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 Incoming links: Signed power-of-two division by biasing

C11 committee draft N1570 2026-10-24

www.open-std.org/jtc1/sc22/wg14/www/docs/n1570.pdf

The 12 April 2011 committee draft specifies C integer division in section 6.5.5, especially paragraphs 5–6 and footnote 105. It fixes the truncation and quotient–remainder relationship used when verifying signed power-of-two division by biasing.

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 1 / e / Solution 2026-10-25

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Use the same bias, then an arithmetic right shift by ten:

div1024:
    asrs r3, r0, #31    // -1 when a<0, otherwise 0
    lsrs r3, r3, #22    // b = 1023 when a<0, otherwise 0
    adds r0, r0, r3     // a+b, representable for every signed input
    asrs r0, r0, #10    // floor((a+b)/1024)
    bx lr

An arithmetic right shift of a two's complement integer computes division rounded down. For

a \geq 0

, this already equals truncation towards zero. For

a < 0

, the signed power-of-two division by biasing identity is

trunc (a /1024) = ⌊ \frac{a + 1023}{1024} ⌋ .

(1)

Indeed, writing

- a = 1024 q + r

with

0 \leq r < 1024

makes the adjusted numerator

- 1024 q + (1023 - r)

; its quotient rounded down is

- q

for every

r

, including zero. Thus the code returns the signed quotient in all cases and contains no data-dependent branch. The final bx lr simply returns to the caller.

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