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    //! Bit fiddling on positive IEEE 754 floats. Negative numbers aren't and needn't be handled.
    //! Normal floating point numbers have a canonical representation as (frac, exp) such that the
    //! value is 2<sup>exp</sup> * (1 + sum(frac[N-i] / 2<sup>i</sup>)) where N is the number of bits.
    //! Subnormals are slightly different and weird, but the same principle applies.
    //!
    //! Here, however, we represent them as (sig, k) with f positive, such that the value is f *
    //! 2<sup>e</sup>. Besides making the "hidden bit" explicit, this changes the exponent by the
    //! so-called mantissa shift.
    //!
    //! Put another way, normally floats are written as (1) but here they are written as (2):
    //!
    //! 1. `1.101100...11 * 2^m`
    //! 2. `1101100...11 * 2^n`
    //!
    //! We call (1) the **fractional representation** and (2) the **integral representation**.
    //!
    //! Many functions in this module only handle normal numbers. The dec2flt routines conservatively
    //! take the universally-correct slow path (Algorithm M) for very small and very large numbers.
    //! That algorithm needs only next_float() which does handle subnormals and zeros.
    use cmp::Ordering::{Less, Equal, Greater};
    use convert::{TryFrom, TryInto};
    use ops::{Add, Mul, Div, Neg};
    use fmt::{Debug, LowerExp};
    use num::diy_float::Fp;
    use num::FpCategory::{Infinite, Zero, Subnormal, Normal, Nan};
    use num::FpCategory;
    use num::dec2flt::num::{self, Big};
    use num::dec2flt::table;

    #[derive(Copy, Clone, Debug)]
    pub struct Unpacked {
        pub sig: u64,
        pub k: i16,
    }

    impl Unpacked {
        pub fn new(sig: u64, k: i16) -> Self {
            Unpacked { sig, k }
        }
    }

    /// A helper trait to avoid duplicating basically all the conversion code for `f32` and `f64`.
    ///
    /// See the parent module's doc comment for why this is necessary.
    ///
    /// Should **never ever** be implemented for other types or be used outside the dec2flt module.
    pub trait RawFloat
        : Copy
        + Debug
        + LowerExp
        + Mul<Output=Self>
        + Div<Output=Self>
        + Neg<Output=Self>
    {
        const INFINITY: Self;
        const NAN: Self;
        const ZERO: Self;

        /// Type used by `to_bits` and `from_bits`.
        type Bits: Add<Output = Self::Bits> + From<u8> + TryFrom<u64>;

        /// Raw transmutation to integer.
        fn to_bits(self) -> Self::Bits;

        /// Raw transmutation from integer.
        fn from_bits(v: Self::Bits) -> Self;

        /// Returns the category that this number falls into.
        fn classify(self) -> FpCategory;

        /// Returns the mantissa, exponent and sign as integers.
        fn integer_decode(self) -> (u64, i16, i8);

        /// Decode the float.
        fn unpack(self) -> Unpacked;

        /// Cast from a small integer that can be represented exactly.  Panic if the integer can't be
        /// represented, the other code in this module makes sure to never let that happen.
        fn from_int(x: u64) -> Self;

        /// Get the value 10<sup>e</sup> from a pre-computed table.
        /// Panics for `e >= CEIL_LOG5_OF_MAX_SIG`.
        fn short_fast_pow10(e: usize) -> Self;

        /// What the name says. It's easier to hard code than juggling intrinsics and
        /// hoping LLVM constant folds it.
        const CEIL_LOG5_OF_MAX_SIG: i16;

        // A conservative bound on the decimal digits of inputs that can't produce overflow or zero or
        /// subnormals. Probably the decimal exponent of the maximum normal value, hence the name.
        const MAX_NORMAL_DIGITS: usize;

        /// When the most significant decimal digit has a place value greater than this, the number
        /// is certainly rounded to infinity.
        const INF_CUTOFF: i64;

