/*
 ==============================================================================

 ok_color.h
 Created: 5 Feb 2022 11:53:07am
 Author:  Marcus W. Hobbs

 ==============================================================================
 */

#pragma once
// Copyright(c) 2021 Björn Ottosson
//
// Permission is hereby granted, free of charge, to any person obtaining a copy of
// this softwareand associated documentation files(the "Software"), to deal in
// the Software without restriction, including without limitation the rights to
// use, copy, modify, merge, publish, distribute, sublicense, and /or sell copies
// of the Software, and to permit persons to whom the Software is furnished to do
// so, subject to the following conditions :
// The above copyright noticeand this permission notice shall be included in all
// copies or substantial portions of the Software.
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
// SOFTWARE.

#include <cmath>
#include <cfloat>

namespace ok_color
{
    struct Lab { float L; float a; float b; };
    struct RGB { float r; float g; float b; };
    struct HSV { float h; float s; float v; };
    struct HSL { float h; float s; float l; };
    struct LC { float L; float C; };

    // Alternative representation of (L_cusp, C_cusp)
    // Encoded so S = C_cusp/L_cusp and T = C_cusp/(1-L_cusp)
    // The maximum value for C in the triangle is then found as fmin(S*L, T*(1-L)), for a given L
    struct ST { float S; float T; };

    constexpr float pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062f;

#if 0
    static float clamp (float x, float min, float max)
    {
        if (x < min)
            return min;
        if (x > max)
            return max;

        return x;
    }

    static float sgn(float x)
    {
        return (float)(0.f < x) - (float)(x < 0.f);
    }
#endif

    static float srgb_transfer_function(float a)
    {
        return .0031308f >= a ? 12.92f * a : 1.055f * powf(a, .4166666666666667f) - .055f;
    }

    static RGB oklab_to_linear_srgb(Lab c)
    {
        float l_ = c.L + 0.3963377774f * c.a + 0.2158037573f * c.b;
        float m_ = c.L - 0.1055613458f * c.a - 0.0638541728f * c.b;
        float s_ = c.L - 0.0894841775f * c.a - 1.2914855480f * c.b;

        float l = l_ * l_ * l_;
        float m = m_ * m_ * m_;
        float s = s_ * s_ * s_;

        return {
            +4.0767416621f * l - 3.3077115913f * m + 0.2309699292f * s,
            -1.2684380046f * l + 2.6097574011f * m - 0.3413193965f * s,
            -0.0041960863f * l - 0.7034186147f * m + 1.7076147010f * s,
        };
    }

    // Finds the maximum saturation possible for a given hue that fits in sRGB
    // Saturation here is defined as S = C/L
    // a and b must be normalized so a^2 + b^2 == 1
    static float compute_max_saturation(float a, float b)
    {
        // Max saturation will be when one of r, g or b goes below zero.

        // Select different coefficients depending on which component goes below zero first
        float k0, k1, k2, k3, k4, wl, wm, ws;

        if (-1.88170328f * a - 0.80936493f * b > 1)
        {
            // Red component
            k0 = +1.19086277f; k1 = +1.76576728f; k2 = +0.59662641f; k3 = +0.75515197f; k4 = +0.56771245f;
            wl = +4.0767416621f; wm = -3.3077115913f; ws = +0.2309699292f;
        }
        else if (1.81444104f * a - 1.19445276f * b > 1)
        {
            // Green component
            k0 = +0.73956515f; k1 = -0.45954404f; k2 = +0.08285427f; k3 = +0.12541070f; k4 = +0.14503204f;
            wl = -1.2684380046f; wm = +2.6097574011f; ws = -0.3413193965f;
        }
        else
        {
            // Blue component
            k0 = +1.35733652f; k1 = -0.00915799f; k2 = -1.15130210f; k3 = -0.50559606f; k4 = +0.00692167f;
            wl = -0.0041960863f; wm = -0.7034186147f; ws = +1.7076147010f;
        }

        // Approximate max saturation using a polynomial:
        float S = k0 + k1 * a + k2 * b + k3 * a * a + k4 * a * b;

        // Do one step Halley's method to get closer
        // this gives an error less than 10e6, except for some blue hues where the dS/dh is close to infinite
        // this should be sufficient for most applications, otherwise do two/three steps

        float k_l = +0.3963377774f * a + 0.2158037573f * b;
        float k_m = -0.1055613458f * a - 0.0638541728f * b;
        float k_s = -0.0894841775f * a - 1.2914855480f * b;

