经典重温，冲冲冲。
Integer
bitXor
/*
* bitXor - x^y using only ~ and &
* Example: bitXor(4, 5) = 1
* Legal ops: ~ &
* Max ops: 14
* Rating: 1
*/
int bitXor(int x, int y)
{
return ~(~(~x & y) & ~(~y & x));
}
解题思路：布尔代数。对于异或，从定义式出发，并使用「德摩根定律」变换：
tmin
/*
* tmin - return minimum two\'s complement integer
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 4
* Rating: 1
*/
int tmin(void)
{
return 1 << 31;
}
解题思路：Nothing special .
isTmax
/*
* isTmax - returns 1 if x is the maximum, two\'s complement number,
* and 0 otherwise
* Legal ops: ! ~ & ^ | +
* Max ops: 10
* Rating: 1
*/
int isTmax(int x)
{
return (!!(x + 1)) & (!(((x + 1) ^ x) + 1));
}
解题思路：isTmax(x) = 1 iff x == 0x7fffffff 。那么 x+1 = 0x80000000 ，显然 x ^ (x+1) == (int)(-1) 。ti需要考虑的特殊情况是 x = -1 ，因此加入 !!(x + 1) 的情况。
allOddBits
/*
* allOddBits - return 1 if all odd-numbered bits in word set to 1
* where bits are numbered from 0 (least significant) to 31 (most significant)
* Examples allOddBits(0xFFFFFFFD) = 0, allOddBits(0xAAAAAAAA) = 1
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 12
* Rating: 2
*/
int allOddBits(int x)
{
int mask = (0xAA << 8) | 0xAA;
mask = (mask << 16) | mask;
return !((x & mask) ^ mask);
}
解题思路：偶数位上的 1 对本题毫无用处，因此构造掩码 mask = 0xAAAAAAAA 将偶数位全部清零，也就是 y = x & mask 。题目要求奇数位上必须全为 1 ，就说明 y == 0xAAAAAAAA == mask ，因此 (y ^ mask) == 0x0 。
negate
/*
* negate - return -x
* Example: negate(1) = -1.
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 5
* Rating: 2
*/
int negate(int x)
{
return (~x) + 1;
}
解题思路：补码的性质，相反数是「按位取反再加一」。
isAsciiDigit
/*
* isAsciiDigit - return 1 if 0x30 <= x <= 0x39 (ASCII codes for characters \'0\' to \'9\')
* Example: isAsciiDigit(0x35) = 1.
* isAsciiDigit(0x3a) = 0.
* isAsciiDigit(0x05) = 0.
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 15
* Rating: 3
*/
int isAsciiDigit(int x)
{
return ((((x + (~0x30) + 1) >> 31) & 1) ^ 1) & ((((0x39 + (~x) + 1) >> 31) & 1) ^ 1);
}
解题思路：思路很明显，就是需要判断 0x30 <= x && x <= 0x30 。下面来看如何判断 a <= b 。
a <= b
-> b - a >= 0
-> b + (~a) + 1 >= 0
-> sign(b + ~a + 1) == 0
-> sign(b + ~a + 1) ^ 1 == 1
sign(x) = (x >> 31) & 1 表示 x 的符号位。
conditional
/*
* conditional - same as x ? y : z
* Example: conditional(2,4,5) = 4
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 16
* Rating: 3
*/
int conditional(int x, int y, int z)
{
x = ~(!!x) + 1;
return (y & x) + (z & (~x));
}
解题思路：原本的想法是构造 y*x + z*(!x) 这种形式作为返回值，其中 x 为 1 或 0 。但是不能用乘法，所以重新思考了一下，构造 (y & x) + (z & ~x) ，其中 x 为 0x0 或者 0xffffffff 。
- 当
x == 0时，需要构造x = 0x0 = -1 + 1 = ~0 + 1。 - 当
x != 0时，需要构造x = 0xffffffff = -1 = negate(1) = negate(!!x) = ~(!!x) + 1。
关键点是要知道 !!x 可以将 int 映射为逻辑真值。
isLessEqual
/*
* isLessOrEqual - if x <= y then return 1, else return 0
* Example: isLessOrEqual(4,5) = 1.
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 24
* Rating: 3
*/
int isLessOrEqual(int x, int y)
{
int signx = (x >> 31) & 1;
int signy = (y >> 31) & 1;
int k1 = (signx ^ signy ^ 1) & (((x + (~y)) >> 31) & 1);
int k2 = (signx & (signy ^ 1));
return k1 | k2;
}
解题思路：看「最大数值」这一小节 。
logicalNeg
/*
* logicalNeg - implement the ! operator, using all of
* the legal operators except !
