bitwise_bounds.h File Reference

#include <big-int/bigint.hh>

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Macros
#define	UINT_X   unsigned
#define	INT_X   int
#define	UINT_FUNC(FUNC, LHS, RHS)
#define	INT_FUNC(FUNC, LHS, RHS)
#define	GET_BIT_INTERVALS(NAME, MIN_UFUNC, MAX_UFUNC, MIN_FUNC, MAX_FUNC)

Functions
UINT_X	unsigned_max_or (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Adaptation of the algorithm in Henry S.
UINT_X	unsigned_min_or (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Adaptation of the algorithm in Henry S.
UINT_X	unsigned_max_and (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_min_and (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_max_xor (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_min_xor (UINT_X a, UINT_X b, UINT_X c, UINT_X d)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_max_not (UINT_X a, UINT_X)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_min_not (UINT_X, UINT_X b)
	Implementation of the algorithm in Henry S.
UINT_X	unsigned_max_left_shift (UINT_X, UINT_X b, UINT_X, UINT_X d)
	Upper bound on interval left shift.
UINT_X	unsigned_min_left_shift (UINT_X a, UINT_X, UINT_X c, UINT_X)
	Lower bound on interval left shift.
UINT_X	unsigned_max_right_shift (UINT_X, UINT_X b, UINT_X c, UINT_X)
	Upper bound on interval right shift.
UINT_X	unsigned_min_right_shift (UINT_X a, UINT_X, UINT_X, UINT_X d)
	Lower bound on interval right shift.
UINT_X	unsigned_2_unsigned_max_truncate (UINT_X a, UINT_X b, UINT_X n)
	Upper bound on unsigned to unsigned interval truncation.
UINT_X	unsigned_2_unsigned_min_truncate (UINT_X a, UINT_X b, UINT_X n)
	Lower bound on unsigned to unsigned interval truncation.
UINT_X	signed_2_unsigned_max_truncate (INT_X a, INT_X b, UINT_X n)
	Upper bound on signed to unsigned interval truncation.
UINT_X	signed_2_unsigned_min_truncate (INT_X a, INT_X b, UINT_X n)
	Lower bound on signed to unsigned interval truncation.
UINT_X	signed_2_unsigned_max_extend (INT_X a, INT_X b, UINT_X n)
	Upper bound on the unsigned extension of an n bit signed interval.
UINT_X	signed_2_unsigned_min_extend (INT_X a, INT_X b, UINT_X n)
	Lower bound on the unsigned extension of an n bit signed interval.
INT_X	signed_max_or (INT_X a, INT_X b, INT_X c, INT_X d)
	Implementation of the algorithm in Henry S.
INT_X	signed_min_or (INT_X a, INT_X b, INT_X c, INT_X d)
	Implementation of the algorithm in Henry S.
INT_X	signed_max_and (INT_X a, INT_X b, INT_X c, INT_X d)
	Implementation of the algorithm in Henry S.
INT_X	signed_min_and (INT_X a, INT_X b, INT_X c, INT_X d)
	Implementation of the algorithm in Henry S.
INT_X	signed_max_xor (INT_X a, INT_X b, INT_X c, INT_X d)
	Original upper bound on signed interval XOR, inspired by the case-based algorithm for signed interval OR in Henry S.
INT_X	signed_min_xor (INT_X a, INT_X b, INT_X c, INT_X d)
	Original lower bound on signed interval XOR, inspired by the case-based algorithm for signed interval OR in Henry S.
INT_X	signed_max_not (INT_X a, INT_X)
	Implementation of the algorithm in Henry S.
INT_X	signed_min_not (INT_X, INT_X b)
	Implementation of the algorithm in Henry S.
INT_X	signed_max_left_shift (INT_X, INT_X b, UINT_X c, UINT_X d)
	Upper bound on interval left shift.
INT_X	signed_min_left_shift (INT_X a, INT_X, UINT_X c, UINT_X d)
	Lower bound on interval left shift.
INT_X	signed_max_right_shift (INT_X, INT_X b, UINT_X c, UINT_X d)
	Upper bound on interval right shift.
INT_X	signed_min_right_shift (INT_X a, INT_X, UINT_X c, UINT_X d)
	Lower bound on interval right shift.
INT_X	unsigned_2_signed (UINT_X x, UINT_X n)
	Cast a limited-width unsigned integer to signed integer.
INT_X	unsigned_2_signed_max_truncate (UINT_X a, UINT_X b, UINT_X n)
	Upper bound on unsigned to signed interval truncation.
INT_X	unsigned_2_signed_min_truncate (UINT_X a, UINT_X b, UINT_X n)
	Lower bound on unsigned to signed interval truncation.
INT_X	signed_2_signed_max_truncate (INT_X a, INT_X b, UINT_X n)
	Upper bound on signed to signed interval truncation.
INT_X	signed_2_signed_min_truncate (INT_X a, INT_X b, UINT_X n)
	Lower bound on signed to signed interval truncation.
INT_X	unsigned_2_signed_max_extend (UINT_X a, UINT_X b, UINT_X n)
	Upper bound on the signed extension of an n bit unsigned interval.
INT_X	unsigned_2_signed_min_extend (UINT_X a, UINT_X b, UINT_X n)
	Lower bound on the signed extension of an n bit unsigned interval.
	GET_BIT_INTERVALS (interval_bitand, unsigned_min_and, unsigned_max_and, signed_min_and, signed_max_and) GET_BIT_INTERVALS(interval_bitor
signed_max_or	GET_BIT_INTERVALS (interval_bitxor, unsigned_min_xor, unsigned_max_xor, signed_min_xor, signed_max_xor) GET_BIT_INTERVALS(interval_logical_right_shift

