Delphi and BASM sample code
PIC specific math@ (including
trigonometric functions sin(), arctan(), etc.),
SX specific math@,
Intel specific math@
Dr Dobbs Journal #271 pg 18
"A Conversation with William
Kahan" William Kahan
"Formulas that are numerically stable are sometimes quite inelegant. Formulas
that give you greater accuracy are sometimes full of cases, ugly cases....
F(x)=(sin x)/x .... F(x)=(sin x)/x if x != 0 and F(0) = 1"
Have exmathematicians become dysfunctional?
See also:

https://www.reenigne.org/blog/multiplyingfasterwithsquares/
Multiplying Faster with Squares (Lookup). Because:
(a+b)^{2} = a^{2} + b^{2} + 2ab
(ab)^{2} = a^{2} + b^{2}  2ab
Subtracting these gives:
4ab = (a+b)^{2}  (ab)^{2}
Or:
ab = (a+b)^{2}/4  (ab)^{2}/4
which reduces the difficult parts from an x*y to an x^{2}/4. With only one variable (and enough ram) lookup tables are suddenly back in frame. Because ab and a+b are always both even or both odd, you only need one table. So if we keep in memory
a table of x^{2}/4 we can do a multiply with an add, two subtractions
and two table lookups. The tables have a+b number of entries rather than
a*b, but must hold values as large as a*b if the full range of possible output
is to be supported.+

http://floatingpointgui.de/ Floating point guide. Detailed
explanation of how floating point works on binary systems and what can go
wrong.+

http://f3.to/portfolio/math/fastatan2.htm
Fast ATAN2. if X>Y [(b*x*y)/(x^2+ a*y^2) + c*sign(x)] else [c*sign(x*y)=
c*sign(x) + (b*x*y)/(y^2+a*x^2)] where a = 0.28088, b = 180/Pi, c = b/2*Pi

http://www.eugeneleeslover.com/VIDEOS/fire_control_computer_1.html
and
part
2^ Shows how mechanical elements are used to compute answers. Understanding
these methods helps to round out our understanding of mathematics.

Online CRC Concise
Encyclopedia of Mathematics from
TreasureTroves.com

Frequently Asked Questions
in Mathematics from The Sci.Math FAQ Team. Editor:
Alex LópezOrtiz

National Institute of Standards and Technology
 Guide to Available Mathematical Software

Netlib is a collection of mathematical software,
papers, and databases

http://farside.ph.utexas.edu/teaching/329/lectures/lectures.html
Nice step by step tutorial in various aspects of computational physics

http://integrals.wolfram.com/
Online integral evaluation with Mathematica.

Scott Dattalo's

SIN / COS approximations

CORDIC functions.

https://github.com/francisrstokes/githublog/blob/main/2024/5/10/cordic.md Why the CORDIC algorithm lives rentfree in my head.
Precompute a table for tan(a), where a is in binary(ish) steps. So for i=0 to 15, each entry is actually atan(2**i), converted to fixed point, so: atan(2**i) * (1 << 16)
To compute a sine or a cosine, convert the angle (e.g. 0.9152) to fixed point: 0.9152 * (1 << 16) = 59978
#Initialize variables:
z = a * (1 << 16) #our input angle offset
x = 39796 #the x point on a unit circle at 0 degrees in fixed point
y = 0 #the y
#The sign of z determines if we rotate clockwise or anticlockwise.
# and as we zero it out, we reach out answers
for (i in range(0,sizeof(table)):
if z >= 0:
x_next = x  (y >> i)
y_next = y + (x >> i)
z = table[i]
else:
x_next = x + (y >> i)
y_next = y  (x >> i)
z += table[i]
x = x_next
y = y_next

http://www.brouhaha.com/~eric/pic/sine.html
(Implementing Sine on the PIC w/ source)

http://www.virtuaweb.com/picprog/links.html
(link pg to all sorts of PIC code)

http://www.taygeta.com/cordic_refs.html
(references to print, periodical and IEEE theory articles)

http://devil.ece.utexas.edu/original.html(references
to print, periodical and IEEE theory articles)

http://ftp.arl.mil/ftp/pub/Gems/src/
(Source code related to the book
"Graphics
Gems" (editor, Andrew S. Glassner, published by Academic Press, Cambridge,
MA, 1990, ISBN 0122861655, 833 pgs.)