* Examples: logicalNeg(3) = 0, logicalNeg(0) = 1
* Legal ops: ~ & ^ | + << >>
* Max ops: 12
* Rating: 4
*/
int logicalNeg(int x)
{
int neg = (x >> 31) & 1;
int plus = ((~x + 1) >> 31) & 1;
return (neg ^ 1) & (plus ^ 1);
}
解题思路：根据「0 既不是正数，也不是负数」这一个规律。所以如果 x 为 0 ，那么 x 和 -x 的符号位都必然是 0 。
howManyBits
/* howManyBits - return the minimum number of bits required to represent x in
* two\'s complement
* Examples: howManyBits(12) = 5
* howManyBits(298) = 10
* howManyBits(-5) = 4
* howManyBits(0) = 1
* howManyBits(-1) = 1
* howManyBits(0x80000000) = 32
* Legal ops: ! ~ & ^ | + << >>
* Max ops: 90
* Rating: 4
*/
int howManyBits(int x)
{
int b16, b8, b4, b2, b1;
x = (x >> 31) ^ x;
b16 = (!!(x >> 16)) << 4, x >>= b16;
b8 = (!!(x >> 8)) << 3, x >>= b8;
b4 = (!!(x >> 4)) << 2, x >>= b4;
b2 = (!!(x >> 2)) << 1, x >>= b2;
b1 = (!!(x >> 1)), x >>= b1;
return b16 + b8 + b4 + b2 + b1 + x + 1;
}
解题思路：有点类似二分查找，本题自己没做出来，是看了几个题解才写出来的。
Float
floatScale2
/*
* floatScale2 - Return bit-level equivalent of expression 2*f for
* floating point argument f.
* Both the argument and result are passed as unsigned int\'s, but
* they are to be interpreted as the bit-level representation of
* single-precision floating point values.
* When argument is NaN, return argument
* Legal ops: Any integer/unsigned operations incl. ||, &&. also if, while
* Max ops: 30
* Rating: 4
*/
unsigned floatScale2(unsigned uf)
{
unsigned sign = (uf & 0x80000000);
unsigned exp = (uf & 0x7f800000);
unsigned frac = (uf & 0x7fffff);
//clear sign bit
uf = uf & (0x7fffffff);
//special cases: INF, NaN, zero
if (exp == 0x7f800000 || uf == 0)
return uf | sign;
if (exp == 0) // uf is a denormalize number
{
if ((frac >> 22) == 1) // 2*uf change into a normalized number
{
exp = (1 << 23);
frac = (frac << 1) & 0x7fffff;
return sign | exp | frac;
}
else // 2*uf is still a denormalized number
return sign | (frac << 1);
}
else // uf is a normalized number, so 2*uf must be a normalized number, too
{
exp += (1 << 23);
return sign | exp | frac;
}
}
解题思路：需要熟悉浮点数编码「IEEE 754」，然后是分类讨论。
-
uf是非规格化数2*uf仍然是非规格化数2*uf是规格化数
-
uf是规格化数那么
2*uf必然是规格化数。 -
其他特殊情况
uf是无穷大，NaN，0 。
floatFloat2Int
/*
* floatFloat2Int - Return bit-level equivalent of expression (int) f
* for floating point argument f.
* Argument is passed as unsigned int, but
* it is to be interpreted as the bit-level representation of a
* single-precision floating point value.
* Anything out of range (including NaN and infinity) should return
* 0x80000000u.
* Legal ops: Any integer/unsigned operations incl. ||, &&. also if, while
* Max ops: 30
* Rating: 4
*/
int floatFloat2Int(unsigned uf)
{
unsigned sign = (uf & 0x80000000);
unsigned exp = (uf >> 23) & 0xff;
unsigned real = (uf & 0x7fffff) | (1 << 23);
unsigned t;
uf = uf & 0x7fffffff;
if (exp < 127) // uf < 1.0
return 0;
if (exp >= 150) // we can keep all 23 frac
{
t = exp - 150;
if (t < 8) // round(uf) is in int range, and we can put no more than 7 zeros after real
{
real = real << t;
return (sign == 0) ? real : (-real);
}
return 0x80000000; // out of range, we can\'t put more than 7 zeros after real
}
real = real >> (150 - exp); // the last serval bits of frac should be thrown
return (sign == 0) ? real : (-real);
}
本题其实仍然是分类讨论，但是分类的依据与某些特殊情况，需要不断测试才能写出代码，所以一定不能畏难，不会就马上找题解，这样收获会很小。上面的代码也是修改了好多次的，我也忘了最初的版本长什么样子了。解法也没什么特别的，就是不断地试，观察不通过的 Test Case ，然后修改。
首先有一个 trick 是把隐含的 1 还原的：
unsigned real = (uf & 0x7fffff) | (1 << 23);
/*
* Here is a sample:
* 155.55 = 0 10000110 00110111000110011001101
* frac = 1.00110111000110011001101
* exp = 10000110 = 134, E = 7
* so the float val is:
* fval = 1.00110111000110011001101 × 2^7
* = 10011011.1000110011001101
