# The Float interface

The generic interface `Float` provides access to the floating-point operations required or recommended by the IEEE floating-point standard. Consult the standard to resolve any fine points in the specification of the procedures. Non-IEEE implementations that have values similar to NaNs and infinities should explain how those values behave in an implementation guide. (NaN is an IEEE term whose informal meaning is `not a number`.)

    GENERIC INTERFACE Float(R);

    IMPORT FloatMode;

    TYPE T = R.T;

    PROCEDURE Scalb(x: T; n: INTEGER): T RAISES {FloatMode.Trap};

Return x..2 n.

    PROCEDURE Logb(x: T): T RAISES {FloatMode.Trap};

Return the exponent of `x`. More precisely, return the unique integer `n` such that the ratio ABS(x) Base n is in the half-open interval `[1..Base)`, unless `x` is denormalized, in which case return the minimum exponent value for `T`.

    PROCEDURE ILogb(x: T): INTEGER;

Like `Logb`, but returns an integer, never raises an exception, and always returns the `n` such that ABS(x) / Base n is in the half-open interval `[1..Base)`, even for denormalized numbers. Special cases: it returns `FIRST(INTEGER)` when `x` = 0.0, `LAST(INTEGER)` when `x` is plus or minus infinity, and zero when `x` is NaN.

    PROCEDURE NextAfter(x, y: T): T RAISES {FloatMode.Trap};

Return the next representable neighbor of `x` in the direction towards `y`. If `x = y`, return `x`.

    PROCEDURE CopySign(x, y: T): T;

Return `x` with the sign of `y`.

    PROCEDURE Finite(x: T): BOOLEAN;

Return `TRUE` if `x` is strictly between minus infinity and plus infinity. This always returns `TRUE` on non-IEEE implementations.

    PROCEDURE IsNaN(x: T): BOOLEAN;

Return `FALSE` if `x` represents a numerical (possibly infinite) value, and `TRUE` if `x` does not represent a numerical value. For example, on IEEE implementations, returns `TRUE` if x is a NaN, `FALSE` otherwise.

NaN (not a number)

    PROCEDURE Sign(x: T): [0..1];

Return the sign bit `x`. For non-IEEE implementations, this is the same as `ORD(x >= 0)`; for IEEE implementations, `Sign(-0) = 1` and `Sign(+0) = 0`.

    PROCEDURE Differs(x, y: T): BOOLEAN;

Return `(x < y OR y < x)`. Thus, for IEEE implementations, `Differs(NaN,x)` is always `FALSE`; for non-IEEE implementations, `Differs(x,y)` is the same as `x # y`.

    PROCEDURE Unordered(x, y: T): BOOLEAN;

Return `NOT (x <= y OR y <= x)`. Thus, for IEEE implementations, `Unordered(NaN, x)` is always `TRUE`; for non-IEEE implementations, `Unordered(x, y)` is always `FALSE`.

    PROCEDURE Sqrt(x: T): T RAISES {FloatMode.Trap};

Return the square root of `T`. This must be correctly rounded if `FloatMode.IEEE` is `TRUE`.

    TYPE IEEEClass =
      {SignalingNaN, QuietNaN, Infinity, Normal, Denormal, Zero};

    PROCEDURE Class(x: T): IEEEClass;

Return the IEEE number class containing `x`. On non-IEEE systems, the result will be `Normal` or `Zero`.

    PROCEDURE FromDecimal(
        sign: [0..1];
        READONLY digits: ARRAY OF [0..9];
        exp: INTEGER): T RAISES {FloatMode.Trap};

Convert from floating-decimal to type `T`.

floating-point conversion from decimal

decimal conversion to floating-point

Let `F` denote the nonnegative, floating-decimal number

     digits[0] . digits[1] ... digits[LAST(digits)] * 10^exp
     = sum(i, digits[i] * 10^(exp - i))

The result of `FromDecimal` is the number `(-1)^sign * F`, rounded to a value of type `T`.

The procedure `FromDecimal` is a floating-point operation, just like `+` and `*`, in the sense that it rounds its ideal result correctly, observing the current rounding mode, and it sets flags and raises traps by the usual rules. On IEEE implementations, it returns minus zero when `F` is sufficiently small and `sign=1`.

    TYPE DecimalApprox = RECORD
        class: IEEEClass;
        sign: [0..1];
        len: [1..R.MaxSignifDigits];
        digits: ARRAY[0..R.MaxSignifDigits-1] OF [0..9];
        exp: INTEGER;
        errorSign: [-1..1]
      END;

    PROCEDURE ToDecimal(x: T): DecimalApprox;

Convert from type `T` to floating-decimal.

floating-point conversion to decimal

decimal conversion from floating-point

Let `D` denote `ToDecimal(x)`. Then, `D.class = Class(x)` and `D.sign = Sign(x)`. The other fields are defined only when `D.class` is either `Normal` or `Denormal`. In those cases, the values `D.len`, `D.digits[0]` through `D.digits[D.len-1]`, and `D.exp` encode a floating-decimal number `F` with the property that `(-1)^D.sign * F` approximates `x` in a sense discussed below. The encoding is such that

     F = digits[0] . digits[1] ... digits[len - 1]  *  10^exp
       = sum(i, digits[i] * 10^(exp - i))

and

     ABS(x) = F * (1 + errorSign * epsilon)

where `epsilon` is small and positive. In particular, `D.errorSign` is `+1`, `0`, or `-1` according as `ABS(x)` is larger than, equal to, or smaller than `F`.

The current rounding mode determines the sense in which the floating-decimal number `(-1)^sign * F` approximates `x`, but in a slightly subtle way. Define the opposite of a directed rounding mode by reversing the direction, as follows:

          Opp(TowardPlusInfinity) := TowardMinusInfinity
         Opp(TowardMinusInfinity) := TowardPlusInfinity
                  Opp(TowardZero) := AwayFromZero

Note that `AwayFromZero` isn't actually a rounding mode, but it is clear what it would mean if it were. For all other rounding modes `M`, we define `Opp(M) = M`. If the current rounding mode is `M`, the call `ToDecimal(x)` returns a floating-decimal number that `FromDecimal` would convert, under rounding mode `Opp(M)`, back to `x`. Among all such numbers, the returned value has as few digits as possible. This implies that both `D.digits[0]` and `D.digits[D.len-1]` are nonzero. If there is a tie for having the fewest digits, the tying number closest to `x` wins. If there is also a tie for being closest to `x`, it must be a two-way tie and the number whose last digit is even wins.

Unlike `FromDecimal`, `ToDecimal` never sets a `FloatMode.Flag` and never raises `FloatMode.Trap`.

The idea of converting to decimal by retaining just as many digits as are necessary to convert back to binary exactly was popularized by Guy L. Steele Jr. and Jon L White [Steele] . David M. Gay pointed out the importance, in this context, of demanding that the conversion to binary handle mid-point cases by a known rule [Gay] . For example, in IEEE double precision, the floating-decimal number `1e23` is precisely halfway between two adjacent floating-binary numbers. If conversion to binary were allowed to go either way in such a mid-point case, conversion to decimal would have to avoid producing the simple number `1e23`, producing instead either `1.0000000000000001e23` or `9.999999999999999e22`. We believe the idea of combining the Steele/White style of automatic precision control with directed rounding by using opposite rounding modes, as above, is new with Lyle Ramshaw.

    END Float.