the blog: *about binary search*

the code:

template <std::size_t N_BITS>
using int_least_t = std::conditional_t<
    N_BITS <= 8, std::uint8_t,
    std::conditional_t<
        N_BITS <= 16, std::uint16_t,
        std::conditional_t<
            N_BITS <= 32, std::uint32_t,
            std::conditional_t<
                N_BITS <= 64, std::uint64_t,
                std::conditional_t<N_BITS <= 128, __uint128_t, void>>>>>;

"I am pretty sure that this method has been explored before" Yes.

Great Blog!

Since most problems allow some relative error in the answer, it is usually not necessary to do the full 64 iterations. For example, if the answer is positive and we can tolerate a $$$10^{-6}$$$ relative error, then we can do binary search on doubles like this:

while (get_int(a) + (1ULL << 32) < get_int(b)) {
    auto m = std::midpoint(get_int(a), get_int(b));
    if (predicate(get_float(m))) {
        a = get_float(m);
    } else {
        b = get_float(m);
    }
}

Outside of annoying edge cases (0, NaN, inf, subnormal, etc.), the condition get_int(a) + (1ULL << 32) >= get_int(b) implies the relative error of $$$a$$$ and $$$b$$$ is at most $$$2^{-20}$$$ < $$$10^{-6}$$$.

 sign bit
 | exponent (11 bits)
 | |           mantissa (52 bits)
 v v           v
+-+-----------+----------------------------------------------------+
|0|10000000000|1000000000000000000000000000000000000000000000000000|
+-+-----------+----------------------------------------------------+

The same could work for any $$$b < 52$$$ for $$$2^{-(52 - b)}$$$ precision. How to modify nor's template to support this is an exercise for the reader.