Variables
	unsigned_min_or
	unsigned_max_or
	signed_min_or
signed_max_or	unsigned_min_right_shift
signed_max_or	unsigned_max_right_shift
signed_max_or	signed_min_right_shift

Macro Definition Documentation

GET_BIT_INTERVALS

#define GET_BIT_INTERVALS	(		NAME,
			MIN_UFUNC,
			MAX_UFUNC,
			MIN_FUNC,
			MAX_FUNC )

Value:

  template <>                                                                  \
  interval_templatet<BigInt> interval_templatet<BigInt>::NAME(                 \
    const interval_templatet<BigInt> &lhs,                                     \
    const interval_templatet<BigInt> &rhs) const                               \
  {                                                                            \
    interval_templatet<BigInt> result;                                         \
    if (!lhs.lower || !lhs.upper || !rhs.lower || !rhs.upper)                  \
      return result;                                                           \
    if (is_unsignedbv_type(lhs.type) && is_unsignedbv_type(rhs.type))          \
    {                                                                          \
      result.set_lower(UINT_FUNC(MIN_UFUNC, lhs, rhs));                        \
      result.set_upper(UINT_FUNC(MAX_UFUNC, lhs, rhs));                        \
    }                                                                          \
    else if (is_signedbv_type(lhs.type) && is_signedbv_type(rhs.type))         \
    {                                                                          \
      result.set_lower(INT_FUNC(MIN_FUNC, lhs, rhs));                          \
      result.set_upper(INT_FUNC(MAX_FUNC, lhs, rhs));                          \
    }                                                                          \
    return result;                                                             \
  }

INT_FUNC

#define INT_FUNC	(		FUNC,
			LHS,
			RHS )

Value:

  FUNC(                                                                        \
    LHS.get_lower().to_int64(),                                                \
    LHS.get_upper().to_int64(),                                                \
    RHS.get_lower().to_int64(),                                                \
    RHS.get_upper().to_int64())

INT_X

#define INT_X   int

UINT_FUNC

#define UINT_FUNC	(		FUNC,
			LHS,
			RHS )

Value:

  FUNC(                                                                        \
    LHS.get_lower().to_uint64(),                                               \
    LHS.get_upper().to_uint64(),                                               \
    RHS.get_lower().to_uint64(),                                               \
    RHS.get_upper().to_uint64())

UINT_X

#define UINT_X   unsigned

Function Documentation

GET_BIT_INTERVALS() [1/2]

GET_BIT_INTERVALS	(	interval_bitand	,
		unsigned_min_and	,
		unsigned_max_and	,
		signed_min_and	,
		signed_max_and	)

GET_BIT_INTERVALS() [2/2]

signed_max_or GET_BIT_INTERVALS	(	interval_bitxor	,
		unsigned_min_xor	,
		unsigned_max_xor	,
		signed_min_xor	,
		signed_max_xor	)

signed_2_signed_max_truncate()

INT_X signed_2_signed_max_truncate	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Upper bound on signed to signed interval truncation.