        {
            float l_ = 1.f + S * k_l;
            float m_ = 1.f + S * k_m;
            float s_ = 1.f + S * k_s;

            float l = l_ * l_ * l_;
            float m = m_ * m_ * m_;
            float s = s_ * s_ * s_;

            float l_dS = 3.f * k_l * l_ * l_;
            float m_dS = 3.f * k_m * m_ * m_;
            float s_dS = 3.f * k_s * s_ * s_;

            float l_dS2 = 6.f * k_l * k_l * l_;
            float m_dS2 = 6.f * k_m * k_m * m_;
            float s_dS2 = 6.f * k_s * k_s * s_;

            float f = wl * l + wm * m + ws * s;
            float f1 = wl * l_dS + wm * m_dS + ws * s_dS;
            float f2 = wl * l_dS2 + wm * m_dS2 + ws * s_dS2;

            S = S - f * f1 / (f1 * f1 - 0.5f * f * f2);
        }

        return S;
    }

    // finds L_cusp and C_cusp for a given hue
    // a and b must be normalized so a^2 + b^2 == 1
    static LC find_cusp(float a, float b)
    {
        // First, find the maximum saturation (saturation S = C/L)
        float S_cusp = compute_max_saturation(a, b);

        // Convert to linear sRGB to find the first point where at least one of r,g or b >= 1:
        RGB rgb_at_max = oklab_to_linear_srgb({ 1, S_cusp * a, S_cusp * b });
        float L_cusp = cbrtf(1.f / fmax(fmax(rgb_at_max.r, rgb_at_max.g), rgb_at_max.b));
        float C_cusp = L_cusp * S_cusp;

        return { L_cusp , C_cusp };
    }

#if 0
    // Finds intersection of the line defined by
    // L = L0 * (1 - t) + t * L1;
    // C = t * C1;
    // a and b must be normalized so a^2 + b^2 == 1
    static float find_gamut_intersection(float a, float b, float L1, float C1, float L0, LC cusp)
    {
        // Find the intersection for upper and lower half seprately
        float t;
        if (((L1 - L0) * cusp.C - (cusp.L - L0) * C1) <= 0.f)
        {
            // Lower half

            t = cusp.C * L0 / (C1 * cusp.L + cusp.C * (L0 - L1));
        }
        else
        {
            // Upper half

            // First intersect with triangle
            t = cusp.C * (L0 - 1.f) / (C1 * (cusp.L - 1.f) + cusp.C * (L0 - L1));

            // Then one step Halley's method
            {
                float dL = L1 - L0;
                float dC = C1;

                float k_l = +0.3963377774f * a + 0.2158037573f * b;
                float k_m = -0.1055613458f * a - 0.0638541728f * b;
                float k_s = -0.0894841775f * a - 1.2914855480f * b;

                float l_dt = dL + dC * k_l;
                float m_dt = dL + dC * k_m;
                float s_dt = dL + dC * k_s;


                // If higher accuracy is required, 2 or 3 iterations of the following block can be used:
                {
                    float L = L0 * (1.f - t) + t * L1;
                    float C = t * C1;

                    float l_ = L + C * k_l;
                    float m_ = L + C * k_m;
                    float s_ = L + C * k_s;

                    float l = l_ * l_ * l_;
                    float m = m_ * m_ * m_;
                    float s = s_ * s_ * s_;

                    float ldt = 3 * l_dt * l_ * l_;
                    float mdt = 3 * m_dt * m_ * m_;
                    float sdt = 3 * s_dt * s_ * s_;

                    float ldt2 = 6 * l_dt * l_dt * l_;
                    float mdt2 = 6 * m_dt * m_dt * m_;
                    float sdt2 = 6 * s_dt * s_dt * s_;

                    float r = 4.0767416621f * l - 3.3077115913f * m + 0.2309699292f * s - 1;
                    float r1 = 4.0767416621f * ldt - 3.3077115913f * mdt + 0.2309699292f * sdt;
                    float r2 = 4.0767416621f * ldt2 - 3.3077115913f * mdt2 + 0.2309699292f * sdt2;

