270: Hexadecimal Floating-Point Constants270: Hexadecimal Floating-Point Constantsby Peter McGoron
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Floating-point numbers are generally in radix 2, but are written by users in radix 10. This SRFI introduces Scheme syntax for hexadecimal floating point constants based on C99's syntax.
Computers generally use binary floating point to store numbers, while humans like decimals. To express binary floating point exactly, one can use hexadecimal floating point constants, which are written in base 16 instead of base 10. They represent binary floating point numbers of any mantissa and exponent size precisely.
Prior art: MIT Scheme supports a version of this syntax. C99 introduced a hexadecimal floating point syntax that was then adopted in IEEE 754-2008.
Non-normative text is in small text.
The following is written in Scheme's BNF, extended with the following: ⟨thing⟩? means zero or one of ⟨thing⟩.
The formal syntax of Scheme is modified to have the following new productions:
. ⟨digit 16⟩+. ⟨digit 16⟩*p ⟨sign⟩? ⟨digit 10⟩+
Without an exactness prefix, a hex float is read as an inexact
number. Like all numbers, hex floats can be read as exact numbers
by prefixing them with #e.
The exponential separator is p because it is ambiguous
whether e would be a decimal separator or a hex digit.
A hexadecimal floating point number is interpreted such that
xnxn−1…x0.x−1x−2…x−mpd
where each x is a hexadecimal digit and d is a base 10 number, is equal to
(Xn×16n + Xn−1×16n−1 + … + X0 + X−1×16−1 + X−2×16−2 + … + X−m×16−m)×2d
where Xi is the numerical value of the digit xi.
On R6RS systems, the above grammar is modified to be:
pPThe R6RS considerations for numbers with ⟨exponent marker⟩ and ⟨mantissa width⟩ apply here.
This grammar is a superset of the previous grammar.
The procedure string->number must understand hexadecimal
floats.
(write-hexadecimal-float x [port])Write an inexact real as a floating point number. If the number is represented by an IEEE 754 binary floating point number, then the number must be represented in a canonical form. If the number is a real, then:
If the number is complex, then each component must be printed with the above rules.
This section is non-normative.
| Expression | Number |
|---|---|
#x9p9 |
9×29 = 4608 |
#x1.2p3 |
(1 + 2×16−1)×23 = 9 |
#xFE.FFp1 |
(15×161 + 14 + 15×16−1 + 15×16−2)×21 = 1044465/2048 ≈ 509.992… |
#x-0.Ap-2 |
(10×16−1)×2−2 = −5/32 = −.15625 |
#x1.2p3+10p1i |
9+256i |
A portable implementation is impossible, as it depends on both modifying the reader and knowing the specific representation of inexact reals.
An implementation-specific version will be written.
Taylor Campbell suggested it to me on #scheme.
© 2026 Peter McGoron.
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