ms

Counts
20 → 49
Calls
20 calls:
Slowest
198.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
185.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
161.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
131.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
36.0ms
(log (sqrt (exp (cbrt (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2)))))))

prune643.0ms

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0.4b

regimes242.0ms

Accuracy

99.5% (0.1b remaining)

Error of 0.4b against oracle of 0.3b and baseline of 14.8b

bsearch198.0ms

end0.0ms

sample8.3s

Algorithm
intervals
Results
4.2s3279×body1280valid
1.8s1832×body640valid
1.5s638×body2560valid
397.0ms802×body320valid
154.0ms1086×body80valid
114.0ms363×body160valid