                    float u_r = r1 / (r1 * r1 - 0.5f * r * r2);
                    float t_r = -r * u_r;

                    float g = -1.2684380046f * l + 2.6097574011f * m - 0.3413193965f * s - 1;
                    float g1 = -1.2684380046f * ldt + 2.6097574011f * mdt - 0.3413193965f * sdt;
                    float g2 = -1.2684380046f * ldt2 + 2.6097574011f * mdt2 - 0.3413193965f * sdt2;

                    float u_g = g1 / (g1 * g1 - 0.5f * g * g2);
                    float t_g = -g * u_g;

                    float b0 = -0.0041960863f * l - 0.7034186147f * m + 1.7076147010f * s - 1;
                    float b1 = -0.0041960863f * ldt - 0.7034186147f * mdt + 1.7076147010f * sdt;
                    float b2 = -0.0041960863f * ldt2 - 0.7034186147f * mdt2 + 1.7076147010f * sdt2;

                    float u_b = b1 / (b1 * b1 - 0.5f * b0 * b2);
                    float t_b = -b0 * u_b;

                    t_r = u_r >= 0.f ? t_r : FLT_MAX;
                    t_g = u_g >= 0.f ? t_g : FLT_MAX;
                    t_b = u_b >= 0.f ? t_b : FLT_MAX;

                    t += fmin(t_r, fmin(t_g, t_b));
                }
            }
        }

        return t;
    }

    static float find_gamut_intersection(float a, float b, float L1, float C1, float L0)
    {
        // Find the cusp of the gamut triangle
        LC cusp = find_cusp(a, b);

        return find_gamut_intersection(a, b, L1, C1, L0, cusp);
    }

    static RGB gamut_clip_preserve_chroma(RGB rgb)
    {
        if (rgb.r < 1 && rgb.g < 1 && rgb.b < 1 && rgb.r > 0 && rgb.g > 0 && rgb.b > 0)
            return rgb;

        Lab lab = linear_srgb_to_oklab(rgb);

        float L = lab.L;
        float eps = 0.00001f;
        float C = fmax(eps, sqrtf(lab.a * lab.a + lab.b * lab.b));
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        float L0 = clamp(L, 0, 1);

        float t = find_gamut_intersection(a_, b_, L, C, L0);
        float L_clipped = L0 * (1 - t) + t * L;
        float C_clipped = t * C;

        return oklab_to_linear_srgb({ L_clipped, C_clipped * a_, C_clipped * b_ });
    }

    static RGB gamut_clip_project_to_0_5(RGB rgb)
    {
        if (rgb.r < 1 && rgb.g < 1 && rgb.b < 1 && rgb.r > 0 && rgb.g > 0 && rgb.b > 0)
            return rgb;

        Lab lab = linear_srgb_to_oklab(rgb);

        float L = lab.L;
        float eps = 0.00001f;
        float C = fmax(eps, sqrtf(lab.a * lab.a + lab.b * lab.b));
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        float L0 = 0.5;

        float t = find_gamut_intersection(a_, b_, L, C, L0);
        float L_clipped = L0 * (1 - t) + t * L;
        float C_clipped = t * C;

        return oklab_to_linear_srgb({ L_clipped, C_clipped * a_, C_clipped * b_ });
    }

    static RGB gamut_clip_project_to_L_cusp(RGB rgb)
    {
        if (rgb.r < 1 && rgb.g < 1 && rgb.b < 1 && rgb.r > 0 && rgb.g > 0 && rgb.b > 0)
            return rgb;

        Lab lab = linear_srgb_to_oklab(rgb);

        float L = lab.L;
        float eps = 0.00001f;
        float C = fmax(eps, sqrtf(lab.a * lab.a + lab.b * lab.b));
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        // The cusp is computed here and in find_gamut_intersection, an optimized solution would only compute it once.
        LC cusp = find_cusp(a_, b_);

        float L0 = cusp.L;

        float t = find_gamut_intersection(a_, b_, L, C, L0);

        float L_clipped = L0 * (1 - t) + t * L;
        float C_clipped = t * C;

        return oklab_to_linear_srgb({ L_clipped, C_clipped * a_, C_clipped * b_ });
    }

    static RGB gamut_clip_adaptive_L0_0_5(RGB rgb, float alpha = 0.05f)
    {
        if (rgb.r < 1 && rgb.g < 1 && rgb.b < 1 && rgb.r > 0 && rgb.g > 0 && rgb.b > 0)
            return rgb;

