[MUG] hfarray --> Maple float

[MUG] Re: hfarray --> Maple float
Author: Maple User Group    Posted: Wed, 7 Aug 2002 16:25:56 -0400 (
>> From: Maple User Group "maple_gr" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 06 Aug 2002 11:50:57 -0700 From: Allan Wittkopf "wittkopf" Subject: hfarray --> Maple float To: "maple-list" In Maple, hardware floats (i.e. hfarrays) contain enough information to determine the exact 53 bit binary floating point number they correspond to, but do not represent that number exactly when being displayed, or being converted to a different datatype (i.e. 100 digit float). When used as a hardware float (in hardware float only computations) they are of course used as the exact binary number. Note that when you look at a hardware float, it actually displays 18 digits, even though accuracy is only guaranteed to 16 digits for double precision floats. The two extra digits give sufficient information to determine the exact 53 bit number you need. The following little routine should do what you need (but I'm sure you could make it more efficient if you wanted): hf_to_arb_binary := proc(hf,digits) local f,s,pow; Digits := digits; pow := 1; s := `if`(hf<0,-1,1); f := s*hf; if f>1.0 then while f*2^pow>1.0 do pow := pow-1; end do; else while f*2^pow<=1.0 do pow := pow+1; end do; pow := pow-1; end if; f := round(f*2^(53+pow)); s*evalf(f/2^(53+pow)); end proc: Examples: > evalhf(4/3); 1.33333333333333326 > hf_to_arb_binary(%,100); 1.3333333333333332593184650249895639717578887939453125000000000000000000000\ 00000000000000000000000000 > evalhf(4/30000000000); -9 0.133333333333333338 10 > hf_to_arb_binary(%,100); 0.1333333333333333381909596420663655438887396087466186145320534706115722656\ -9 250000000000000000000000000 10 Notice the last two digits in the hardware float are not what one would expect from the computation (i.e. the first case we would expect '33' instead of '26') but that is an artifact of the limited 53 bit representation for hardware floats. The higher precision number computed from hf_to_arb_binary matches these results. Note also that the provided routine could also be used to determine the nearest hardware float representation of an object, just like evalhf: > Digits := 20: > f := evalf(4/3); f := 1.3333333333333333333 > hf_to_arb_binary(%,100); 1.3333333333333332593184650249895639717578887939453125000000000000000000000\ 00000000000000000000000000 > evalhf(%%); 1.33333333333333326 Note the agreement in the '26' and '259...' - Allan Wittkopf -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 6 Aug 2002 19:01:32 -0700 (PDT) From: Peter Montgomery "pmontgom" To: "maple-list" Subject: hfarray --> Maple float I haven't used the BINARY option. But I suggest that your C program output rational numbers (mantissa) * 2^(exponent) (not necessarily in lowest terms). Input these as rational numbers, and convert them to floating yourself. The C library function frexp gives mantissa with absolute value in [1/2, 1). You should be able to multiply this by 2^53, and output an exact signed integer. | >> From: "Dr. George Corliss" "George.Corliss" | | Question: Does Maple do hfarray --> 100 digit | number without rounding? | | I am using Maple to check results of a C program | that I know is sensitive to rounding errors. | In pseudocode, the C program: | double x, y; | x = something I construct; | y = f(x); for some function f | write AS BINARY x, y; | | In the Maple program, I want something like | Digits := 100; | fd := open (fileName, READ, BINARY); | x := hfarray(1..1); | y := hfarray(1..1); | x := readbytes(fd, x); # Binary read | y := readbytes(fd, y); # Binary read | z := f(x); # Some function f | # Compare y and z | # Expect abs(y - z) / z to be about 10^(-16) | | What I WANT is that in Maple, the binary number | x is exactly representable in its hardware floating | point form. Before the function f is evaluated | in Maple's 100 digit arithmetic, I want x to be | converted to the exact 100 digit value of the | binary x. Then I am evaluating f at the exact | argument for f as used by my C program, and z is | the correct value (to nearly 100 digits) of f(x). | | I could imagine Maple doing the hfarray --> 100 digit | number exactly. | | I could imagine that Maple might take the binary | value, convert it to about 16-18 decimal digits | (possibly committing a rounding error), then | append 82-84 zeros to get a 100 digit number with | rounding errors in about the 16th place. | | Does anyone know which (or something else) it is? | How can I tell for certain?

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