Miscalculating Area and Angles of a Needle-like Triangle
( from Lecture Notes for Introductory Numerical Analysis Classes )
Prof. W. Kahan, 19 July 1999
Abstract:
This case study drawn from an elementary numerical analysis course is aimed at
computer language designers and implementors who took no competent course on the
subject or forgot it and consequently subscribe to principles of language design
inimical to the best interests of writers and users of software with a little
floating-point arithmetic in it. Though triangles rarely matter to computer
language designers and implementors, recollections of their high-school
trigonometry and calculus will enable them to follow the discussion, which is
intended primarily to expose and correct common misconceptions. The first of
these is that classical trigonometric formulas taught in schools and found in
handbooks and software must have passed the Test of Time. Actually they have
withstood it; they are unnecessarily inaccurate, sometimes badly, for some
data some of which is unexceptionable. Better formulas are supplied here. Even
if these are impeccably accurate and cost no more than the classical formulas
they should supplant, the expectation that the better formulas will supplant
the worse is based upon misconceptions too (see pp. 5, 9, 16 and 21).
Other misconceptions addressed herein ( on indicated pages ) are ...
= That subtractive cancellation always causes numerical inaccuracy, or is the
only cause of it. (3, 5)
= That a singularity always degrades accuracy when data approach it.
(4, 5, 7, 8, 10, 12-16)
= That algorithms known to be numerically unstable should never be used. (10)
= That arithmetic much more precise than the data it operates upon is pointless.
(7, 9-12, 14, 17-21)
= That modern "Backward Error-Analysis" explains everything, or excuses it.
(12, 14-17)
= That bad results are due to bad data or bad programmers, never to a
programming language. (9-12, 18-20)
Misconceptions like these hinder programming languages from conveying to owners
of by far the majority of computers now on desk-tops the benefits of their
hardware's superior floating-point semantics. (9-11, 19-20)
Contents:
$1. Classical Formulas for Area and Angle C page 2
$2. How to compute Area 3
$3. How to compute C 3
$4. Why Cancellation Cannot Hurt 3
$5. Examples 4
Table 1: Area and Angle C 4
$6. Opposite Side c and Adjacent Angle B 5
$7. How to compute c 5
$8. How to compute B 5
$9. How Accurate is B ? 7
Table 2: Angle B , New Formula B(...) vs. Classical BS(...) 7
Table 3: Uncertainty dB due to Roundoff 9
$10. Must the Classical Formulas be Amended ? 9
Table 4: Formats of IEEE Std. 754 for Bin. Floating-Pt. Arith. 9
$11. What Extra Precision Does for B 11
$12. How Much Accuracy do Data Deserve ? 12
$13. A Triangle is an Object 13
Table 5: Side c , cb(a,A,b) vs. cB(a,A,B(a,A,b)) 13
$14. Three Caveats and a Picture 14
$15. Proper Precision Management 18
$16. Different Floating-Point Semantics 19
$17. Conclusion 21
$18. Acknowledgments and Bibliography 22
$19. Footnote about Examples 22
22 pp. Acrobat Reader .pdf file:
http://http.cs.berkeley.edu/~wkahan/Triangle.pdf