Testing Error for `pow(x,1/2.4)` for sRGB Conversion

//https://math.stackexchange.com/a/3494710
//Compile with C++20

#include <bit>
#include <cmath>
#include <cstdint>
#include <cstdio>

float test(float x) {
    auto negtwelfth_root = [](float a) {
        float r = std::bit_cast<float>( 0x44c3cf20 - (std::bit_cast<uint32_t>(a)>>16)*0x1555 );
        float r3=r*r*r; float r6=r3*r3; float r12=r6*r6;
        float h = 0x1.ff078cp-1f - r12*a;
        r = (0x1.61bf5ap-4f + 0x1.905614p-5f*h)*h*r + r;
        return r;
    };
    return std::sqrt(x) * negtwelfth_root(x);
}
double gt(float x) {
    return std::pow( (double)x, 1.0/2.4 );
}

int main() {
    uint32_t start = std::bit_cast<uint32_t>( std::nextafter(0.0031308f,9999.0f) );
    uint32_t stop  = std::bit_cast<uint32_t>( 1.0f );

    double maxerr    = 0.0;
    double maxrelerr = 0.0;
    for (uint32_t a=start;a<=stop;++a) {
        float x = std::bit_cast<float>(a);

        float  res = test(x);
        double ref = gt(x);
        double err    = std::abs( res-ref );
        double relerr = std::abs( (res-ref) / ref );
        if (err   >maxerr   ) maxerr   =err   ;
        if (relerr>maxrelerr) maxrelerr=relerr;
    }
    printf("maxerr    = %11.4e\n",maxerr   );
    printf("maxrelerr = %11.4e\n",maxrelerr);
    getchar();

    return EXIT_SUCCESS;
}