Split Int into Int and MinInt

`MinInt` contains the basic methods that are only needed by integers
involved in widening operations, i.e. big integers. `Int` retains all
other operations and convenience methods.

src/float/add.rs

@@ -1,5 +1,5 @@
 use crate::float::Float;
-use crate::int::{CastInto, Int};
+use crate::int::{CastInto, Int, MinInt};
 
 /// Returns `a + b`
 fn add<F: Float>(a: F, b: F) -> F
@@ -57,17 +57,17 @@ where
         }
 
         // zero + anything = anything
-        if a_abs == Int::ZERO {
+        if a_abs == MinInt::ZERO {
             // but we need to get the sign right for zero + zero
-            if b_abs == Int::ZERO {
+            if b_abs == MinInt::ZERO {
                 return F::from_repr(a.repr() & b.repr());
             } else {
                 return b;
             }
         }
 
         // anything + zero = anything
-        if b_abs == Int::ZERO {
+        if b_abs == MinInt::ZERO {
             return a;
         }
     }
@@ -113,10 +113,10 @@ where
     // Shift the significand of b by the difference in exponents, with a sticky
     // bottom bit to get rounding correct.
     let align = a_exponent.wrapping_sub(b_exponent).cast();
-    if align != Int::ZERO {
+    if align != MinInt::ZERO {
         if align < bits {
             let sticky =
-                F::Int::from_bool(b_significand << bits.wrapping_sub(align).cast() != Int::ZERO);
+                F::Int::from_bool(b_significand << bits.wrapping_sub(align).cast() != MinInt::ZERO);
             b_significand = (b_significand >> align.cast()) | sticky;
         } else {
             b_significand = one; // sticky; b is known to be non-zero.
@@ -125,8 +125,8 @@ where
     if subtraction {
         a_significand = a_significand.wrapping_sub(b_significand);
         // If a == -b, return +zero.
-        if a_significand == Int::ZERO {
-            return F::from_repr(Int::ZERO);
+        if a_significand == MinInt::ZERO {
+            return F::from_repr(MinInt::ZERO);
         }
 
         // If partial cancellation occured, we need to left-shift the result
@@ -143,8 +143,8 @@ where
 
         // If the addition carried up, we need to right-shift the result and
         // adjust the exponent:
-        if a_significand & implicit_bit << 4 != Int::ZERO {
-            let sticky = F::Int::from_bool(a_significand & one != Int::ZERO);
+        if a_significand & implicit_bit << 4 != MinInt::ZERO {
+            let sticky = F::Int::from_bool(a_significand & one != MinInt::ZERO);
             a_significand = a_significand >> 1 | sticky;
             a_exponent += 1;
         }
@@ -160,7 +160,7 @@ where
         // need to shift the significand.
         let shift = (1 - a_exponent).cast();
         let sticky =
-            F::Int::from_bool((a_significand << bits.wrapping_sub(shift).cast()) != Int::ZERO);
+            F::Int::from_bool((a_significand << bits.wrapping_sub(shift).cast()) != MinInt::ZERO);
         a_significand = a_significand >> shift.cast() | sticky;
         a_exponent = 0;
     }

src/float/cmp.rs

@@ -1,7 +1,7 @@
 #![allow(unreachable_code)]
 
 use crate::float::Float;
-use crate::int::Int;
+use crate::int::MinInt;
 
 #[derive(Clone, Copy)]
 enum Result {

src/float/div.rs

@@ -3,7 +3,9 @@
 #![allow(clippy::needless_return)]
 
 use crate::float::Float;
-use crate::int::{CastInto, DInt, HInt, Int};
+use crate::int::{CastInto, DInt, HInt, Int, MinInt};
+
+use super::HalfRep;
 