        /// When the most significant decimal digit has a place value less than this, the number
        /// is certainly rounded to zero.
        const ZERO_CUTOFF: i64;

        /// The number of bits in the exponent.
        const EXP_BITS: u8;

        /// The number of bits in the significand, *including* the hidden bit.
        const SIG_BITS: u8;

        /// The number of bits in the significand, *excluding* the hidden bit.
        const EXPLICIT_SIG_BITS: u8;

        /// The maximum legal exponent in fractional representation.
        const MAX_EXP: i16;

        /// The minimum legal exponent in fractional representation, excluding subnormals.
        const MIN_EXP: i16;

        /// `MAX_EXP` for integral representation, i.e., with the shift applied.
        const MAX_EXP_INT: i16;

        /// `MAX_EXP` encoded (i.e., with offset bias)
        const MAX_ENCODED_EXP: i16;

        /// `MIN_EXP` for integral representation, i.e., with the shift applied.
        const MIN_EXP_INT: i16;

        /// The maximum normalized significand in integral representation.
        const MAX_SIG: u64;

        /// The minimal normalized significand in integral representation.
        const MIN_SIG: u64;
    }

    // Mostly a workaround for #34344.
    macro_rules! other_constants {
        ($type: ident) => {
            const EXPLICIT_SIG_BITS: u8 = Self::SIG_BITS - 1;
            const MAX_EXP: i16 = (1 << (Self::EXP_BITS - 1)) - 1;
            const MIN_EXP: i16 = -Self::MAX_EXP + 1;
            const MAX_EXP_INT: i16 = Self::MAX_EXP - (Self::SIG_BITS as i16 - 1);
            const MAX_ENCODED_EXP: i16 = (1 << Self::EXP_BITS) - 1;
            const MIN_EXP_INT: i16 = Self::MIN_EXP - (Self::SIG_BITS as i16 - 1);
            const MAX_SIG: u64 = (1 << Self::SIG_BITS) - 1;
            const MIN_SIG: u64 = 1 << (Self::SIG_BITS - 1);

            const INFINITY: Self = $crate::$type::INFINITY;
            const NAN: Self = $crate::$type::NAN;
            const ZERO: Self = 0.0;
        }
    }

    impl RawFloat for f32 {
        type Bits = u32;

        const SIG_BITS: u8 = 24;
        const EXP_BITS: u8 = 8;
        const CEIL_LOG5_OF_MAX_SIG: i16 = 11;
        const MAX_NORMAL_DIGITS: usize = 35;
        const INF_CUTOFF: i64 = 40;
        const ZERO_CUTOFF: i64 = -48;
        other_constants!(f32);

        /// Returns the mantissa, exponent and sign as integers.
        fn integer_decode(self) -> (u64, i16, i8) {
            let bits = self.to_bits();
            let sign: i8 = if bits >> 31 == 0 { 1 } else { -1 };
            let mut exponent: i16 = ((bits >> 23) & 0xff) as i16;
            let mantissa = if exponent == 0 {
                (bits & 0x7fffff) << 1
            } else {
                (bits & 0x7fffff) | 0x800000
            };
            // Exponent bias + mantissa shift
            exponent -= 127 + 23;
            (mantissa as u64, exponent, sign)
        }

        fn unpack(self) -> Unpacked {
            let (sig, exp, _sig) = self.integer_decode();
            Unpacked::new(sig, exp)
        }

        fn from_int(x: u64) -> f32 {
            // rkruppe is uncertain whether `as` rounds correctly on all platforms.
            debug_assert!(x as f32 == fp_to_float(Fp { f: x, e: 0 }));
            x as f32
        }

        fn short_fast_pow10(e: usize) -> Self {
            table::F32_SHORT_POWERS[e]
        }

        fn classify(self) -> FpCategory { self.classify() }
        fn to_bits(self) -> Self::Bits { self.to_bits() }
        fn from_bits(v: Self::Bits) -> Self { Self::from_bits(v) }
    }

    impl RawFloat for f64 {
        type Bits = u64;

        const SIG_BITS: u8 = 53;
        const EXP_BITS: u8 = 11;
        const CEIL_LOG5_OF_MAX_SIG: i16 = 23;
        const MAX_NORMAL_DIGITS: usize = 305;
        const INF_CUTOFF: i64 = 310;
        const ZERO_CUTOFF: i64 = -326;
        other_constants!(f64);

