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 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101``` ```# exp(x) = 2^hi + 2^hi (2^lo - 1) # where hi+lo = log2e*x with 128bit precision # exact log2e*x calculation depends on nearest rounding mode # using the exact multiplication method of Dekker and Veltkamp .global expl .type expl,@function expl: fldt 8(%rsp) # interesting case: 0x1p-32 <= |x| < 16384 # check if (exponent|0x8000) is in [0xbfff-32, 0xbfff+13] mov 16(%rsp), %ax or \$0x8000, %ax sub \$0xbfdf, %ax cmp \$45, %ax jbe 2f test %ax, %ax fld1 js 1f # if |x|>=0x1p14 or nan return 2^trunc(x) fscale fstp %st(1) ret # if |x|<0x1p-32 return 1+x 1: faddp ret # should be 0x1.71547652b82fe178p0L == 0x3fff b8aa3b29 5c17f0bc # it will be wrong on non-nearest rounding mode 2: fldl2e subq \$48, %rsp # hi = log2e_hi*x # 2^hi = exp2l(hi) fmul %st(1),%st fld %st(0) fstpt (%rsp) fstpt 16(%rsp) fstpt 32(%rsp) call exp2l@PLT # if 2^hi == inf return 2^hi fld %st(0) fstpt (%rsp) cmpw \$0x7fff, 8(%rsp) je 1f fldt 32(%rsp) fldt 16(%rsp) # fpu stack: 2^hi x hi # exact mult: x*log2e fld %st(1) # c = 0x1p32+1 movq \$0x41f0000000100000,%rax pushq %rax fldl (%rsp) # xh = x - c*x + c*x # xl = x - xh fmulp fld %st(2) fsub %st(1), %st faddp fld %st(2) fsub %st(1), %st # yh = log2e_hi - c*log2e_hi + c*log2e_hi movq \$0x3ff7154765200000,%rax pushq %rax fldl (%rsp) # fpu stack: 2^hi x hi xh xl yh # lo = hi - xh*yh + xl*yh fld %st(2) fmul %st(1), %st fsubp %st, %st(4) fmul %st(1), %st faddp %st, %st(3) # yl = log2e_hi - yh movq \$0x3de705fc2f000000,%rax pushq %rax fldl (%rsp) # fpu stack: 2^hi x lo xh xl yl # lo += xh*yl + xl*yl fmul %st, %st(2) fmulp %st, %st(1) fxch %st(2) faddp faddp # log2e_lo movq \$0xbfbe,%rax pushq %rax movq \$0x82f0025f2dc582ee,%rax pushq %rax fldt (%rsp) addq \$40,%rsp # fpu stack: 2^hi x lo log2e_lo # lo += log2e_lo*x # return 2^hi + 2^hi (2^lo - 1) fmulp %st, %st(2) faddp f2xm1 fmul %st(1), %st faddp 1: addq \$48, %rsp ret ```