Author

Edoardo Manino.

Maximum value of signed truncate(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

signed_2_signed_min_truncate()

INT_X signed_2_signed_min_truncate	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Lower bound on signed to signed interval truncation.

Author

Edoardo Manino.

Minimum value of signed truncate(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

signed_2_unsigned_max_extend()

UINT_X signed_2_unsigned_max_extend	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Upper bound on the unsigned extension of an n bit signed interval.

Author

Edoardo Manino.

Maximum value of unsigned extend(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the input

signed_2_unsigned_max_truncate()

UINT_X signed_2_unsigned_max_truncate	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Upper bound on signed to unsigned interval truncation.

Author

Edoardo Manino.

Maximum value of unsigned truncate(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

signed_2_unsigned_min_extend()

UINT_X signed_2_unsigned_min_extend	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Lower bound on the unsigned extension of an n bit signed interval.

Author

Edoardo Manino.

Minimum value of unsigned extend(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the input

signed_2_unsigned_min_truncate()

UINT_X signed_2_unsigned_min_truncate	(	INT_X	a,
		INT_X	b,
		UINT_X	n )

Lower bound on signed to unsigned interval truncation.

Author

Edoardo Manino.

Minimum value of unsigned truncate(x, n) given n bits and signed x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

signed_max_and()

INT_X signed_max_and	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of x & y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_max_left_shift()

INT_X signed_max_left_shift	(	INT_X	,
		INT_X	b,
		UINT_X	c,
		UINT_X	d )

Upper bound on interval left shift.

Author

Edoardo Manino.

Maximum value of x << y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_max_not()

INT_X signed_max_not	(	INT_X	a,
		INT_X	)

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of ~x given x in [a,b], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x

signed_max_or()

INT_X signed_max_or	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of x | y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_max_right_shift()

INT_X signed_max_right_shift	(	INT_X	,
		INT_X	b,
		UINT_X	c,
		UINT_X	d )

Upper bound on interval right shift.

Author

Edoardo Manino.

Maximum value of x >> y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_max_xor()

INT_X signed_max_xor	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Original upper bound on signed interval XOR, inspired by the case-based algorithm for signed interval OR in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of x ^ y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_min_and()

INT_X signed_min_and	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of x & y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_min_left_shift()

INT_X signed_min_left_shift	(	INT_X	a,
		INT_X	,
		UINT_X	c,
		UINT_X	d )

Lower bound on interval left shift.

Author

Edoardo Manino.

Minimum value of x << y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_min_not()

INT_X signed_min_not	(	INT_X	,
		INT_X	b )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of ~x given x in [a,b], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x

signed_min_or()

INT_X signed_min_or	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of x | y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_min_right_shift()

INT_X signed_min_right_shift	(	INT_X	a,
		INT_X	,
		UINT_X	c,
		UINT_X	d )

Lower bound on interval right shift.

Author

Edoardo Manino.

Minimum value of x >> y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

signed_min_xor()

INT_X signed_min_xor	(	INT_X	a,
		INT_X	b,
		INT_X	c,
		INT_X	d )

Original lower bound on signed interval XOR, inspired by the case-based algorithm for signed interval OR in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of x ^ y given x in [a,b] and y in [c,d], potentially negative

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_2_signed()

INT_X unsigned_2_signed	(	UINT_X	x,
		UINT_X	n )

Cast a limited-width unsigned integer to signed integer.

Author

Edoardo Manino.

Compute cas(x, n) given an n-bit unsigned input x

Parameters

x	Value to be converted
n	Number of bits of the result

unsigned_2_signed_max_extend()

INT_X unsigned_2_signed_max_extend	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Upper bound on the signed extension of an n bit unsigned interval.

