/* * integer.c * * functions for integer cells. * * (c) 2017 Simon Brooke * Licensed under GPL version 2.0, or, at your option, any later version. */ #define _GNU_SOURCE #include #include #include #include /* * wide characters */ #include #include #include "memory/conspage.h" #include "memory/consspaceobject.h" #include "debug.h" #include "ops/equal.h" #include "ops/lispops.h" #include "arith/peano.h" /** * hexadecimal digits for printing numbers. */ const char *hex_digits = "0123456789ABCDEF"; /* * Doctrine from here on in is that ALL integers are bignums, it's just * that integers less than 65 bits are bignums of one cell only. */ /** * Allocate an integer cell representing this `value` and return a cons_pointer to it. * @param value an integer value; * @param more `NIL`, or a pointer to the more significant cell(s) of this number. * *NOTE* that if `more` is not `NIL`, `value` *must not* exceed `MAX_INTEGER`. */ struct cons_pointer make_integer( int64_t value, struct cons_pointer more ) { struct cons_pointer result = NIL; debug_print( L"Entering make_integer\n", DEBUG_ALLOC ); if ( integerp( more ) || nilp( more ) ) { result = allocate_cell( INTEGERTV ); struct cons_space_object *cell = &pointer2cell( result ); cell->payload.integer.value = value; cell->payload.integer.more = more; } debug_print( L"make_integer: returning\n", DEBUG_ALLOC ); debug_dump_object( result, DEBUG_ALLOC ); return result; } /** * Low level integer arithmetic, do not use elsewhere. * * @param c a pointer to a cell, assumed to be an integer cell; * @param op a character representing the operation: expectedto be either * '+' or '*'; behaviour with other values is undefined. * @param is_first_cell true if this is the first cell in a bignum * chain, else false. * \see multiply_integers * \see add_integers */ __int128_t cell_value( struct cons_pointer c, char op, bool is_first_cell ) { long int val = nilp( c ) ? 0 : pointer2cell( c ).payload.integer.value; long int carry = is_first_cell ? 0 : ( MAX_INTEGER + 1 ); __int128_t result = ( __int128_t ) integerp( c ) ? ( val == 0 ) ? carry : val : op == '*' ? 1 : 0; debug_printf( DEBUG_ARITH, L"cell_value: raw value is %ld, is_first_cell = %s; '%4.4s'; returning ", val, is_first_cell ? "true" : "false", pointer2cell( c ).tag.bytes ); debug_print_128bit( result, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); return result; } /** * Overwrite the value field of the integer indicated by `new` with * the least significant 60 bits of `val`, and return the more significant * bits (if any) right-shifted by 60 places. Destructive, primitive, do not * use in any context except primitive operations on integers. * * @param val the value to represent; * @param less_significant the less significant words of this bignum, if any, * else NIL; * @param new a newly created integer, which will be destructively changed. * @return carry, if any, else 0. */ __int128_t int128_to_integer( __int128_t val, struct cons_pointer less_significant, struct cons_pointer new ) { __int128_t carry = 0; if ( MAX_INTEGER >= val ) { carry = 0; } else { carry = val >> INTEGER_BIT_SHIFT; debug_printf( DEBUG_ARITH, L"int128_to_integer: 64 bit overflow; setting carry to %ld\n", ( int64_t ) carry ); val &= MAX_INTEGER; } struct cons_space_object *newc = &pointer2cell( new ); newc->payload.integer.value = val; if ( integerp( less_significant ) ) { struct cons_space_object *lsc = &pointer2cell( less_significant ); inc_ref( new ); lsc->payload.integer.more = new; } return carry; } struct cons_pointer make_integer_128( __int128_t val, struct cons_pointer less_significant ) { struct cons_pointer result = NIL; do { if ( MAX_INTEGER >= val ) { result = make_integer( ( long int ) val, less_significant ); } else { less_significant = make_integer( ( long int ) val & MAX_INTEGER, less_significant ); val = val >> INTEGER_BIT_SHIFT; } } while ( nilp( result ) ); return result; } /** * Return a pointer to an integer representing the sum of the integers * pointed to by `a` and `b`. If either isn't an integer, will