 fn div32<F: Float>(a: F, b: F) -> F
 where
@@ -454,15 +456,20 @@ where
 
 fn div64<F: Float>(a: F, b: F) -> F
 where
-    u32: CastInto<F::Int>,
     F::Int: CastInto<u32>,
-    i32: CastInto<F::Int>,
     F::Int: CastInto<i32>,
-    u64: CastInto<F::Int>,
+    F::Int: CastInto<HalfRep<F>>,
+    F::Int: From<HalfRep<F>>,
+    F::Int: From<u8>,
     F::Int: CastInto<u64>,
-    i64: CastInto<F::Int>,
     F::Int: CastInto<i64>,
-    F::Int: HInt,
+    F::Int: HInt + DInt,
+    u16: CastInto<F::Int>,
+    i32: CastInto<F::Int>,
+    i64: CastInto<F::Int>,
+    u32: CastInto<F::Int>,
+    u64: CastInto<F::Int>,
+    u64: CastInto<HalfRep<F>>,
 {
     const NUMBER_OF_HALF_ITERATIONS: usize = 3;
     const NUMBER_OF_FULL_ITERATIONS: usize = 1;
@@ -471,7 +478,7 @@ where
     let one = F::Int::ONE;
     let zero = F::Int::ZERO;
     let hw = F::BITS / 2;
-    let lo_mask = u64::MAX >> hw;
+    let lo_mask = F::Int::MAX >> hw;
 
     let significand_bits = F::SIGNIFICAND_BITS;
     let max_exponent = F::EXPONENT_MAX;
@@ -616,21 +623,23 @@ where
 
     let mut x_uq0 = if NUMBER_OF_HALF_ITERATIONS > 0 {
         // Starting with (n-1) half-width iterations
-        let b_uq1_hw: u32 =
-            (CastInto::<u64>::cast(b_significand) >> (significand_bits + 1 - hw)) as u32;
+        let b_uq1_hw: HalfRep<F> = CastInto::<HalfRep<F>>::cast(
+            CastInto::<u64>::cast(b_significand) >> (significand_bits + 1 - hw),
+        );
 
         // C is (3/4 + 1/sqrt(2)) - 1 truncated to W0 fractional bits as UQ0.HW
         // with W0 being either 16 or 32 and W0 <= HW.
         // That is, C is the aforementioned 3/4 + 1/sqrt(2) constant (from which
         // b/2 is subtracted to obtain x0) wrapped to [0, 1) range.
 
         // HW is at least 32. Shifting into the highest bits if needed.
-        let c_hw = (0x7504F333_u64 as u32).wrapping_shl(hw.wrapping_sub(32));
+        let c_hw = (CastInto::<HalfRep<F>>::cast(0x7504F333_u64)).wrapping_shl(hw.wrapping_sub(32));
 
         // b >= 1, thus an upper bound for 3/4 + 1/sqrt(2) - b/2 is about 0.9572,
         // so x0 fits to UQ0.HW without wrapping.
-        let x_uq0_hw: u32 = {
-            let mut x_uq0_hw: u32 = c_hw.wrapping_sub(b_uq1_hw /* exact b_hw/2 as UQ0.HW */);
+        let x_uq0_hw: HalfRep<F> = {
+            let mut x_uq0_hw: HalfRep<F> =
+                c_hw.wrapping_sub(b_uq1_hw /* exact b_hw/2 as UQ0.HW */);
             // dbg!(x_uq0_hw);
             // An e_0 error is comprised of errors due to
             // * x0 being an inherently imprecise first approximation of 1/b_hw
@@ -661,8 +670,9 @@ where
                 // no overflow occurred earlier: ((rep_t)x_UQ0_hw * b_UQ1_hw >> HW) is
                 // expected to be strictly positive because b_UQ1_hw has its highest bit set
                 // and x_UQ0_hw should be rather large (it converges to 1/2 < 1/b_hw <= 1).
-                let corr_uq1_hw: u32 =
-                    0.wrapping_sub(((x_uq0_hw as u64).wrapping_mul(b_uq1_hw as u64)) >> hw) as u32;
+                let corr_uq1_hw: HalfRep<F> = CastInto::<HalfRep<F>>::cast(zero.wrapping_sub(
+                    ((F::Int::from(x_uq0_hw)).wrapping_mul(F::Int::from(b_uq1_hw))) >> hw,
+                ));
                 // dbg!(corr_uq1_hw);
 