Author

Edoardo Manino.

Maximum value of signed extend(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the input

unsigned_2_signed_max_truncate()

INT_X unsigned_2_signed_max_truncate	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Upper bound on unsigned to signed interval truncation.

Author

Edoardo Manino.

Maximum value of signed truncate(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

unsigned_2_signed_min_extend()

INT_X unsigned_2_signed_min_extend	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Lower bound on the signed extension of an n bit unsigned interval.

Author

Edoardo Manino.

Minimum value of signed extend(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the input

unsigned_2_signed_min_truncate()

INT_X unsigned_2_signed_min_truncate	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Lower bound on unsigned to signed interval truncation.

Author

Edoardo Manino.

Minimum value of signed truncate(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

unsigned_2_unsigned_max_truncate()

UINT_X unsigned_2_unsigned_max_truncate	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Upper bound on unsigned to unsigned interval truncation.

Author

Edoardo Manino.

Maximum value of unsigned truncate(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

unsigned_2_unsigned_min_truncate()

UINT_X unsigned_2_unsigned_min_truncate	(	UINT_X	a,
		UINT_X	b,
		UINT_X	n )

Lower bound on unsigned to unsigned interval truncation.

Author

Edoardo Manino.

Minimum value of unsigned truncate(x, n) given n bits and unsigned x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x
n	Number of bits of the result

unsigned_max_and()

UINT_X unsigned_max_and	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of x & y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_max_left_shift()

UINT_X unsigned_max_left_shift	(	UINT_X	,
		UINT_X	b,
		UINT_X	,
		UINT_X	d )

Upper bound on interval left shift.

Author

Edoardo Manino.

Maximum value of x << y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_max_not()

UINT_X unsigned_max_not	(	UINT_X	a,
		UINT_X	)

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of ~x given x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x

unsigned_max_or()

UINT_X unsigned_max_or	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Adaptation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003 for unsigned integers of any width, by scanning from the least significant bit.

Author

Edoardo Manino.

Maximum value of x | y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_max_right_shift()

UINT_X unsigned_max_right_shift	(	UINT_X	,
		UINT_X	b,
		UINT_X	c,
		UINT_X	)

Upper bound on interval right shift.

Author

Edoardo Manino.

Maximum value of x >> y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_max_xor()

UINT_X unsigned_max_xor	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Maximum value of x ^ y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_min_and()

UINT_X unsigned_min_and	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of x & y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_min_left_shift()

UINT_X unsigned_min_left_shift	(	UINT_X	a,
		UINT_X	,
		UINT_X	c,
		UINT_X	)

Lower bound on interval left shift.

Author

Edoardo Manino.

Minimum value of x << y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_min_not()

UINT_X unsigned_min_not	(	UINT_X	,
		UINT_X	b )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of ~x given x in [a,b]

Parameters

a	Min value of variable x
b	Max value of variable x

unsigned_min_or()

UINT_X unsigned_min_or	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Adaptation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003 for unsigned integers of any width, by scanning from the least significant bit.

Author

Edoardo Manino.

Minimum value of x | y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_min_right_shift()

UINT_X unsigned_min_right_shift	(	UINT_X	a,
		UINT_X	,
		UINT_X	,
		UINT_X	d )

Lower bound on interval right shift.

Author

Edoardo Manino.

Minimum value of x >> y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

unsigned_min_xor()

UINT_X unsigned_min_xor	(	UINT_X	a,
		UINT_X	b,
		UINT_X	c,
		UINT_X	d )

Implementation of the algorithm in Henry S.

Warren Jr., "Hacker's Delight", 2003

Author

Edoardo Manino.

Minimum value of x ^ y given x in [a,b] and y in [c,d]

Parameters

a	Min value of variable x
b	Max value of variable x
c	Min value of variable y
d	Max value of variable y

Variable Documentation

signed_min_or

signed_min_or

signed_min_right_shift

signed_max_or signed_min_right_shift

unsigned_max_or

unsigned_max_or

unsigned_max_right_shift

signed_max_or unsigned_max_right_shift

unsigned_min_or

unsigned_min_or

unsigned_min_right_shift

signed_max_or unsigned_min_right_shift