* when fval cast into int, the \'1000110011001101\' will be thrown:
* ival = 0...0 10011011 = 155
*/
此时，我们默认小数点的位置在第 23 bit 与 第 22 bit 之间。
后面我们要解决的就是小数点移动到后面的哪一位的问题，这就要根据 E = exp - 127 的具体数值来决定：
- 如果
E >= 23，那么real后面的所有小数位都得以保留，并且需要添加E - 23个 0 。 - 如果
E <= 22，那么说明real的小数点最远就只能移动到第 1 bit 与第 0 bit 之间，小数点后的位都需要「丢弃」。
「丢弃」和「补 0」都通过移位来完成。
下面解析代码中的几个 if 条件。
-
exp < 127对于非规格化数，一定是小于 1.0 的，因为其值为 \\(denormalied = 0.frac \\times 2^{-126}\\) 。对于规格化数，\\(normalized = 1.frac \\times 2^{exp - 127}\\) ，要使
uf < 1.0，就要令 \\(exp - 127 \\le -1\\) ，所以可得uf < 1.0的等价条件为exp <= 126。 -
exp >= 150此情况是可保留 23 位
frac。令E = exp - 127 >= 23，即可得：exp >= 150。超出 23 的部分就要补上exp - 150个 0 ，但是最多只能补 7 个 0 ，因为real本身就有 24 位，int的符号位占 1 位。/* * For an example: * 999999999.0 = 0 10011100 11011100110101100101000 * frac = 11011100110101100101000 * real = 1.11011100110101100101000 * exp = 10011100 = 0x9c = 156, E = 29 * so the float val is: * fval = 1.11011100110101100101000 × 2^(29) * = 111011100110101100101000 × 2^(6) * = 111011100110101100101000 000000 * when cast into int val: * ival = 111011100110101100101000 000000 * ival has 24 + 6 = 30 bits, put it into 32 bits is: * ival = (00 111011100110101100101000 000000)2 = 1000000000 * the number \'1000000000\' is decimal */ -
127 <= exp <= 149这里是就是单纯的「舍弃」小数点后面的数值。看上面提到的例子：
/* * Here is a sample: * 155.55 = 0 10000110 00110111000110011001101 * real = 1.00110111000110011001101 * exp = 10000110 = 134, E = 7 * so the float val is: * fval = 1.00110111000110011001101 × 2^7 * = 10011011.1000110011001101 * when fval cast into int, the \'1000110011001101\' will be thrown: * ival = 0...0 10011011 = 155 */
floatPower2
/*
* floatPower2 - Return bit-level equivalent of the expression 2.0^x
* (2.0 raised to the power x) for any 32-bit integer x.
*
* The unsigned value that is returned should have the identical bit
* representation as the single-precision floating-point number 2.0^x.
* If the result is too small to be represented as a denorm, return
* 0. If too large, return +INF.
*
* Legal ops: Any integer/unsigned operations incl. ||, &&. Also if, while
* Max ops: 30
* Rating: 4
*/
unsigned floatPower2(int x)
{
int exp = x + 127;
if (1 <= exp && exp <= 254) // 2^x can be a normalized number
return exp << 23;
else if (exp >= 255) // 2^x out of range
return 0x7f800000;
else // x <= -127 (exp <= 0), 2^x may be a denormalized number, maybe zero
{
if (x >= -149)
return 1 << (149 + x);
else
return 0;
}
}
本题要求实现 \\(2^x\\) 。显然，这就是浮点数的一种十分特殊的情况：\\(1.0...0 \\times 2^{exp - 127}\\) 。但是 \\(exp \\in [1,254]\\) ，这需要对 exp = E + 127 = x + 127 进行讨论。注意 exp 必须要使用有符号数进行存储，因为 x + 127 仍可能为负数 。
-
1 <= exp && exp <= 254说明 \\(2^x\\) 在规格化数的范围内，即 \\(val = 1.0\\cdot\\cdot\\cdot0 \\times 2^x\\) ，所以
frac = 0...0, exp = x + 127。 -
exp >= 255说明 \\(2^x\\) 超过了
float的有效范围。 -
exp <= 0这时候 \\(2^x\\) 就必须采用非规格化数来存储。非规格化数 \\(denormalized = 0.frac \\times 2^{-126}\\) ，所以：
-
\\(2^x\\) 的最大值为 \\(max = 0.10\\cdot\\cdot\\cdot0 \\times 2^{-126} = 2^{-127}, x=-127\\)
-
\\(2^x\\) 的最小值为 \\(min = 0.0\\cdot\\cdot\\cdot01 \\times 2^{-126} = 2^{-149}, x=-149\\)
\\(2^x\\) 这种形式决定了
frac中必定只有一个 1 ，而这个 1 的后面 0 的数目取决于 \\(x\\) 的大小，实际上不难发现是 \\(149 + x\\) 。 -
Evaluation
unix > ./driver.pl
Correctness Results Perf Results
Points Rating Errors Points Ops Puzzle
1 1 0 2 8 bitXor
1 1 0 2 1 tmin
1 1 0 2 8 isTmax
2 2 0 2 7 allOddBits
2 2 0 2 2 negate
3 3 0 2 13 isAsciiDigit
3 3 0 2 8 conditional
3 3 0 2 14 isLessOrEqual
4 4 0 2 9 logicalNeg
4 4 0 2 32 howManyBits
4 4 0 2 22 floatScale2
4 4 0 2 18 floatFloat2Int
4 4 0 2 10 floatPower2
Score = 62/62 [36/36 Corr + 26/26 Perf] (152 total operators)
Summary
Enjoy the fun of writing !