                 // Now, we should multiply UQ0.HW and UQ1.(HW-1) numbers, naturally
@@ -677,7 +687,9 @@ where
                 // The fact corr_UQ1_hw was virtually round up (due to result of
                 // multiplication being **first** truncated, then negated - to improve
                 // error estimations) can increase x_UQ0_hw by up to 2*Ulp of x_UQ0_hw.
-                x_uq0_hw = ((x_uq0_hw as u64).wrapping_mul(corr_uq1_hw as u64) >> (hw - 1)) as u32;
+                x_uq0_hw = ((F::Int::from(x_uq0_hw)).wrapping_mul(F::Int::from(corr_uq1_hw))
+                    >> (hw - 1))
+                    .cast();
                 // dbg!(x_uq0_hw);
                 // Now, either no overflow occurred or x_UQ0_hw is 0 or 1 in its half_rep_t
                 // representation. In the latter case, x_UQ0_hw will be either 0 or 1 after
@@ -707,7 +719,7 @@ where
             // be not below that value (see g(x) above), so it is safe to decrement just
             // once after the final iteration. On the other hand, an effective value of
             // divisor changes after this point (from b_hw to b), so adjust here.
-            x_uq0_hw.wrapping_sub(1_u32)
+            x_uq0_hw.wrapping_sub(HalfRep::<F>::ONE)
         };
 
         // Error estimations for full-precision iterations are calculated just
@@ -717,7 +729,7 @@ where
         // Simulating operations on a twice_rep_t to perform a single final full-width
         // iteration. Using ad-hoc multiplication implementations to take advantage
         // of particular structure of operands.
-        let blo: u64 = (CastInto::<u64>::cast(b_uq1)) & lo_mask;
+        let blo: F::Int = b_uq1 & lo_mask;
         // x_UQ0 = x_UQ0_hw * 2^HW - 1
         // x_UQ0 * b_UQ1 = (x_UQ0_hw * 2^HW) * (b_UQ1_hw * 2^HW + blo) - b_UQ1
         //
@@ -726,19 +738,20 @@ where
         // +            [  x_UQ0_hw *  blo  ]
         // -                      [      b_UQ1       ]
         // = [      result       ][.... discarded ...]
-        let corr_uq1 = negate_u64(
-            (x_uq0_hw as u64) * (b_uq1_hw as u64) + (((x_uq0_hw as u64) * (blo)) >> hw) - 1,
-        ); // account for *possible* carry
-        let lo_corr = corr_uq1 & lo_mask;
-        let hi_corr = corr_uq1 >> hw;
+        let corr_uq1: F::Int = (F::Int::from(x_uq0_hw) * F::Int::from(b_uq1_hw)
+            + ((F::Int::from(x_uq0_hw) * blo) >> hw))
+            .wrapping_sub(one)
+            .wrapping_neg(); // account for *possible* carry
+        let lo_corr: F::Int = corr_uq1 & lo_mask;
+        let hi_corr: F::Int = corr_uq1 >> hw;
         // x_UQ0 * corr_UQ1 = (x_UQ0_hw * 2^HW) * (hi_corr * 2^HW + lo_corr) - corr_UQ1
-        let mut x_uq0: <F as Float>::Int = ((((x_uq0_hw as u64) * hi_corr) << 1)
-            .wrapping_add(((x_uq0_hw as u64) * lo_corr) >> (hw - 1))
-            .wrapping_sub(2))
-        .cast(); // 1 to account for the highest bit of corr_UQ1 can be 1
-                 // 1 to account for possible carry
-                 // Just like the case of half-width iterations but with possibility
-                 // of overflowing by one extra Ulp of x_UQ0.
+        let mut x_uq0: F::Int = ((F::Int::from(x_uq0_hw) * hi_corr) << 1)
+            .wrapping_add((F::Int::from(x_uq0_hw) * lo_corr) >> (hw - 1))
+            .wrapping_sub(F::Int::from(2u8));
+        // 1 to account for the highest bit of corr_UQ1 can be 1
+        // 1 to account for possible carry
+        // Just like the case of half-width iterations but with possibility
+        // of overflowing by one extra Ulp of x_UQ0.
         x_uq0 -= one;
         // ... and then traditional fixup by 2 should work
 
@@ -755,8 +768,8 @@ where
         x_uq0
     } else {
         // C is (3/4 + 1/sqrt(2)) - 1 truncated to 64 fractional bits as UQ0.n
-        let c: <F as Float>::Int = (0x7504F333 << (F::BITS - 32)).cast();
-        let x_uq0: <F as Float>::Int = c.wrapping_sub(b_uq1);
+        let c: F::Int = (0x7504F333 << (F::BITS - 32)).cast();
+        let x_uq0: F::Int = c.wrapping_sub(b_uq1);
         // E_0 <= 3/4 - 1/sqrt(2) + 2 * 2^-64
         x_uq0
     };
@@ -806,7 +819,7 @@ where
     // Now 1/b - (2*P) * 2^-W < x < 1/b
     // FIXME Is x_UQ0 still >= 0.5?
 
