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+// Adapted from https://github.com/Alexhuszagh/rust-lexical.
+
+//! Defines rounding schemes for floating-point numbers.
+
+use super::float::ExtendedFloat;
+use super::num::*;
+use super::shift::*;
+use core::mem;
+
+// MASKS
+
+/// Calculate a scalar factor of 2 above the halfway point.
+#[inline]
+pub(crate) fn nth_bit(n: u64) -> u64 {
+ let bits: u64 = mem::size_of::<u64>() as u64 * 8;
+ debug_assert!(n < bits, "nth_bit() overflow in shl.");
+
+ 1 << n
+}
+
+/// Generate a bitwise mask for the lower `n` bits.
+#[inline]
+pub(crate) fn lower_n_mask(n: u64) -> u64 {
+ let bits: u64 = mem::size_of::<u64>() as u64 * 8;
+ debug_assert!(n <= bits, "lower_n_mask() overflow in shl.");
+
+ if n == bits {
+ u64::max_value()
+ } else {
+ (1 << n) - 1
+ }
+}
+
+/// Calculate the halfway point for the lower `n` bits.
+#[inline]
+pub(crate) fn lower_n_halfway(n: u64) -> u64 {
+ let bits: u64 = mem::size_of::<u64>() as u64 * 8;
+ debug_assert!(n <= bits, "lower_n_halfway() overflow in shl.");
+
+ if n == 0 {
+ 0
+ } else {
+ nth_bit(n - 1)
+ }
+}
+
+/// Calculate a bitwise mask with `n` 1 bits starting at the `bit` position.
+#[inline]
+pub(crate) fn internal_n_mask(bit: u64, n: u64) -> u64 {
+ let bits: u64 = mem::size_of::<u64>() as u64 * 8;
+ debug_assert!(bit <= bits, "internal_n_halfway() overflow in shl.");
+ debug_assert!(n <= bits, "internal_n_halfway() overflow in shl.");
+ debug_assert!(bit >= n, "internal_n_halfway() overflow in sub.");
+
+ lower_n_mask(bit) ^ lower_n_mask(bit - n)
+}
+
+// NEAREST ROUNDING
+
+// Shift right N-bytes and round to the nearest.
+//
+// Return if we are above halfway and if we are halfway.
+#[inline]
+pub(crate) fn round_nearest(fp: &mut ExtendedFloat, shift: i32) -> (bool, bool) {
+ // Extract the truncated bits using mask.
+ // Calculate if the value of the truncated bits are either above
+ // the mid-way point, or equal to it.
+ //
+ // For example, for 4 truncated bytes, the mask would be b1111
+ // and the midway point would be b1000.
+ let mask: u64 = lower_n_mask(shift as u64);
+ let halfway: u64 = lower_n_halfway(shift as u64);
+
+ let truncated_bits = fp.mant & mask;
+ let is_above = truncated_bits > halfway;
+ let is_halfway = truncated_bits == halfway;
+
+ // Bit shift so the leading bit is in the hidden bit.
+ overflowing_shr(fp, shift);
+
+ (is_above, is_halfway)
+}
+
+// Tie rounded floating point to event.
+#[inline]
+pub(crate) fn tie_even(fp: &mut ExtendedFloat, is_above: bool, is_halfway: bool) {
+ // Extract the last bit after shifting (and determine if it is odd).
+ let is_odd = fp.mant & 1 == 1;
+
+ // Calculate if we need to roundup.
+ // We need to roundup if we are above halfway, or if we are odd
+ // and at half-way (need to tie-to-even).
+ if is_above || (is_odd && is_halfway) {
+ fp.mant += 1;
+ }
+}
+
+// Shift right N-bytes and round nearest, tie-to-even.
+//
+// Floating-point arithmetic uses round to nearest, ties to even,
+// which rounds to the nearest value, if the value is halfway in between,
+// round to an even value.
+#[inline]
+pub(crate) fn round_nearest_tie_even(fp: &mut ExtendedFloat, shift: i32) {
+ let (is_above, is_halfway) = round_nearest(fp, shift);
+ tie_even(fp, is_above, is_halfway);
+}
+
+// DIRECTED ROUNDING
+
+// Shift right N-bytes and round towards a direction.
+//
+// Return if we have any truncated bytes.
+#[inline]
+fn round_toward(fp: &mut ExtendedFloat, shift: i32) -> bool {
+ let mask: u64 = lower_n_mask(shift as u64);
+ let truncated_bits = fp.mant & mask;
+
+ // Bit shift so the leading bit is in the hidden bit.
