Wow in QB64, this power estimator is on par with the ^ operator!
_TITLE "Power Function 2 by bplus"
'QB64 X 64 version 1.2 20180228/86 from git b301f92
' started 2018-07-23 Naalaa has no power function (or operator), so I wrote a power function for it.
' ''Power pack the short version.txt
' ''written for Naalaa 6 by bplus posted 2018-07-23
' ''extracted from large test of fractions, random number functions, string... in Power test.txt
' 2018-07-24 Power Function 2 split is replaced with two much smaller and very handy string functions.
' OMG the crazy thing worked! It produced decent estimates of roots given the limitations of precision...
' Now I want to see how well it works with far greater precision available. So here we are, looking to see
' how this function compares to the regualar ^ operator in QB64.
'from Naalaa comments:
' The main purpose of this code: to demo a power function for real numbers,
' I had an idea for how real numbers to the power of real numbers might be done ie x ^ y = ?
' This means that not only can you take SQR of a number, you can get cube or cube root, quartic, 5th 6th... roots and any multiple
' It came from this simple idea
' 2 ^ 3.5 = 2 ^ 3 * 2 ^ .5 = 8 * Sqr(2)
' 3 ^ 3.5 = 3 ^ 3 * 3 ^ .5 = 27 * Sqr(3)
' so 2 ^ 3.25 = 2 ^ 3 * 2 ^ .25
' what is 2 ^ .25 ? It is sqr(sqr(2)) !
' likewise 2 ^ 3.125 = 2 ^ 3 * 2 ^ 1/8
' what is 2 ^ 1/8 ? It is sqr(sqr(sqr(2))) !
' any decimal can be written as a sum of fraction powers of 2 ie 1/2^n, as any integer can be written in powers of 2.
' in binary expansions
' 1/2 = .1 or .5 base 10
' 1/4 = .01 or .25 base 10
' 1/8 = .001 or .125 base 10
' 1/16 = .0001 or .0625 base 10
' So with binary expansion of decimal, we can SQR and multiply our way to an estimate
' of any real number to the power of another real number using binary expansion of
' the decimal parts as long as we stay in Naalaa integer limits and are mindful of precision.
CONST wW = 800
CONST wH = 600
SCREEN _NEWIMAGE(wW, wH, 32)
_SCREENMOVE 360, 60
_DEFINE A-Z AS _FLOAT
DO
PRINT "Testing the power(x, pow) function:"
INPUT "(nothing) quits, Please enter a real number to raise to some power. x = "; x
IF x = 0 THEN EXIT DO
INPUT "(nothing) quits, Please enter a real number for the power. pow = ", pw
IF pw = 0 THEN EXIT DO
result = power(x, pw)
PRINT result; " is what we estimate for"; x; " raised to power of"; pw
PRINT x ^ pw; " is what the ^ operator gives us."
PRINT
LOOP
PRINT
PRINT "This is matching the ^ operator very well! This code is clear proof of concept!"
PRINT " OMG, it worked!!!"
SLEEP
' A power function for real numbers (small ones but still!)
' x to the power of pow
FUNCTION power## (x AS _FLOAT, pow AS _FLOAT)
'this sub needs: bExpand60$, leftOf$, rightOf$
DIM build AS _FLOAT
r$ = "0" + STR$(pow) 'in case pow starts with decimal
integer$ = leftOf$(r$, ".")
build = 1.0
IF integer$ <> "0" THEN
p = VAL(integer$)
FOR i = 1 TO p
build = build * x
NEXT
END IF
'that takes care of integer part
n$ = rightOf$(r$, ".")
IF n$ = "" THEN power = build: EXIT SUB
'remove 0's to right of main digits
ld = LEN(n$)
WHILE RIGHT$(n$, 1) = "0"
n$ = LEFT$(n$, ld - 1)
ld = LEN(n$)
WEND
'note: we are pretending that the ^ operator is not available, so this is hand made integer power
denom& = 10
FOR i = 2 TO ld
denom& = denom& * 10
NEXT
'OK for bExpand60$ don't have to simplify fraction and that saves us having to extract n and d again from n/d
bs$ = bExpand60$(VAL(n$), denom&)
'at moment we haven't taken any sqr of x
runningXSQR = x
'run through all the 0's and 1's in the bianry expansion, bs$, the fraction part of the power float
FOR i = 1 TO LEN(bs$)
'this is the matching sqr of the sqr of the sqr... of x
runningXSQR = SQR(runningXSQR)
'for every 1 in the expansion, multiple our build with the running sqr of ... sqr of x
IF MID$(bs$, i, 1) = "1" THEN build = build * runningXSQR
NEXT
'our build should now be an estimate or x to power of pow
power = build
END FUNCTION
'write a series of 1s and 0s that represent the decimal fraction n/d in binary 60 places long
FUNCTION bExpand60$ (nOver&, d&)
DIM b AS _FLOAT, r AS _FLOAT
' b for base
b = 0.5
' r for remainder
r = nOver& / d&
' s for string$ 0's and 1's that we will build and return for function value
s$ = ""
' f for flag to stop
f% = 0
' c for count to track how far we are, don't want to go past 20
c% = 0
WHILE f% = 0
IF r < b THEN
s$ = s$ + "0"
ELSE
s$ = s$ + "1"
IF r > b THEN
r = r - b
ELSE
f% = 1
END IF
END IF
c% = c% + 1
IF c% >= 60 THEN f% = 1
b = b * 0.5
WEND
bExpand60$ = s$
END FUNCTION
FUNCTION leftOf$ (source$, of$)
posOf = INSTR(source$, of$)
IF posOf > 0 THEN leftOf$ = MID$(source$, 1, posOf - 1)
END FUNCTION
FUNCTION rightOf$ (source$, of$)
posOf = INSTR(source$, of$)
IF posOf > 0 THEN rightOf$ = MID$(source$, posOf + LEN(of$))
END FUNCTION
