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C++ integer arithmetic
- 1. Integers in C++ arvid@libtorrent.org The leading provider of blockchain technologies, on the forefront of work in financial cryptography and distributed systems.
- 2. Agenda • values and storage bits vs. values • Integer arithmetic bitwise vs. arithmetic operators • Integer overflow • Integral promotion rules unsigned integer promotion
- 3. discourage frivolous use of unsigned integers
- 4. Integer arithmetic Integers are values, not collections of bits. Two separate abstraction levels Bits Values Storage Integer values with algebraic properties
- 5. Agenda • values and storage bits vs. Values • Integer arithmetic bitwise vs. arithmetic operators • Integer overflow • Integral promotion rules unsigned integer promotion
- 6. Normal arithmetic Integer arithmetic 0 1 2 3 4 5 6 7-8 -7 -6 -5 -4 -3 -2 -1 8
- 8. Modular arithmetic Integer arithmetic 11 + 3 2 = 1 11 3 2 = 6 2 +
- 13. Integer arithmetic negative & positive only positive Normal arithmetic int Modular arithmetic unsigned int This only cost you a single bit! If you need it, get more bits This is danger zone!
- 14. Integer arithmetic • division /, addition +, subtraction -, multiplication *, modulus % operates on integers values • AND &, OR |, NOT ~, XOR ^ operates on independent bits (storage) • shift left <<, shift right >> technically arithmetic operators, but want to be bitwise
- 15. Integer arithmetic right-shift on integers are defined in terms of mathematical properties E1 >> E2
- 16. Integer arithmetic right-shift on integers are defined in terms of mathematical properties E1 >> E2 If E1 has an unsigned type or if E1 has a signed type and a non-negative value, the value of the result is the integral part of the quotient of E1/2E2.
- 17. Integer arithmetic Implementation defined for negative integers E1 >> E2 If E1 has an unsigned type or if E1 has a signed type and a non-negative value, the value of the result is the integral part of the quotient of E1/2E2. If E1 has a signed type and a negative value, the resulting value is implementation-defined.
- 18. Integer arithmetic Implementation defined for negative integers E1 >> E2 0
- 19. Integer arithmetic if E1 has a signed type and non-negative value, and E1×2E2 is representable in the corresponding unsigned type of the result type, then that value, converted to the result type, is the resulting value; otherwise, the behavior is undefined. left-shift has undefined behavior on negative integers and implementation defined behavior when shifting out-of-range E1 << E2
- 20. Integer arithmetic left-shift has undefined behavior on negative integers and implementation defined behavior when shifting out-of-range E1 << E2 010 . . . . . .
- 21. Integer arithmetic shift- and bitwise operators have non-trivial behavior on signed integers. Bitwise operators on signed integers depend on the implementation defined representation Bits Signed Values Arithmetic operators Bitwise operators Implementation defined mapping
- 22. Integer arithmetic • unary - on unsigned integers unsigned int a = 0; unsigned int b = -a; b == ?
- 23. Integer arithmetic • unary - on unsigned integers unsigned int a = 0; unsigned int b = -a; b == 232 - 0 The negative of an unsigned quantity is computed by subtracting its value from 2n, where n is the number of bits in the promoted operand.
- 24. Integer arithmetic Arithmetic operators Bitwise operators Unary - << >> + - * / % | & ^ ~ Operate on values Operate on bits
- 26. Integer arithmetic Signed integer Unsigned integer Arithmetic operators Bitwise operators Impractical or unintuitive semantics. Potentially poor performance. Implementation- & undefined defined behavior. Questionable semantics on values
- 27. Agenda • values and storage bits vs. values • Integer arithmetic bitwise vs. arithmetic operators • Integer overflow • Integral promotion rules unsigned integer promotion
- 28. Integer overflow • Signed integer overflow has undefined behavior • Any defined behavior is worse • You'll get the wrong answer either way If during the evaluation of an expression, the result is not mathematically defined or not in the range of representable values for its type, the behavior is undefined.
- 29. Integer overflow • Signed integer overflow has undefined behavior • Any defined behavior is worse • You'll get the wrong answer either way • Unsigned integers don’t have overflow, as they use modular arithmetic If during the evaluation of an expression, the result is not mathematically defined or not in the range of representable values for its type, the behavior is undefined.