        Lab lab = linear_srgb_to_oklab(rgb);

        float L = lab.L;
        float eps = 0.00001f;
        float C = fmax(eps, sqrtf(lab.a * lab.a + lab.b * lab.b));
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        float Ld = L - 0.5f;
        float e1 = 0.5f + fabs(Ld) + alpha * C;
        float L0 = 0.5f * (1.f + sgn(Ld) * (e1 - sqrtf(e1 * e1 - 2.f * fabs(Ld))));

        float t = find_gamut_intersection(a_, b_, L, C, L0);
        float L_clipped = L0 * (1.f - t) + t * L;
        float C_clipped = t * C;

        return oklab_to_linear_srgb({ L_clipped, C_clipped * a_, C_clipped * b_ });
    }

    static RGB gamut_clip_adaptive_L0_L_cusp(RGB rgb, float alpha = 0.05f)
    {
        if (rgb.r < 1 && rgb.g < 1 && rgb.b < 1 && rgb.r > 0 && rgb.g > 0 && rgb.b > 0)
            return rgb;

        Lab lab = linear_srgb_to_oklab(rgb);

        float L = lab.L;
        float eps = 0.00001f;
        float C = fmax(eps, sqrtf(lab.a * lab.a + lab.b * lab.b));
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        // The cusp is computed here and in find_gamut_intersection, an optimized solution would only compute it once.
        LC cusp = find_cusp(a_, b_);

        float Ld = L - cusp.L;
        float k = 2.f * (Ld > 0 ? 1.f - cusp.L : cusp.L);

        float e1 = 0.5f * k + fabs(Ld) + alpha * C / k;
        float L0 = cusp.L + 0.5f * (sgn(Ld) * (e1 - sqrtf(e1 * e1 - 2.f * k * fabs(Ld))));

        float t = find_gamut_intersection(a_, b_, L, C, L0);
        float L_clipped = L0 * (1.f - t) + t * L;
        float C_clipped = t * C;

        return oklab_to_linear_srgb({ L_clipped, C_clipped * a_, C_clipped * b_ });
    }
#endif

    static float toe_inv(float x)
    {
        constexpr float k_1 = 0.206f;
        constexpr float k_2 = 0.03f;
        constexpr float k_3 = (1.f + k_1) / (1.f + k_2);
        return (x * x + k_1 * x) / (k_3 * (x + k_2));
    }

    static ST to_ST(LC cusp)
    {
        float L = cusp.L;
        float C = cusp.C;
        return { C / L, C / (1 - L) };
    }

    static RGB okhsv_to_srgb(HSV hsv)
    {
        float h = hsv.h;
        float s = hsv.s;
        float v = hsv.v;

        float a_ = cosf(2.f * pi * h);
        float b_ = sinf(2.f * pi * h);

        LC cusp = find_cusp(a_, b_);
        ST ST_max = to_ST(cusp);
        float S_max = ST_max.S;
        float T_max = ST_max.T;
        float S_0 = 0.5f;
        float k = 1 - S_0 / S_max;

        // first we compute L and V as if the gamut is a perfect triangle:

        // L, C when v==1:
        float L_v = 1     - s * S_0 / (S_0 + T_max - T_max * k * s);
        float C_v = s * T_max * S_0 / (S_0 + T_max - T_max * k * s);

        float L = v * L_v;
        float C = v * C_v;

        // then we compensate for both toe and the curved top part of the triangle:
        float L_vt = toe_inv(L_v);
        float C_vt = C_v * L_vt / L_v;

        float L_new = toe_inv(L);
        C = C * L_new / L;
        L = L_new;

        RGB rgb_scale = oklab_to_linear_srgb({ L_vt, a_ * C_vt, b_ * C_vt });
        float scale_L = cbrtf(1.f / fmax(fmax(rgb_scale.r, rgb_scale.g), fmax(rgb_scale.b, 0.f)));

        L = L * scale_L;
        C = C * scale_L;

        RGB rgb = oklab_to_linear_srgb({ L, C * a_, C * b_ });
        return {
            srgb_transfer_function(rgb.r),
            srgb_transfer_function(rgb.g),
            srgb_transfer_function(rgb.b),
        };
    }

