Average Error: 37.1 → 0.5
Time: 17.5s
Precision: 64
Internal Precision: 128
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2264245780452515 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.7206170988045887 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.9
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 < -1.2264245780452515e-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 -1.2264245780452515e-08 < eps < 1.7206170988045887e-16

    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. Simplified0.2

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

    if 1.7206170988045887e-16 < eps

    1. Initial program 29.5

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

    3. Applied sin-sum1.1

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

    4. Applied associate--l+1.2

      \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2264245780452515 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.7206170988045887 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + x\right) + \varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +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: 16.1s)Debug log

sample227.0ms

Algorithm
intervals
Results
256×(pre true 80)
115×(body real 1280)
60×(body real 640)
27×(body real 2560)
25×(body real 80)
17×(body real 320)
12×(body real 160)

simplify7.0ms

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

prune46.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.4b

localize17.0ms

Local error

Found 2 expressions with local error:

3.8b
(sin (+ x eps))
1.7b
(- (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))
16.0ms
(sin (+ x eps))

simplify831.0ms

Counts
33 → 46
Calls
33 calls:
Slowest
386.0ms
(- eps (+ (* 1/2 (* x (pow eps 2))) (* 1/2 (* (pow x 2) eps))))
44.0ms
(- (+ x eps) (* 1/6 (pow x 3)))
36.0ms
(fma 1 (sin (+ x eps)) (- (* (sin x) 1)))
25.0ms
(fma (* (cbrt (sin (+ x eps))) (cbrt (sin (+ x eps)))) (cbrt (sin (+ x eps))) (- (* (cbrt (sin x)) (* (cbrt (sin x)) (cbrt (sin x))))))
24.0ms
(fma (* (cbrt (sin (+ x eps))) (cbrt (sin (+ x eps)))) (cbrt (sin (+ x eps))) (- (* (sin x) 1)))

prune507.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.4b

localize47.0ms

Local error

Found 2 expressions with local error:

4.0b
(cos (/ (+ (+ x x) eps) 2))
0.3b
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))

rewrite22.0ms

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
12.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
9.0ms
(cos (/ (+ (+ x x) eps) 2))

series111.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(* (cos (/ (+ (+ x x) eps) 2)) (sin (/ eps 2)))
32.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify626.0ms

Counts
14 → 34
Calls
14 calls:
Slowest
299.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
126.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
72.0ms
(* (* (* (cos (/ (+ (+ x x) eps) 2)) (cos (/ (+ (+ x x) eps) 2))) (cos (/ (+ (+ x x) eps) 2))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))
30.0ms
(* (cos (* 1/2 (+ (* 2 x) eps))) (sin (* 1/2 eps)))
28.0ms
(cos (* 1/2 (+ (* 2 x) eps)))

prune448.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.3b

localize15.0ms

Local error

Found 4 expressions with local error:

4.0b
(cos (/ (+ (+ x x) eps) 2))
0.6b
(log (exp (cos (/ (+ (+ x x) eps) 2))))
0.3b
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
0.1b
(exp (cos (/ (+ (+ x x) eps) 2)))

rewrite55.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×exp-prod
8×log-pow
8×pow1
7×associate-*l*
6×add-exp-log
6×add-cbrt-cube
5×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*r*
3×log-prod
1×rem-exp-log
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×rem-log-exp
Counts
4 → 62
Calls
4 calls:
Slowest
27.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
12.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))
11.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
5.0ms
(cos (/ (+ (+ x x) eps) 2))

series155.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
69.0ms
(* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (sin (/ eps 2)))
36.0ms
(exp (cos (/ (+ (+ x x) eps) 2)))
28.0ms
(cos (/ (+ (+ x x) eps) 2))
22.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))

simplify1.5s

Counts
34 → 74
Calls
34 calls:
Slowest
348.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
313.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
182.0ms
(- E (+ (* 1/2 (* x (* E eps))) (* 1/2 (* (pow x 2) E))))
164.0ms
(- (* 1/2 eps) (+ (* 1/4 (* x (pow eps 2))) (* 1/4 (* (pow x 2) eps))))
67.0ms
(* (* (* (log (exp (cos (/ (+ (+ x x) eps) 2)))) (log (exp (cos (/ (+ (+ x x) eps) 2))))) (log (exp (cos (/ (+ (+ x x) eps) 2))))) (* (* (sin (/ eps 2)) (sin (/ eps 2))) (sin (/ eps 2))))

prune1.0s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.3b

localize57.0ms

Local error

Found 4 expressions with local error:

4.0b
(cos (/ (+ x (+ x eps)) 2))
4.0b
(cos (/ (+ x (+ x eps)) 2))
4.0b
(cos (/ (+ (+ x x) eps) 2))
0.6b
(log (exp (cos (/ (+ (+ x x) eps) 2))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×log-pow
4×add-exp-log
4×add-cbrt-cube
4×expm1-log1p-u
3×exp-prod
3×log-prod
1×rem-log-exp
Counts
4 → 44
Calls
4 calls:
Slowest
18.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))
9.0ms
(cos (/ (+ (+ x x) eps) 2))
5.0ms
(cos (/ (+ x (+ x eps)) 2))
5.0ms
(cos (/ (+ x (+ x eps)) 2))

series113.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
30.0ms
(log (exp (cos (/ (+ (+ x x) eps) 2))))
28.0ms
(cos (/ (+ x (+ x eps)) 2))
28.0ms
(cos (/ (+ x (+ x eps)) 2))
27.0ms
(cos (/ (+ (+ x x) eps) 2))

simplify1.6s

Counts
19 → 56
Calls
19 calls:
Slowest
349.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
346.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
323.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
306.0ms
(- 1 (+ (* 1/2 (* x eps)) (* 1/2 (pow x 2))))
25.0ms
(cos (* 1/2 (+ (* 2 x) eps)))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.3b

regimes289.0ms

Accuracy

98.9% (0.2b remaining)

Error of 0.5b against oracle of 0.4b and baseline of 14.9b

bsearch286.0ms

end0.0ms

sample7.0s

Algorithm
intervals
Results
8000×(pre true 80)
3311×(body real 1280)
1789×(body real 640)
1090×(body real 80)
772×(body real 320)
664×(body real 2560)
374×(body real 160)