http://devil.ece.utexas.edu/code.html

Scott Dattalo's
Sine Waves

"ArcTan as Fast as You Can" by Dave Van Ess, has C code to implement
arctan2(x,y) at various size/speed tradeoffs.

http://www.coranac.com/documents/arctangent/

Randy Hyde's UCR
Standard Library

Reciprocal Division
(BCD math)

http://www.opferman.com/Computers/Math/index.shtml
Toby Opferman's Math Tutorials For  Unit Circle & Trig Functions *
Linear Algerbra  Vectors  Matrices * Calculus  Limits  Derivatives
 Integrals (AntiDerivatives)

https://www.embeddedrelated.com/showarticle/760.php
a few good algorithems. Russian Peasant Multiplication, The SinglePole LowPass
Filter, Welford's Method (And Friends), Topological Sort

Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method

Part 6: Greenâ€™s Theorem and SweptArea Detection

http://babbage.cs.qc.edu/courses/cs341/IEEE754.html
Convert between decimal and IEEE754 formats

http://www.optonline.com/comptons/ceo/04099_A.html
ROMAN NUMERALS

http://www.mathcom.com/nafaq/index.html
FAQ: Numerical Analysis & Associated Fields Resource Guide

http://www.geocities.com/SiliconValley/2499/essays.html#TWOS
Two's Complement and Binary Math
Books

"Statistical Distributions" by NAJ Hastings and J.B.Peacock John Wiley 1974
ISBN
0470358890 This rare book is essential for anyone doing MonteCarlo
simulations or using Random Number sequences. It is a collection of simple
algorithms for generating random numbers with specific distributions given
a simple random number generator with good first and second order distribution.
Math:
If you've got a basket with 3 oranges in it and you take 5 out, then you
have to put 2 oranges in again in order for it to be empty.  Peter Gutmann
Questions:


https://www.poshenloh.com/quadraticdetail/
Simple way to factor quadratic equations of the form x^{2} + Bx +
C = 0.
 Convert from Ax^{2} + Bx + C = 0 by dividing all terms by A.
 Find two numbers with sum B and product C.
 They will sum to B if they are the average of B plus or minus some number:
B/2 + u in other words: B/2 = u and B/2 = u.
 And (B/2 + u)(B/2  u) = C or B/2  u^{2} = C or u^{2}
= C  B/2
 So u = sqrt(C  B/2).
Summary: x^2 + Bx + C = 0. u = sqrt(C  B/2), r = B/2 + u, and w = B/2
 u. So (r + x)(w  x) = 0.

taking the square (x^2) with
PICspecific math

Nicolaas Gerhardus Scheepers of
ElectronCraft asks:
Hi there, I am having trouble with arithmetic in assembler. I want to do
the following calculation:
Y = ((XXmin)((YmaxYmin)/(XmaxXmin))) + Ymin This is a formula to scale
a point between Xmax and Xmin to a point between Ymax and Ymin. The Y points
is 8bit but the X points can be up to 16bit values. You can already see
my problem because I cannot get any 16bit divided by 16bit algorith or
for that matter 24bit divided by 16bit that replies with the fractional
part. So I came up with the next formula to try and use a 16bit divided
by 8bit number to give back fractions:
Y = ((XXmin)(1/(XmaxXmin)/(YmaxYmin))) + Ymin But it just gets more complex
as I go along. My second problem is that say I work out the division and
now I must multiply the answer (which is fractional , how the hell do you
do it with the fractional part. Do you first multiply both numbers so they
are integer and then divide it again later with the same number or what.
Am I even on the right track here or am I going bananas? Please help me I
am now three months into the game and have learned a lot in a short space
of time but the learning just seems to grow and not level out, not that I
am complaining just a remark:)
Rearrange the formula so that the top and bottom of the fraction is kept
seperate. What you really want is:
Y = ((XXmin)(YmaxYmin) + Ymin(XmaxXmin)) over XmaxXmin.
Now you can do the top and bottom seperatly. Multiplying by a fraction is
a seperate job but can still be done without division. It is a good
idea to reduce the fraction after each operation to prevent overflow. A method
for doing that is available but I can't find it at the moment.
Once you have the final answer, then you do the division.
See:
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