use std pkg math = const flt32fromflt64 : (f : flt64 -> flt32) const flt64fromflt32 : (x : flt32 -> flt64) /* For use in various normalizations */ const find_first1_64 : (b : uint64, start : int64 -> int64) const find_first1_64_hl : (h : uint64, l : uint64, start : int64 -> int64) /* >> and <<, but without wrapping when the shift is >= 64 */ const shr : (u : uint64, s : int64 -> uint64) const shl : (u : uint64, s : int64 -> uint64) /* Whether RN() requires incrementing after truncating */ const need_round_away : (h : uint64, l : uint64, bitpos_last : int64 -> bool) /* Multiply x * y to z1 + z2 */ const two_by_two : (x : flt64, y : flt64 -> (flt64, flt64)) /* Return (s, t) such that s + t = a + b, with s = rn(a + b). */ generic fast2sum : (x : @f, y : @f -> (@f, @f)) :: floating, numeric @f /* Rounds a + b (as flt64s) to a flt32. */ const round_down : (a : flt64, b : flt64 -> flt32) ;; /* Split precision down the middle */ const twentysix_bits_mask = (0xffffffffffffffff << 27) const flt64fromflt32 = {f : flt32 var n, e, s (n, e, s) = std.flt32explode(f) var xs : uint64 = (s : uint64) var xe : int64 = (e : int64) if e == 128 -> std.flt64assem(n, 1024, xs) elif e == -127 /* All subnormals in single precision (except 0.0s) can be upgraded to double precision, since the exponent range is so much wider. */ var first1 = find_first1_64(xs, 23) if first1 < 0 -> std.flt64assem(n, -1023, 0) ;; xs = xs << (52 - (first1 : uint64)) xe = -126 - (23 - first1) -> std.flt64assem(n, xe, xs) ;; -> std.flt64assem(n, xe, xs << (52 - 23)) } const flt32fromflt64 = {f : flt64 var n : bool, e : int64, s : uint64 (n, e, s) = std.flt64explode(f) var ts : uint32 var te : int32 = (e : int32) if e >= 128 if e == 1023 && s != 0 /* NaN */ -> std.flt32assem(n, 128, 1) else /* infinity */ -> std.flt32assem(n, 128, 0) ;; ;; if e >= -127 /* normal */ ts = ((s >> (52 - 23)) : uint32) if need_round_away(0, s, 52 - 23) ts++ if ts & (1 << 24) != 0 ts >>= 1 te++ ;; ;; if te >= -126 -> std.flt32assem(n, te, ts) ;; ;; /* subnormal already, will have to go to 0 */ if e == -1023 -> std.flt32assem(n, -127, 0) ;; /* subnormal (at least, it will be) */ te = -127 var shift : int64 = (52 - 23) + (-126 - e) var ts1 = shr(s, shift) ts = (ts1 : uint32) if need_round_away(0, s, shift) ts++ if ts & (1 << 23) != 0 /* false alarm, it's normal again */ te++ ;; ;; -> std.flt32assem(n, te, ts) } /* >> and <<, but without wrapping when the shift is >= 64 */ const shr = {u : uint64, s : int64 if (s : uint64) >= 64 -> 0 else -> u >> (s : uint64) ;; } const shl = {u : uint64, s : int64 if (s : uint64) >= 64 -> 0 else -> u << (s : uint64) ;; } /* Find the first 1 bit in a bitstring */ const find_first1_64 = {b : uint64, start : int64 for var j = start; j >= 0; --j var m = shl(1, j) if b & m != 0 -> j ;; ;; -> -1 } const find_first1_64_hl = {h, l, start var first1_h = find_first1_64(h, start - 64) if first1_h >= 0 -> first1_h + 64 ;; -> find_first1_64(l, 63) } /* For [ h ][ l ], where bitpos_last is the position of the last bit that was included in the truncated result (l's last bit has position 0), decide whether rounding up/away is needed. This is true if - following bitpos_last is a 1, then a non-zero sequence, or - following bitpos_last is a 1, then a zero sequence, and the round would be to even */ const need_round_away = {h : uint64, l : uint64, bitpos_last : int64 var first_omitted_is_1 = false var nonzero_beyond = false if bitpos_last > 64 first_omitted_is_1 = h & shl(1, bitpos_last - 1 - 64) != 0 nonzero_beyond = nonzero_beyond || h & shr((-1 : uint64), 2 + 64 - (bitpos_last - 64)) != 0 nonzero_beyond = nonzero_beyond || (l != 0) else first_omitted_is_1 = l & shl(1, bitpos_last - 1) != 0 nonzero_beyond = nonzero_beyond || l & shr((-1 : uint64), 1 + 64 - bitpos_last) != 0 ;; if !first_omitted_is_1 -> false ;; if nonzero_beyond -> true ;; var hl_is_odd = false if bitpos_last >= 64 hl_is_odd = h & shl(1, bitpos_last - 64) != 0 else hl_is_odd = l & shl(1, bitpos_last) != 0 ;; -> hl_is_odd } /* Perform high-prec multiplication: x * y = z1 + z2. */ const two_by_two = {x : flt64, y : flt64 var xh : flt64 = std.flt64frombits(std.flt64bits(x) & twentysix_bits_mask) var xl : flt64 = x - xh var yh : flt64 = std.flt64frombits(std.flt64bits(y) & twentysix_bits_mask) var yl : flt64 = y - yh /* Multiply out */ var a1 : flt64 = xh * yh var a2 : flt64 = xh * yl var a3 : flt64 = xl * yh var a4 : flt64 = xl * yl /* By-hand compensated summation */ var yy, u, t, v, z, s, c if a2 < a3 std.swap(&a3, &a2) ;; s = a1 c = 0.0 /* a2 */ (s, c) = fast2sum(s, a2) /* a3 */ (yy, u) = fast2sum(c, a3) (t, v) = fast2sum(s, yy) z = u + v (s, c) = fast2sum(t, z) /* a4 */ (yy, u) = fast2sum(c, a4) (t, v) = fast2sum(s, yy) z = u + v (s, c) = fast2sum(t, z) -> (s, c) } /* Return (s, t) such that s + t = a + b, with s = rn(a + b). */ generic fast2sum = {a : @f, b : @f :: floating, numeric @f var s = a + b var z = s - a var t = b - z -> (s, t) } /* Round a + b to a flt32. Only notable if round(a) is a rounding tie, and b is non-zero */ const round_down = {a : flt64, b : flt64 var au : uint64 = std.flt64bits(a) if au & 0x0000000070000000 == 0x0000000070000000 if b > 0.0 au++ elif b < 0.0 au-- ;; -> (std.flt64frombits(au) : flt32) ;; -> (a : flt32) }