        /// Returns the mantissa, exponent and sign as integers.
        fn integer_decode(self) -> (u64, i16, i8) {
            let bits = self.to_bits();
            let sign: i8 = if bits >> 63 == 0 { 1 } else { -1 };
            let mut exponent: i16 = ((bits >> 52) & 0x7ff) as i16;
            let mantissa = if exponent == 0 {
                (bits & 0xfffffffffffff) << 1
            } else {
                (bits & 0xfffffffffffff) | 0x10000000000000
            };
            // Exponent bias + mantissa shift
            exponent -= 1023 + 52;
            (mantissa, exponent, sign)
        }

        fn unpack(self) -> Unpacked {
            let (sig, exp, _sig) = self.integer_decode();
            Unpacked::new(sig, exp)
        }

        fn from_int(x: u64) -> f64 {
            // rkruppe is uncertain whether `as` rounds correctly on all platforms.
            debug_assert!(x as f64 == fp_to_float(Fp { f: x, e: 0 }));
            x as f64
        }

        fn short_fast_pow10(e: usize) -> Self {
            table::F64_SHORT_POWERS[e]
        }

        fn classify(self) -> FpCategory { self.classify() }
        fn to_bits(self) -> Self::Bits { self.to_bits() }
        fn from_bits(v: Self::Bits) -> Self { Self::from_bits(v) }
    }

    /// Convert an Fp to the closest machine float type.
    /// Does not handle subnormal results.
    pub fn fp_to_float<T: RawFloat>(x: Fp) -> T {
        let x = x.normalize();
        // x.f is 64 bit, so x.e has a mantissa shift of 63
        let e = x.e + 63;
        if e > T::MAX_EXP {
            panic!("fp_to_float: exponent {} too large", e)
        }  else if e > T::MIN_EXP {
            encode_normal(round_normal::<T>(x))
        } else {
            panic!("fp_to_float: exponent {} too small", e)
        }
    }

    /// Round the 64-bit significand to T::SIG_BITS bits with half-to-even.
    /// Does not handle exponent overflow.
    pub fn round_normal<T: RawFloat>(x: Fp) -> Unpacked {
        let excess = 64 - T::SIG_BITS as i16;
        let half: u64 = 1 << (excess - 1);
        let (q, rem) = (x.f >> excess, x.f & ((1 << excess) - 1));
        assert_eq!(q << excess | rem, x.f);
        // Adjust mantissa shift
        let k = x.e + excess;
        if rem < half {
            Unpacked::new(q, k)
        } else if rem == half && (q % 2) == 0 {
            Unpacked::new(q, k)
        } else if q == T::MAX_SIG {
            Unpacked::new(T::MIN_SIG, k + 1)
        } else {
            Unpacked::new(q + 1, k)
        }
    }

    /// Inverse of `RawFloat::unpack()` for normalized numbers.
    /// Panics if the significand or exponent are not valid for normalized numbers.
    pub fn encode_normal<T: RawFloat>(x: Unpacked) -> T {
        debug_assert!(T::MIN_SIG <= x.sig && x.sig <= T::MAX_SIG,
            "encode_normal: significand not normalized");
        // Remove the hidden bit
        let sig_enc = x.sig & !(1 << T::EXPLICIT_SIG_BITS);
        // Adjust the exponent for exponent bias and mantissa shift
        let k_enc = x.k + T::MAX_EXP + T::EXPLICIT_SIG_BITS as i16;
        debug_assert!(k_enc != 0 && k_enc < T::MAX_ENCODED_EXP,
            "encode_normal: exponent out of range");
        // Leave sign bit at 0 ("+"), our numbers are all positive
        let bits = (k_enc as u64) << T::EXPLICIT_SIG_BITS | sig_enc;
        T::from_bits(bits.try_into().unwrap_or_else(|_| unreachable!()))
    }