#if 0
    static float toe(float x)
    {
        constexpr float k_1 = 0.206f;
        constexpr float k_2 = 0.03f;
        constexpr float k_3 = (1.f + k_1) / (1.f + k_2);
        return 0.5f * (k_3 * x - k_1 + sqrtf((k_3 * x - k_1) * (k_3 * x - k_1) + 4 * k_2 * k_3 * x));
    }

    static float srgb_transfer_function_inv(float a)
    {
        return .04045f < a ? powf((a + .055f) / 1.055f, 2.4f) : a / 12.92f;
    }

    static Lab linear_srgb_to_oklab(RGB c)
    {
        float l = 0.4122214708f * c.r + 0.5363325363f * c.g + 0.0514459929f * c.b;
        float m = 0.2119034982f * c.r + 0.6806995451f * c.g + 0.1073969566f * c.b;
        float s = 0.0883024619f * c.r + 0.2817188376f * c.g + 0.6299787005f * c.b;

        float l_ = cbrtf(l);
        float m_ = cbrtf(m);
        float s_ = cbrtf(s);

        return {
            0.2104542553f * l_ + 0.7936177850f * m_ - 0.0040720468f * s_,
            1.9779984951f * l_ - 2.4285922050f * m_ + 0.4505937099f * s_,
            0.0259040371f * l_ + 0.7827717662f * m_ - 0.8086757660f * s_,
        };
    }

    // Returns a smooth approximation of the location of the cusp
    // This polynomial was created by an optimization process
    // It has been designed so that S_mid < S_max and T_mid < T_max
    static ST get_ST_mid(float a_, float b_)
    {
        float S = 0.11516993f + 1.f / (
                                       +7.44778970f + 4.15901240f * b_
                                       + a_ * (-2.19557347f + 1.75198401f * b_
                                               + a_ * (-2.13704948f - 10.02301043f * b_
                                                       + a_ * (-4.24894561f + 5.38770819f * b_ + 4.69891013f * a_
                                                               )))
                                       );

        float T = 0.11239642f + 1.f / (
                                       +1.61320320f - 0.68124379f * b_
                                       + a_ * (+0.40370612f + 0.90148123f * b_
                                               + a_ * (-0.27087943f + 0.61223990f * b_
                                                       + a_ * (+0.00299215f - 0.45399568f * b_ - 0.14661872f * a_
                                                               )))
                                       );

        return { S, T };
    }

    struct Cs { float C_0; float C_mid; float C_max; };
    static Cs get_Cs(float L, float a_, float b_)
    {
        LC cusp = find_cusp(a_, b_);

        float C_max = find_gamut_intersection(a_, b_, L, 1, L, cusp);
        ST ST_max = to_ST(cusp);

        // Scale factor to compensate for the curved part of gamut shape:
        float k = C_max / fmin((L * ST_max.S), (1 - L) * ST_max.T);

        float C_mid;
        {
            ST ST_mid = get_ST_mid(a_, b_);

            // Use a soft minimum function, instead of a sharp triangle shape to get a smooth value for chroma.
            float C_a = L * ST_mid.S;
            float C_b = (1.f - L) * ST_mid.T;
            C_mid = 0.9f * k * sqrtf(sqrtf(1.f / (1.f / (C_a * C_a * C_a * C_a) + 1.f / (C_b * C_b * C_b * C_b))));
        }

        float C_0;
        {
            // for C_0, the shape is independent of hue, so ST are constant. Values picked to roughly be the average values of ST.
            float C_a = L * 0.4f;
            float C_b = (1.f - L) * 0.8f;

            // Use a soft minimum function, instead of a sharp triangle shape to get a smooth value for chroma.
            C_0 = sqrtf(1.f / (1.f / (C_a * C_a) + 1.f / (C_b * C_b)));
        }

        return { C_0, C_mid, C_max };
    }

    static RGB okhsl_to_srgb(HSL hsl)
    {
        float h = hsl.h;
        float s = hsl.s;
        float l = hsl.l;

        if (l == 1.0f)
        {
            return { 1.f, 1.f, 1.f };
        }

        else if (l == 0.f)
        {
            return { 0.f, 0.f, 0.f };
        }

        float a_ = cosf(2.f * pi * h);
        float b_ = sinf(2.f * pi * h);
        float L = toe_inv(l);

