From: sgk@REMOVEtroutmask.apl.washington.edu   
      
   On Sun, 19 Nov 2023 13:28:18 +0000, Thomas Koenig wrote:   
      
   > David Jones  schrieb:   
   >   
   >> There seems no reason why the standard might not be extended to allow   
   >> the two different types of representations of complex variables to   
   >> exist in the same program, as separate data-types, and to interact when   
   >> required. Two major questions are:   
   >>   
   >> (i) whether there are any applications that would be more readily and   
   >> usefully programmed using the modulus-phase representation?   
   >>   
   >> (ii) the relative speed of both addition and multiplication in the two   
   >> representations.?   
   >   
   > Multiplication and especially division would likely be faster - you   
   > would have to multiply the two moduli and add and normalize the modulus   
   > to lie between 0 and 2*pi.   
   >   
   > However, the normalization step can have unintended execution speed   
   > consequences if the processor implements it via branches, and branches   
   > can be quite expensive if mispredicted.   
   >   
   > _Addition_ is very expensive indeed in polar notation.  You have to   
   > calculate the sin() and cos() of each number, add them, and then call   
   > atan2() (with a normalization) to get back the original representation.   
   >   
   > If you're doing a lot of multiplication, and not a lot of addition,   
   > that could actually pay off.   
      
   If a vendor used magnitude and phase as the internal representation,   
   then that vendor would not be around very long.  Consider cmplx(0,1).   
   The magnitude is easy.  It is 1.  Mathematically, the phase is   
   pi/2, which is of course not exactly representable.   
      
   % tlibm acos -f -a 0.   
      x =  0.00000000e+00f, /* 0x00000000 */   
   libm =  1.57079637e+00f, /* 0x3fc90fdb */   
   mpfr =  1.57079637e+00f, /* 0x3fc90fdb */   
    ULP = 0.36668   
   % tlibm cos -f -a 1.57079637   
      x =  1.57079625e+00f, /* 0x3fc90fda */   
   libm =  7.54979013e-08f, /* 0x33a22169 */   
   mpfr =  7.54979013e-08f, /* 0x33a22169 */   
    ULP = 0.24138   
      
   7.549... is significantly different when compared to 0.   
      
   --   
   steve   
      
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