-    let mut quotient: <F as Float>::Int = x_uq0.widen_mul(a_significand << 1).hi();
+    let mut quotient: F::Int = x_uq0.widen_mul(a_significand << 1).hi();
     // Now, a/b - 4*P * 2^-W < q < a/b for q=<quotient_UQ1:dummy> in UQ1.(SB+1+W).
 
     // quotient_UQ1 is in [0.5, 2.0) as UQ1.(SB+1),
@@ -868,7 +881,7 @@ where
     // r = a - b * q
     let abs_result = if written_exponent > 0 {
         let mut ret = quotient & significand_mask;
-        ret |= ((written_exponent as u64) << significand_bits).cast();
+        ret |= written_exponent.cast() << significand_bits;
         residual <<= 1;
         ret
     } else {

src/float/extend.rs

@@ -1,5 +1,5 @@
 use crate::float::Float;
-use crate::int::{CastInto, Int};
+use crate::int::{CastInto, Int, MinInt};
 
 /// Generic conversion from a narrower to a wider IEEE-754 floating-point type
 fn extend<F: Float, R: Float>(a: F) -> R

src/float/mod.rs

@@ -1,6 +1,6 @@
 use core::ops;
 
-use super::int::Int;
+use crate::int::{DInt, Int, MinInt};
 
 pub mod add;
 pub mod cmp;
@@ -12,6 +12,9 @@ pub mod pow;
 pub mod sub;
 pub mod trunc;
 
+/// Wrapper to extract the integer type half of the float's size
+pub(crate) type HalfRep<F> = <<F as Float>::Int as DInt>::H;
+
 public_test_dep! {
 /// Trait for some basic operations on floats
 #[allow(dead_code)]
@@ -60,7 +63,7 @@ pub(crate) trait Float:
     /// A mask for the significand
     const SIGNIFICAND_MASK: Self::Int;
 
-    // The implicit bit of the float format
+    /// The implicit bit of the float format
     const IMPLICIT_BIT: Self::Int;
 
     /// A mask for the exponent

src/float/mul.rs

@@ -1,5 +1,5 @@
 use crate::float::Float;
-use crate::int::{CastInto, DInt, HInt, Int};
+use crate::int::{CastInto, DInt, HInt, Int, MinInt};
 
 fn mul<F: Float>(a: F, b: F) -> F
 where

src/float/trunc.rs

@@ -1,5 +1,5 @@
 use crate::float::Float;
-use crate::int::{CastInto, Int};
+use crate::int::{CastInto, Int, MinInt};
 
 fn trunc<F: Float, R: Float>(a: F) -> R
 where

src/int/addsub.rs

@@ -1,6 +1,6 @@
-use crate::int::{DInt, Int};
+use crate::int::{DInt, Int, MinInt};
 
-trait UAddSub: DInt {
+trait UAddSub: DInt + Int {
     fn uadd(self, other: Self) -> Self {
         let (lo, carry) = self.lo().overflowing_add(other.lo());
         let hi = self.hi().wrapping_add(other.hi());
@@ -22,7 +22,7 @@ impl UAddSub for u128 {}
 
 trait AddSub: Int
 where
-    <Self as Int>::UnsignedInt: UAddSub,
+    <Self as MinInt>::UnsignedInt: UAddSub,
 {
     fn add(self, other: Self) -> Self {
         Self::from_unsigned(self.unsigned().uadd(other.unsigned()))
@@ -37,7 +37,7 @@ impl AddSub for i128 {}
 
 trait Addo: AddSub
 where
-    <Self as Int>::UnsignedInt: UAddSub,
+    <Self as MinInt>::UnsignedInt: UAddSub,
 {
     fn addo(self, other: Self) -> (Self, bool) {
         let sum = AddSub::add(self, other);
@@ -50,7 +50,7 @@ impl Addo for u128 {}
 
 trait Subo: AddSub
 where
-    <Self as Int>::UnsignedInt: UAddSub,
+    <Self as MinInt>::UnsignedInt: UAddSub,
 {
     fn subo(self, other: Self) -> (Self, bool) {
         let sum = AddSub::sub(self, other);