        Cs cs = get_Cs(L, a_, b_);
        float C_0 = cs.C_0;
        float C_mid = cs.C_mid;
        float C_max = cs.C_max;

        float mid = 0.8f;
        float mid_inv = 1.25f;

        float C, t, k_0, k_1, k_2;

        if (s < mid)
        {
            t = mid_inv * s;

            k_1 = mid * C_0;
            k_2 = (1.f - k_1 / C_mid);

            C = t * k_1 / (1.f - k_2 * t);
        }
        else
        {
            t = (s - mid)/ (1 - mid);

            k_0 = C_mid;
            k_1 = (1.f - mid) * C_mid * C_mid * mid_inv * mid_inv / C_0;
            k_2 = (1.f - (k_1) / (C_max - C_mid));

            C = k_0 + t * k_1 / (1.f - k_2 * t);
        }

        RGB rgb = oklab_to_linear_srgb({ L, C * a_, C * b_ });
        return {
            srgb_transfer_function(rgb.r),
            srgb_transfer_function(rgb.g),
            srgb_transfer_function(rgb.b),
        };
    }

    static HSL srgb_to_okhsl(RGB rgb)
    {
        Lab lab = linear_srgb_to_oklab({
            srgb_transfer_function_inv(rgb.r),
            srgb_transfer_function_inv(rgb.g),
            srgb_transfer_function_inv(rgb.b)
        });

        float C = sqrtf(lab.a * lab.a + lab.b * lab.b);
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        float L = lab.L;
        float h = 0.5f + 0.5f * atan2f(-lab.b, -lab.a) / pi;

        Cs cs = get_Cs(L, a_, b_);
        float C_0 = cs.C_0;
        float C_mid = cs.C_mid;
        float C_max = cs.C_max;

        // Inverse of the interpolation in okhsl_to_srgb:

        float mid = 0.8f;
        float mid_inv = 1.25f;

        float s;
        if (C < C_mid)
        {
            float k_1 = mid * C_0;
            float k_2 = (1.f - k_1 / C_mid);

            float t = C / (k_1 + k_2 * C);
            s = t * mid;
        }
        else
        {
            float k_0 = C_mid;
            float k_1 = (1.f - mid) * C_mid * C_mid * mid_inv * mid_inv / C_0;
            float k_2 = (1.f - (k_1) / (C_max - C_mid));

            float t = (C - k_0) / (k_1 + k_2 * (C - k_0));
            s = mid + (1.f - mid) * t;
        }

        float l = toe(L);
        return { h, s, l };
    }

    static HSV srgb_to_okhsv(RGB rgb)
    {
        Lab lab = linear_srgb_to_oklab({
            srgb_transfer_function_inv(rgb.r),
            srgb_transfer_function_inv(rgb.g),
            srgb_transfer_function_inv(rgb.b)
        });

        float C = sqrtf(lab.a * lab.a + lab.b * lab.b);
        float a_ = lab.a / C;
        float b_ = lab.b / C;

        float L = lab.L;
        float h = 0.5f + 0.5f * atan2f(-lab.b, -lab.a) / pi;

        LC cusp = find_cusp(a_, b_);
        ST ST_max = to_ST(cusp);
        float S_max = ST_max.S;
        float T_max = ST_max.T;
        float S_0 = 0.5f;
        float k = 1 - S_0 / S_max;

        // first we find L_v, C_v, L_vt and C_vt

        float t = T_max / (C + L * T_max);
        float L_v = t * L;
        float C_v = t * C;

        float L_vt = toe_inv(L_v);
        float C_vt = C_v * L_vt / L_v;

        // we can then use these to invert the step that compensates for the toe and the curved top part of the triangle:
        RGB rgb_scale = oklab_to_linear_srgb({ L_vt, a_ * C_vt, b_ * C_vt });
        float scale_L = cbrtf(1.f / fmax(fmax(rgb_scale.r, rgb_scale.g), fmax(rgb_scale.b, 0.f)));

        L = L / scale_L;
        C = C / scale_L;

        C = C * toe(L) / L;
        L = toe(L);

        // we can now compute v and s:

        float v = L / L_v;
        float s = (S_0 + T_max) * C_v / ((T_max * S_0) + T_max * k * C_v);

        return { h, s, v };
    }

#endif
} // namespace ok_color