return nil. */ struct cons_pointer add_integers( struct cons_pointer a, struct cons_pointer b ) { struct cons_pointer result = NIL; struct cons_pointer cursor = NIL; debug_print( L"add_integers: a = ", DEBUG_ARITH ); debug_print_object( a, DEBUG_ARITH ); debug_print( L"; b = ", DEBUG_ARITH ); debug_print_object( b, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); __int128_t carry = 0; bool is_first_cell = true; if ( integerp( a ) && integerp( b ) ) { debug_print( L"add_integers: \n", DEBUG_ARITH ); debug_dump_object( a, DEBUG_ARITH ); debug_print( L" plus \n", DEBUG_ARITH ); debug_dump_object( b, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); while ( !nilp( a ) || !nilp( b ) || carry != 0 ) { __int128_t av = cell_value( a, '+', is_first_cell ); __int128_t bv = cell_value( b, '+', is_first_cell ); __int128_t rv = av + bv + carry; debug_print( L"add_integers: av = ", DEBUG_ARITH ); debug_print_128bit( av, DEBUG_ARITH ); debug_print( L"; bv = ", DEBUG_ARITH ); debug_print_128bit( bv, DEBUG_ARITH ); debug_print( L"; carry = ", DEBUG_ARITH ); debug_print_128bit( carry, DEBUG_ARITH ); debug_print( L"; rv = ", DEBUG_ARITH ); debug_print_128bit( rv, DEBUG_ARITH ); debug_print( L"\n", DEBUG_ARITH ); struct cons_pointer new = make_integer( 0, NIL ); carry = int128_to_integer( rv, cursor, new ); cursor = new; if ( nilp( result ) ) { result = cursor; } a = pointer2cell( a ).payload.integer.more; b = pointer2cell( b ).payload.integer.more; is_first_cell = false; } } debug_print( L"add_integers returning: ", DEBUG_ARITH ); debug_print_object( result, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); return result; } struct cons_pointer base_partial( int depth ) { struct cons_pointer result = NIL; for ( int i = 0; i < depth; i++ ) { result = make_integer( 0, result ); } return result; } /** * destructively modify this `partial` by appending this `digit`. */ struct cons_pointer append_digit( struct cons_pointer partial, struct cons_pointer digit ) { struct cons_pointer c = partial; struct cons_pointer result = partial; if ( nilp( partial ) ) { result = digit; } else { while ( !nilp( pointer2cell( c ).payload.integer.more ) ) { c = pointer2cell( c ).payload.integer.more; } ( &pointer2cell( c ) )->payload.integer.more = digit; } return result; } /** * Return a pointer to an integer representing the product of the integers * pointed to by `a` and `b`. If either isn't an integer, will return nil. * * Yes, this is one of Muhammad ibn Musa al-Khwarizmi's original recipes, so * you'd think it would be easy; the reason that each step is documented is * because I did not find it so. * * @param a an integer; * @param b an integer. */ struct cons_pointer multiply_integers( struct cons_pointer a, struct cons_pointer b ) { struct cons_pointer result = make_integer( 0, NIL ); bool neg = is_negative( a ) != is_negative( b ); bool is_first_b = true; int i = 0; debug_print( L"multiply_integers: a = ", DEBUG_ARITH ); debug_print_object( a, DEBUG_ARITH ); debug_print( L"; b = ", DEBUG_ARITH ); debug_print_object( b, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); if ( integerp( a ) && integerp( b ) ) { /* for each digit in a, starting with the least significant (ai) */ for ( struct cons_pointer ai = a; !nilp( ai ); ai = pointer2cell( ai ).payload.integer.more ) { /* set carry to 0 */ __int128_t carry = 0; /* set least significant digits for result ri for this iteration * to i zeros */ struct cons_pointer ri = base_partial( i++ ); /* for each digit in b, starting with the least significant (bj) */ for ( struct cons_pointer bj = b; !nilp( bj ); bj = pointer2cell( bj ).payload.integer.more ) { debug_printf( DEBUG_ARITH, L"multiply_integers: a[i] = %Ld, b[j] = %Ld, i = %d\n", pointer2cell( ai ).payload.integer.value, pointer2cell( bj ).payload.integer.value, i ); /* multiply ai with bj and add the carry, resulting in a * value xj which may exceed one digit */ __int128_t xj = pointer2cell( ai ).payload.integer.value * pointer2cell( bj ).payload.integer.value; xj += carry; /* if xj exceeds