notes

Integer division and modulo rounding

Programming languages implement division and modulo with various rounding modes. Most languages use truncated division by default. More information and additional languages can be found on the Wikipedia article for modulo.

Truncated (rounding toward zero)

• C: /, %
• C++: /, %
• Coq: Z.quot, Z.rem, Z.quotrem, Infix "÷" : Z_scope
• egglog: /, %
• Go: /, %, big.Int.Quo, big.Int.Rem, big.Int.QuoRem
• GMP: mpz_tdiv_q, mpz_tdiv_r, mpz_tdiv_qr
• Haskell: quot, rem, quotRem (Haskell 98 and 2010)
• LLVM: sdiv, srem
• num (Rust): /, %
• jq %
• Rug (Rust): /, %, div_trunc, rem_trunc, div_rem
• Rust: /, %, wrapping_div, wrapping_rem, overflowing_div, overflowing_rem, checked_div, checked_rem, strict_div (unstable), strict_rem (unstable), saturating_div

Floored (rounding down)

• Coq: Z.div, Z.modulo, Z.div_eucl, Infix "/" : Z_scope, Infix "mod" : Z_scope, N.div, N.modulo, N.div_eucl, Infix "/" : N_scope, Infix "mod" : N_scope
• GMP: mpz_fdiv_q, mpz_fdiv_r, mpz_fdiv_qr
• Haskell: div, mod, divMod (Haskell 98 and 2010)
• Rug (Rust): div_floor, rem_floor, div_rem_floor
• Rust: div_floor

Euclidean (non-negative remainder)

• Go: big.Int.Div, big.Int.Mod, big.Int.DivMod
• Rug (Rust): div_euc, rem_euc, div_rem_euc, mod_u
• Rust: div_euclid, rem_euclid, wrapping_div_euclid, wrapping_rem_euclid, overflowing_div_euclid, overflowing_rem_euclid, checked_div_euclid, checked_rem_euclid, strict_div_euclid (unstable), strict_rem_euclid (unstable)
• SMT-LIB: (div Int Int), (mod Int Int)

Ceiling (rounding up)

• GMP: mpz_cdiv_q, mpz_cdiv_r, mpz_cdiv_qr
• Rug (Rust): div_ceil, rem_ceil, div_rem_ceil
• Rust: div_ceil

Rounding half away from zero

• Rug (Rust): div_rem_round (not in GMP)

Non-negative remainder

• GMP: mpz_mod

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