- 30. Integer overflow • For symbolic simplification of expressions, they must implement regular algebra. e.g. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation: x = (-2 - b) / 2 + 10
- 31. Integer overflow x = (-2 - b) / 2 + 10 x = -2 / 2 - b / 2 + 10 • For symbolic simplification of expressions, they must implement regular algebra. e.g. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation:
- 32. Integer overflow x = (-2 - b) / 2 + 10 x = -2 / 2 - b / 2 + 10 x = -1 - b / 2 + 10 • For symbolic simplification of expressions, they must implement regular algebra. e.g. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation:
- 33. Integer overflow x = (-2 - b) / 2 + 10 x = -2 / 2 - b / 2 + 10 x = -1 - b / 2 + 10 x = 9 - b / 2 • For symbolic simplification of expressions, they must implement regular algebra. e.g. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation:
- 34. • For symbolic simplification of expressions, they must implement regular algebra. e.g. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation: Integer overflow x = (-2 - b) / 2 + 10 x = -2 / 2 - b / 2 + 10 x = -1 - b / 2 + 10 x = 9 - b / 2 x = 9 - (b >> 1) If the hardware supports shifting negative numbers
- 35. Integer overflow • Modular arithmetic • introduces discontinuities in the value space • Symbolic simplification of expressions much harder. Less likely to be optimized • unsigned integers have these properties
- 36. A is promoted to unsigned. (a - b) / 2 is not continuous. cannot be factored out to: a / 2 - b / 2 Integer overflow • Same example. If b is unsigned. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation: x = (-2 - b) / 2 + 10 x = -2 / 2 - b / 2 + 10
- 37. • Same example. If b is unsigned. x = (a - b) / c + 10 Given a=-2, c=2 through constant propagation: Actual expression with modulo semantics (i.e. unsigned) Integer overflow x = ((((-2 - b) % 2n) / 2) % 2n) + 10) % 2n
- 38. Integer overflow • Graph illustration - signed f(b) = (-2 - b)/2 + 10 -240 -160 -80 0 80 160 240 320 -160 -80 80 160
- 39. Integer overflow • Graph illustration - unsigned f(b) = (-2 - b)/2 + 10 -240 -160 -80 0 80 160 240 320 -160 -80 80 160
- 40. Integer overflow • Undefined behavior allows trapping overflow • -ftrapv • --sanitize=undefined
- 41. • Use signed integers for values that require normal arithmetic. • Use unsigned integers for: • flags (without arithmetic operators) • IDs (without arithmetic operators) • enumerations (enum class) • bits (only used with bitwise operators) Integer arithmetic
- 42. Agenda • values and storage bits vs. values • Integer arithmetic bitwise vs. arithmetic operators • Integer overflow • Integral promotion rules unsigned integer promotion
- 43. Integral promotion Integers are promoted to int before applying operators
- 44. Integral promotion Integers are promoted to int before applying operators uint16_t a = …; uint16_t b = …; uint16_t c = a + b;
- 45. Integral promotion Integers are promoted to int before applying operators uint16_t a = …; uint16_t b = …; uint16_t c = a + b; uint16_t c = int(a) + int(b); As if all built-in operators only take int parameters
- 46. Integral promotion • Assuming int ≡ std::int32_t. this causes an overflow and fail int32_t c1 = 2’000’000’000; int32_t c2 = 3; int32_t c3 = 4; int64_t result = c1 * c2 / c3; Intermediate type is int32_t OVERFLOW!
- 47. Integral promotion • ints and wider integral types are promoted to the highest ranking type • wider types rank higher • At same width, unsigned has higher rank
- 48. Integral promotion • ints and wider integral types are promoted to the highest ranking type • wider types rank higher • At same width, unsigned has higher rank Pitfall! This leads to modular arithmetic
- 49. Integral promotion • can cause surprising behavior. e.g. uint8_t a = 1; int b = ~a; std::cout << std::hex << b << "n";
- 50. • can cause surprising behavior. e.g. Integral promotion uint8_t a = 1; int b = ~a; std::cout << std::hex << b << "n"; 0xfffffffe Promoted to int zero-extended bitwise inverted
- 51. unsigned promotion values may not be preserved in signed ⟷ unsigned conversions
- 52. unsigned promotion • When wider than int, unsigned types are viral • can cause surprising behavior if they sneak into expressions.
- 53. unsigned promotion • When wider than int, unsigned types are viral • can cause surprising behavior if they sneak into expressions. e.g. int32_t a = -1; uint32_t b = 1; if (a > b) printf("wat");
- 54. unsigned promotion int32_t a = -1; uint32_t b = 1; if (a > b) printf("wat"); wat • When wider than int, unsigned types are viral • can cause surprising behavior if they sneak into expressions. e.g. Promoted to unsigned integer (0xffffffff)
- 55. Avoid implicit sign conversions -Wsign-conversion -Werror Integral promotion
- 56. • Pick signed/unsigned primarily based on the arithmetic you want • Signed integers ↔ arithmetic ops • Unsigned integers ↔ bitwise ops • Avoid signed → unsigned promotion Summary negative & positive only positive Normal arithmetic int Modular arithmetic unsigned int Signed integer Unsigned integer Arithmetic operators Bitwise & shift operators