one digit, break it into the digit dj and * the carry */ carry = xj >> INTEGER_BIT_SHIFT; struct cons_pointer dj = make_integer( xj & MAX_INTEGER, NIL ); /* destructively modify ri by appending dj */ ri = append_digit( ri, dj ); } /* end for bj */ /* if carry is not equal to zero, append it as a final digit * to ri */ if ( carry != 0 ) { ri = append_digit( ri, make_integer( carry, NIL ) ); } /* add ri to result */ result = add_integers( result, ri ); debug_print( L"multiply_integers: result is ", DEBUG_ARITH ); debug_print_object( result, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); } /* end for ai */ } debug_print( L"multiply_integers returning: ", DEBUG_ARITH ); debug_print_object( result, DEBUG_ARITH ); debug_println( DEBUG_ARITH ); return result; } /** * don't use; private to integer_to_string, and somewhat dodgy. */ struct cons_pointer integer_to_string_add_digit( int digit, int digits, struct cons_pointer tail ) { wint_t character = btowc( hex_digits[digit] ); return ( digits % 3 == 0 ) ? make_string( L',', make_string( character, tail ) ) : make_string( character, tail ); } /** * The general principle of printing a bignum is that you print the least * significant digit in whatever base you're dealing with, divide through * by the base, print the next, and carry on until you've none left. * Obviously, that means you print from right to left. Given that we build * strings from right to left, 'printing' an integer to a lisp string * would seem reasonably easy. The problem is when you jump from one integer * object to the next. 64 bit integers don't align with decimal numbers, so * when we get to the last digit from one integer cell, we have potentially * to be looking to the next. H'mmmm. */ struct cons_pointer integer_to_string( struct cons_pointer int_pointer, int base ) { struct cons_pointer result = NIL; if ( integerp( int_pointer ) ) { struct cons_pointer next = pointer2cell( int_pointer ).payload.integer.more; __int128_t accumulator = llabs( pointer2cell( int_pointer ).payload.integer.value ); bool is_negative = pointer2cell( int_pointer ).payload.integer.value < 0; int digits = 0; if ( accumulator == 0 && nilp( next ) ) { result = c_string_to_lisp_string( L"0" ); } else { while ( accumulator > 0 || !nilp( next ) ) { if ( accumulator < MAX_INTEGER && !nilp( next ) ) { accumulator += ( pointer2cell( next ).payload.integer.value << INTEGER_BIT_SHIFT ); next = pointer2cell( next ).payload.integer.more; } int offset = ( int ) ( accumulator % base ); debug_printf( DEBUG_IO, L"integer_to_string: digit is %ld, hexadecimal is %c, accumulator is: ", offset, hex_digits[offset] ); debug_print_128bit( accumulator, DEBUG_IO ); debug_print( L"; result is: ", DEBUG_IO ); debug_print_object( result, DEBUG_IO ); debug_println( DEBUG_IO ); result = integer_to_string_add_digit( offset, ++digits, result ); accumulator = accumulator / base; } if ( stringp( result ) && pointer2cell( result ).payload.string.character == L',' ) { /* if the number of digits in the string is divisible by 3, there will be * an unwanted comma on the front. */ result = pointer2cell( result ).payload.string.cdr; } if ( is_negative ) { result = make_string( L'-', result ); } } } return result; } /** * true if a and be are both integers whose value is the same value. */ bool equal_integer_integer( struct cons_pointer a, struct cons_pointer b ) { bool result = false; if ( integerp( a ) && integerp( b ) ) { struct cons_space_object *cell_a = &pointer2cell( a ); struct cons_space_object *cell_b = &pointer2cell( b ); result = cell_a->payload.integer.value == cell_b->payload.integer.value; } return result; } /** * true if `a` is an integer, and `b` is a real number whose value is the * value of that integer. */ bool equal_integer_real( struct cons_pointer a, struct cons_pointer b ) { bool result = false; if ( integerp( a ) && realp( b ) ) { long double bv = pointer2cell( b ).payload.real.value; if ( floor( bv ) == bv ) { result = pointer2cell( a ).payload.integer.value == ( int64_t ) bv; } } return result; }