+ overflowing_shr(fp, shift);
+
+ truncated_bits != 0
+}
+
+// Round down.
+#[inline]
+fn downard(_: &mut ExtendedFloat, _: bool) {}
+
+// Shift right N-bytes and round toward zero.
+//
+// Floating-point arithmetic defines round toward zero, which rounds
+// towards positive zero.
+#[inline]
+pub(crate) fn round_downward(fp: &mut ExtendedFloat, shift: i32) {
+ // Bit shift so the leading bit is in the hidden bit.
+ // No rounding schemes, so we just ignore everything else.
+ let is_truncated = round_toward(fp, shift);
+ downard(fp, is_truncated);
+}
+
+// ROUND TO FLOAT
+
+// Shift the ExtendedFloat fraction to the fraction bits in a native float.
+//
+// Floating-point arithmetic uses round to nearest, ties to even,
+// which rounds to the nearest value, if the value is halfway in between,
+// round to an even value.
+#[inline]
+pub(crate) fn round_to_float<F, Algorithm>(fp: &mut ExtendedFloat, algorithm: Algorithm)
+where
+ F: Float,
+ Algorithm: FnOnce(&mut ExtendedFloat, i32),
+{
+ // Calculate the difference to allow a single calculation
+ // rather than a loop, to minimize the number of ops required.
+ // This does underflow detection.
+ let final_exp = fp.exp + F::DEFAULT_SHIFT;
+ if final_exp < F::DENORMAL_EXPONENT {
+ // We would end up with a denormal exponent, try to round to more
+ // digits. Only shift right if we can avoid zeroing out the value,
+ // which requires the exponent diff to be < M::BITS. The value
+ // is already normalized, so we shouldn't have any issue zeroing
+ // out the value.
+ let diff = F::DENORMAL_EXPONENT - fp.exp;
+ if diff <= u64::FULL {
+ // We can avoid underflow, can get a valid representation.
+ algorithm(fp, diff);
+ } else {
+ // Certain underflow, assign literal 0s.
+ fp.mant = 0;
+ fp.exp = 0;
+ }
+ } else {
+ algorithm(fp, F::DEFAULT_SHIFT);
+ }
+
+ if fp.mant & F::CARRY_MASK == F::CARRY_MASK {
+ // Roundup carried over to 1 past the hidden bit.
+ shr(fp, 1);
+ }
+}
+
+// AVOID OVERFLOW/UNDERFLOW
+
+// Avoid overflow for large values, shift left as needed.
+//
+// Shift until a 1-bit is in the hidden bit, if the mantissa is not 0.
+#[inline]
+pub(crate) fn avoid_overflow<F>(fp: &mut ExtendedFloat)
+where
+ F: Float,
+{
+ // Calculate the difference to allow a single calculation
+ // rather than a loop, minimizing the number of ops required.
+ if fp.exp >= F::MAX_EXPONENT {
+ let diff = fp.exp - F::MAX_EXPONENT;
+ if diff <= F::MANTISSA_SIZE {
+ // Our overflow mask needs to start at the hidden bit, or at
+ // `F::MANTISSA_SIZE+1`, and needs to have `diff+1` bits set,
+ // to see if our value overflows.
+ let bit = (F::MANTISSA_SIZE + 1) as u64;
+ let n = (diff + 1) as u64;
+ let mask = internal_n_mask(bit, n);
+ if (fp.mant & mask) == 0 {
+ // If we have no 1-bit in the hidden-bit position,
+ // which is index 0, we need to shift 1.
+ let shift = diff + 1;
+ shl(fp, shift);
+ }
+ }
+ }
+}
+
+// ROUND TO NATIVE
+
+// Round an extended-precision float to a native float representation.
+#[inline]
+pub(crate) fn round_to_native<F, Algorithm>(fp: &mut ExtendedFloat, algorithm: Algorithm)
+where
+ F: Float,
+ Algorithm: FnOnce(&mut ExtendedFloat, i32),
+{
+ // Shift all the way left, to ensure a consistent representation.
+ // The following right-shifts do not work for a non-normalized number.
+ fp.normalize();
+
+ // Round so the fraction is in a native mantissa representation,
+ // and avoid overflow/underflow.
+ round_to_float::<F, _>(fp, algorithm);
+ avoid_overflow::<F>(fp);
+}