/********************************************************
 * Calculating 1/sqrt(x)
 *
 ********************************************************/

// Chose the constant which minimizes the desired error
//
//                                                Error
// Optimal Norm                         1-Norm    2-Norm    OO-Norm
// 1-Norm          1597203179           0.01594   0.02224   0.05055
// 2-Norm          1597294787           0.01715   0.02093   0.04482
// Infinity-Norm   1597465647           0.02339   0.02528   0.03421

// Minimize 1-norm
// int const INV_SQRT_N = 1597203179;
// Minimize 2-norm
// int const INV_SQRT_N = 1597294787;
// Minimize oo-norm
int const INV_SQRT_N = 1597465647;

float inv_sqrt( float x ) {
	int xi = *reinterpret_cast<int *>( &x );
	xi = INV_SQRT_N - (xi >> 1);
	return *reinterpret_cast<float *>( &xi );
}

// Chose the constant which minimizes the desired error
//
//                                                Error
// Optimal Norm                         1-Norm    2-Norm    OO-Norm
// 1-Norm          1597292357           0.0006520 0.001078  0.002988
// 2-Norm          1597376322           0.0007246 0.0009483 0.002338
// Infinity-Norm   1597463175           0.0009549 0.001118  0.001751

// Minimize 1-norm
// int const INV_SQRT_NEWTON_N = 1597292357;
// Minimize 2-norm
// int const INV_SQRT_NEWTON_N = 1597376322;
// Minimize oo-norm
int const INV_SQRT_NEWTON_N = 1597463175;

float inv_sqrt_newton( float x ) {
	float const half_x0 = 0.5f*x;
	int xi = *reinterpret_cast<int *>( &x );
	xi = INV_SQRT_NEWTON_N - (xi >> 1);
	x = *reinterpret_cast<float *>( &xi );
	return x*(1.5f - half_x0*x*x);
}

// Chose the constant which minimizes the desired error
//
//                                           Error
// 1-Norm          1.000363245811462       0.0005151
// 2-Norm          1.000724768371582       0.0006122
// Infinity-Norm   1.000876311302185f      0.0008765

// Minimize 1-norm
// float const MULTIPLIER = 1.000363245811462f;
// Minimize 2-norm
// float const MULTIPLIER = 1.000724768371582f;
// Minimize oo-norm
float const MULTIPLIER = 1.000876311302185f;

float inv_sqrt_multiplier( float x ) {
	float const mx = 0.5f*MULTIPLIER*x;
	int xi = *reinterpret_cast<int *>( &x );
	xi = INV_SQRT_NEWTON_N - (xi >> 1);
	x = *reinterpret_cast<float *>( &xi );
	return x*(1.5f*MULTIPLIER - mx*x*x);
}