/*---------------------------------------------------------------------------+ | poly_l2.c | | | | Compute the base 2 log of a FPU_REG, using a polynomial approximation. | | | | Copyright (C) 1992,1993 | | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, | | Australia. E-mail billm@vaxc.cc.monash.edu.au | | | | | +---------------------------------------------------------------------------*/ #include "exception.h" #include "reg_constant.h" #include "fpu_emu.h" #include "control_w.h" #define HIPOWER 9 static unsigned short const lterms[HIPOWER][4] = { /* Ideal computation with these coeffs gives about 64.6 bit rel accuracy. */ { 0xe177, 0xb82f, 0x7652, 0x7154 }, { 0xee0f, 0xe80f, 0x2770, 0x7b1c }, { 0x0fc0, 0xbe87, 0xb143, 0x49dd }, { 0x78b9, 0xdadd, 0xec54, 0x34c2 }, { 0x003a, 0x5de9, 0x628b, 0x2909 }, { 0x5588, 0xed16, 0x4abf, 0x2193 }, { 0xb461, 0x85f7, 0x347a, 0x1c6a }, { 0x0975, 0x87b3, 0xd5bf, 0x1876 }, { 0xe85c, 0xcec9, 0x84e7, 0x187d } }; /*--- poly_l2() -------------------------------------------------------------+ | Base 2 logarithm by a polynomial approximation. | +---------------------------------------------------------------------------*/ void poly_l2(FPU_REG const *arg, FPU_REG *result) { short exponent; char zero; /* flag for an Xx == 0 */ unsigned short bits, shift; unsigned long long Xsq; FPU_REG accum, denom, num, Xx; exponent = arg->exp - EXP_BIAS; accum.tag = TW_Valid; /* set the tags to Valid */ if ( arg->sigh > (unsigned)0xb504f334 ) { /* This is good enough for the computation of the polynomial sum, but actually results in a loss of precision for the computation of Xx. This will matter only if exponent becomes zero. */ exponent++; accum.sign = 1; /* sign to negative */ num.exp = EXP_BIAS; /* needed to prevent errors in div routine */ reg_u_div(&CONST_1, arg, &num, FULL_PRECISION); } else { accum.sign = 0; /* set the sign to positive */ num.sigl = arg->sigl; /* copy the mantissa */ num.sigh = arg->sigh; } /* shift num left, lose the ms bit */ num.sigh <<= 1; if ( num.sigl & 0x80000000 ) num.sigh |= 1; num.sigl <<= 1; denom.sigl = num.sigl; denom.sigh = num.sigh; poly_div4(&significand(&denom)); denom.sigh += 0x80000000; /* set the msb */ Xx.exp = EXP_BIAS; /* needed to prevent errors in div routine */ reg_u_div(&num, &denom, &Xx, FULL_PRECISION); zero = !(Xx.sigh | Xx.sigl); mul64(&significand(&Xx), &significand(&Xx), &Xsq); poly_div16(&Xsq); accum.exp = -1; /* exponent of accum */ /* Do the basic fixed point polynomial evaluation */ polynomial((unsigned *)&accum.sigl, (unsigned *)&Xsq, lterms, HIPOWER-1); if ( !exponent ) { /* If the exponent is zero, then we would lose precision by sticking to fixed point computation here */ /* We need to re-compute Xx because of loss of precision. */ FPU_REG lXx; char sign; sign = accum.sign; accum.sign = 0; /* make accum compatible and normalize */ accum.exp = EXP_BIAS + accum.exp; normalize(&accum); if ( zero ) { reg_move(&CONST_Z, result); } else { /* we need to re-compute lXx to better accuracy */ num.tag = TW_Valid; /* set the tags to Vaild */ num.sign = 0; /* set the sign to positive */ num.exp = EXP_BIAS - 1; if ( sign ) { /* The argument is of the form 1-x */ /* Use 1-1/(1-x) = x/(1-x) */ significand(&num) = - significand(arg); normalize(&num); reg_div(&num, arg, &num, FULL_PRECISION); } else { normalize(&num); } denom.tag = TW_Valid; /* set the tags to Valid */ denom.sign = SIGN_POS; /* set the sign to positive */ denom.exp = EXP_BIAS; reg_div(&num, &denom, &lXx, FULL_PRECISION); reg_u_mul(&lXx, &accum, &accum, FULL_PRECISION); reg_u_add(&lXx, &accum, result, FULL_PRECISION); normalize(result); } result->sign = sign; return; } mul64(&significand(&accum), &significand(&Xx), &significand(&accum)); significand(&accum) += significand(&Xx); if ( Xx.sigh > accum.sigh ) { /* There was an overflow */ poly_div2(&significand(&accum)); accum.sigh |= 0x80000000; accum.exp++; } /* When we add the exponent to the accum result later, we will require that their signs are the same. Here we ensure that this is so. */ if ( exponent && ((exponent < 0) ^ (accum.sign)) ) { /* signs are different */ accum.sign = !accum.sign; /* An exceptional case is when accum is zero */ if ( accum.sigl | accum.sigh ) { /* find 1-accum */ /* Shift to get exponent == 0 */ if ( accum.exp < 0 ) { poly_div2(&significand(&accum)); accum.exp++; } /* Just negate, but throw away the sign */ significand(&accum) = - significand(&accum); if ( exponent < 0 ) exponent++; else exponent--; } } shift = exponent >= 0 ? exponent : -exponent ; bits = 0; if ( shift ) { if ( accum.exp ) { accum.exp++; poly_div2(&significand(&accum)); } while ( shift ) { poly_div2(&significand(&accum)); if ( shift & 1) accum.sigh |= 0x80000000; shift >>= 1; bits++; } } /* Convert to 64 bit signed-compatible */ accum.exp += bits + EXP_BIAS - 1; reg_move(&accum, result); normalize(result); return; } /*--- poly_l2p1() -----------------------------------------------------------+ | Base 2 logarithm by a polynomial approximation. | | log2(x+1) | +---------------------------------------------------------------------------*/ int poly_l2p1(FPU_REG const *arg, FPU_REG *result) { char sign = 0; unsigned long long Xsq; FPU_REG arg_pl1, denom, accum, local_arg, poly_arg; sign = arg->sign; reg_add(arg, &CONST_1, &arg_pl1, FULL_PRECISION); if ( (arg_pl1.sign) | (arg_pl1.tag) ) { /* We need a valid positive number! */ return 1; } reg_add(&CONST_1, &arg_pl1, &denom, FULL_PRECISION); reg_div(arg, &denom, &local_arg, FULL_PRECISION); local_arg.sign = 0; /* Make the sign positive */ /* Now we need to check that |local_arg| is less than 3-2*sqrt(2) = 0.17157.. = .0xafb0ccc0 * 2^-2 */ if ( local_arg.exp >= EXP_BIAS - 3 ) { if ( (local_arg.exp > EXP_BIAS - 3) || (local_arg.sigh > (unsigned)0xafb0ccc0) ) { /* The argument is large */ poly_l2(&arg_pl1, result); return 0; } } /* Make a copy of local_arg */ reg_move(&local_arg, &poly_arg); /* Get poly_arg bits aligned as required */ shrx((unsigned *)&(poly_arg.sigl), -(poly_arg.exp - EXP_BIAS + 3)); mul64(&significand(&poly_arg), &significand(&poly_arg), &Xsq); poly_div16(&Xsq); /* Do the basic fixed point polynomial evaluation */ polynomial(&(accum.sigl), (unsigned *)&Xsq, lterms, HIPOWER-1); accum.tag = TW_Valid; /* set the tags to Valid */ accum.sign = SIGN_POS; /* and make accum positive */ /* make accum compatible and normalize */ accum.exp = EXP_BIAS - 1; normalize(&accum); reg_u_mul(&local_arg, &accum, &accum, FULL_PRECISION); reg_u_add(&local_arg, &accum, result, FULL_PRECISION); /* Multiply the result by 2 */ result->exp++; result->sign = sign; return 0; }