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# 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)
# special cases: 2*x is +-inf, nan or |x| < 0x1p-32
# check (exponent|0x8000)+2 < 0xbfff+2-32
movw 16(%rsp), %ax
movw %ax, %dx
orw $0x8000, %dx
addw $2, %dx
cmpw $0xbfff-30, %dx
jnb 3f
cmpw $1, %dx
jbe 1f
# if |x|<0x1p-32 return 1+x
fld1
jmp 2f
1: testw %ax, %ax
jns 1f
# if 2*x == -inf,-nan return -0/x
fldz
fchs
fdivp
ret
# if 2*x == inf,nan return 2*x
1: fld %st(0)
2: faddp
ret
# should be 0x1.71547652b82fe178p0 == 0x3fff b8aa3b29 5c17f0bc
# it will be wrong on non-nearest rounding mode
3: 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
# 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
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