There is not a simple formula for computing the correlated color temperature of an XYZ color. Instead, an algorithm is used to inverse interpolate one data table, and then forward interpolate a second table. This method was developed by A. R. Robertson. Below is an ANSI 'C' implementation of Robertson's method.
/*
* Name: XYZtoCorColorTemp.c
*
* Author: Bruce Justin Lindbloom
*
* Copyright (c) 2003 Bruce Justin Lindbloom. All rights reserved.
*
* Input: xyz = pointer to the input array of X, Y and Z color components (in that order).
* temp = pointer to where the computed correlated color temperature should be placed.
*
* Output: *temp = correlated color temperature, if successful.
* = unchanged if unsuccessful.
*
* Return: 0 if successful, else -1.
*
* Description:
* This is an implementation of Robertson's method of computing the correlated color
* temperature of an XYZ color. It can compute correlated color temperatures in the
* range [1666.7K, infinity].
*
* Reference:
* "Color Science: Concepts and Methods, Quantitative Data and Formulae", Second Edition,
* Gunter Wyszecki and W. S. Stiles, John Wiley & Sons, 1982, pp. 227, 228.
*/
#include <float.h>
#include <math.h>
/* LERP(a,b,c) = linear interpolation macro, is 'a' when c == 0.0 and 'b' when c == 1.0 */
#define LERP(a,b,c) (((b) - (a)) * (c) + (a))
typedef struct UVT {
double u;
double v;
double t;
} UVT;
double rt[31] = { /* reciprocal temperature (K) */
DBL_MIN, 10.0e-6, 20.0e-6, 30.0e-6, 40.0e-6, 50.0e-6,
60.0e-6, 70.0e-6, 80.0e-6, 90.0e-6, 100.0e-6, 125.0e-6,
150.0e-6, 175.0e-6, 200.0e-6, 225.0e-6, 250.0e-6, 275.0e-6,
300.0e-6, 325.0e-6, 350.0e-6, 375.0e-6, 400.0e-6, 425.0e-6,
450.0e-6, 475.0e-6, 500.0e-6, 525.0e-6, 550.0e-6, 575.0e-6,
600.0e-6
};
UVT uvt[31] = {
{0.18006, 0.26352, -0.24341},
{0.18066, 0.26589, -0.25479},
{0.18133, 0.26846, -0.26876},
{0.18208, 0.27119, -0.28539},
{0.18293, 0.27407, -0.30470},
{0.18388, 0.27709, -0.32675},
{0.18494, 0.28021, -0.35156},
{0.18611, 0.28342, -0.37915},
{0.18740, 0.28668, -0.40955},
{0.18880, 0.28997, -0.44278},
{0.19032, 0.29326, -0.47888},
{0.19462, 0.30141, -0.58204},
{0.19962, 0.30921, -0.70471},
{0.20525, 0.31647, -0.84901},
{0.21142, 0.32312, -1.0182},
{0.21807, 0.32909, -1.2168},
{0.22511, 0.33439, -1.4512},
{0.23247, 0.33904, -1.7298},
{0.24010, 0.34308, -2.0637},
{0.24792, 0.34655, -2.4681}, /* Note: 0.24792 is a corrected value for the error found in W&S as 0.24702 */
{0.25591, 0.34951, -2.9641},
{0.26400, 0.35200, -3.5814},
{0.27218, 0.35407, -4.3633},
{0.28039, 0.35577, -5.3762},
{0.28863, 0.35714, -6.7262},
{0.29685, 0.35823, -8.5955},
{0.30505, 0.35907, -11.324},
{0.31320, 0.35968, -15.628},
{0.32129, 0.36011, -23.325},
{0.32931, 0.36038, -40.770},
{0.33724, 0.36051, -116.45}
};
int XYZtoCorColorTemp(double *xyz, double *temp)
{
double us, vs, p, di, dm;
int i;
if ((xyz[0] < 1.0e-20) && (xyz[1] < 1.0e-20) && (xyz[2] < 1.0e-20))
return(-1); /* protect against possible divide-by-zero failure */
us = (4.0 * xyz[0]) / (xyz[0] + 15.0 * xyz[1] + 3.0 * xyz[2]);
vs = (6.0 * xyz[1]) / (xyz[0] + 15.0 * xyz[1] + 3.0 * xyz[2]);
dm = 0.0;
for (i = 0; i < 31; i++) {
di = (vs - uvt[i].v) - uvt[i].t * (us - uvt[i].u);
if ((i > 0) && (((di < 0.0) && (dm >= 0.0)) || ((di >= 0.0) && (dm < 0.0))))
break; /* found lines bounding (us, vs) : i-1 and i */
dm = di;
}
if (i == 31)
return(-1); /* bad XYZ input, color temp would be less than minimum of 1666.7 degrees, or too far towards blue */
di = di / sqrt(1.0 + uvt[i ].t * uvt[i ].t);
dm = dm / sqrt(1.0 + uvt[i - 1].t * uvt[i - 1].t);
p = dm / (dm - di); /* p = interpolation parameter, 0.0 : i-1, 1.0 : i */
p = 1.0 / (LERP(rt[i - 1], rt[i], p));
*temp = p;
return(0); /* success */
}