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Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

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Appendix I / Utilities / Trigonometrical Functions<br />

_ _<br />

22.11 Trigonometric Functions<br />

Comparisons between 64 bit computations and function approximations<br />

were performed on a DEC ALPHA 4000-610 using the AXP V6.2<br />

Operating system. The constants used in the approximations,<br />

22.11.1 Sine Approximation<br />

( π π 2 , 3 ⋅ π 2 , 2 ⋅ π )<br />

, : =<br />

( 3.<br />

14159 , 1.<br />

57080 , 4.<br />

71240 , 6.<br />

28318 )<br />

22.11-1<br />

Equation 22.11-1<br />

SIN_F takes angle (XI) in the range [-2π , 2π], and returns its sine in the<br />

range [-1 , 1].<br />

y<br />

: = SIN_F<br />

2<br />

2<br />

X ⋅<br />

( ) ( 1 − X ) ⋅ ( 2.<br />

8444 − X )<br />

X : =<br />

I<br />

0.<br />

9048 + 0.<br />

273344 ⋅ X<br />

] π , 2 ⋅ π ] ⇒ X : = ( X π ) − 2<br />

XI ∈<br />

I<br />

X<br />

I<br />

[ − π , π ] ⇒ X : = ( X π )<br />

∈ I<br />

[ − 2 ⋅ π,<br />

− π [ ⇒ X : = ( X π ) + 2<br />

XI ∈<br />

I<br />

2<br />

Equation 22.11-2<br />

Equation 22.11-3<br />

Equation 22.11-4<br />

Equation 22.11-5<br />

The error in this function is in the range [-250 , 250] ppm. A more accurate<br />

approximation is provided by SIN_G,<br />

y<br />

: =<br />

X ⋅<br />

y : = SIN_G<br />

( XI<br />

)<br />

Equation 22.11-6<br />

2<br />

2<br />

2<br />

( 1 − X ) ⋅ ( 3.<br />

03209 − X ) ⋅ ( 1.<br />

28452 + X )<br />

2<br />

2<br />

( 0.<br />

7412 + 0.<br />

36569 ⋅ X ) ⋅ ( 1.<br />

67279 + X )<br />

Equation 22.11-7<br />

The error in this function is [-50 , 50] ppm as shown in Figure 22-10.

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