    /// Construct a subnormal. A mantissa of 0 is allowed and constructs zero.
    pub fn encode_subnormal<T: RawFloat>(significand: u64) -> T {
        assert!(significand < T::MIN_SIG, "encode_subnormal: not actually subnormal");
        // Encoded exponent is 0, the sign bit is 0, so we just have to reinterpret the bits.
        T::from_bits(significand.try_into().unwrap_or_else(|_| unreachable!()))
    }

    /// Approximate a bignum with an Fp. Rounds within 0.5 ULP with half-to-even.
    pub fn big_to_fp(f: &Big) -> Fp {
        let end = f.bit_length();
        assert!(end != 0, "big_to_fp: unexpectedly, input is zero");
        let start = end.saturating_sub(64);
        let leading = num::get_bits(f, start, end);
        // We cut off all bits prior to the index `start`, i.e., we effectively right-shift by
        // an amount of `start`, so this is also the exponent we need.
        let e = start as i16;
        let rounded_down = Fp { f: leading, e }.normalize();
        // Round (half-to-even) depending on the truncated bits.
        match num::compare_with_half_ulp(f, start) {
            Less => rounded_down,
            Equal if leading % 2 == 0 => rounded_down,
            Equal | Greater => match leading.checked_add(1) {
                Some(f) => Fp { f, e }.normalize(),
                None => Fp { f: 1 << 63, e: e + 1 },
            }
        }
    }

    /// Find the largest floating point number strictly smaller than the argument.
    /// Does not handle subnormals, zero, or exponent underflow.
    pub fn prev_float<T: RawFloat>(x: T) -> T {
        match x.classify() {
            Infinite => panic!("prev_float: argument is infinite"),
            Nan => panic!("prev_float: argument is NaN"),
            Subnormal => panic!("prev_float: argument is subnormal"),
            Zero => panic!("prev_float: argument is zero"),
            Normal => {
                let Unpacked { sig, k } = x.unpack();
                if sig == T::MIN_SIG {
                    encode_normal(Unpacked::new(T::MAX_SIG, k - 1))
                } else {
                    encode_normal(Unpacked::new(sig - 1, k))
                }
            }
        }
    }

    // Find the smallest floating point number strictly larger than the argument.
    // This operation is saturating, i.e., next_float(inf) == inf.
    // Unlike most code in this module, this function does handle zero, subnormals, and infinities.
    // However, like all other code here, it does not deal with NaN and negative numbers.
    pub fn next_float<T: RawFloat>(x: T) -> T {
        match x.classify() {
            Nan => panic!("next_float: argument is NaN"),
            Infinite => T::INFINITY,
            // This seems too good to be true, but it works.
            // 0.0 is encoded as the all-zero word. Subnormals are 0x000m...m where m is the mantissa.
            // In particular, the smallest subnormal is 0x0...01 and the largest is 0x000F...F.
            // The smallest normal number is 0x0010...0, so this corner case works as well.
            // If the increment overflows the mantissa, the carry bit increments the exponent as we
            // want, and the mantissa bits become zero. Because of the hidden bit convention, this
            // too is exactly what we want!
            // Finally, f64::MAX + 1 = 7eff...f + 1 = 7ff0...0 = f64::INFINITY.
            Zero | Subnormal | Normal => {
                T::from_bits(x.to_bits() + T::Bits::from(1u8))
            }
        }
    }

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