java - 长数组的精确和

Question

为了获得 long[] 的精确总和，我使用了以下代码段。

public static BigInteger sum(long[] a) {
    long low = 0;
    long high = 0;
    for (final long x : a) {
        low += (x & 0xFFFF_FFFFL);
        high += (x >> 32);
    }
    return BigInteger.valueOf(high).shiftLeft(32).add(BigInteger.valueOf(low));
}

它可以很好地处理分成两半的数字，最后合并部分和。令人惊讶的是，这种方法也有效:

public static BigInteger fastestSum(long[] a) {
    long low = 0;
    long high = 0;
    for (final long x : a) {
        low += x;
        high += (x >> 32);
    }
    // We know that low has the lowest 64 bits of the exact sum.
    // We also know that BigInteger.valueOf(high).shiftLeft(32) differs from the exact sum by less than 2**63.
    // So the upper half of high is off by at most one.
    high >>= 32;
    if (low < 0) ++high; // Surprisingly, this is enough to fix it.
    return BigInteger.valueOf(high).shiftLeft(64).add(BigInteger.valueOf(low));
}

我不相信fastestSum应该按原样工作。我相信它可以工作，但在最后一步必须做更多的事情。但是，它通过了我所有的测试(包括大型随机测试)。所以我要问:有人可以证明它有效或找到反例吗？

Answer 1

fastestSum(new long[]{+1, -1})  => -18446744073709551616