1. rename your current DE() routine to DEInner()
2. use the following as your DE().
   (this is Apple Metal code, but you get the idea)


bool    control.doInversion
float3  control.InvCenter  // each field -5 ... 5
float    control.InvRadius  // 0.01 ... 10
float    control.InvAngle   // -10 ... 10


float DE(float3 pos,device Control &control,thread float4 &orbitTrap) {
    if(control.doInversion) {
        pos = pos - control.InvCenter;
        float r = length(pos);
        float r2 = r*r;

        pos = (control.InvRadius * control.InvRadius / r2 ) * pos + control.InvCenter;
       
        float an = atan2(pos.y,pos.x) + control.InvAngle;
        float ra = sqrt(pos.y * pos.y + pos.x * pos.x);

        pos.x = cos(an)*ra;
        pos.y = sin(an)*ra;

        float de = DE_Inner(pos,control,orbitTrap);

        de = r2 * de / (control.InvRadius * control.InvRadius + r * de);
        return de;
    }
   
    return DE_Inner(pos,control,orbitTrap);
}

// image is Apollonian after inversion

Linkback: https://fractalforums.org/programming/11/add-spherical-inversion-to-any-3d-fractal/3633/

it is the standard sphere inversion BUT has a rotation included



      float an = atan2(pos.y,pos.x) + control.InvAngle;
        float ra = sqrt(pos.y * pos.y + pos.x * pos.x);

        pos.x = cos(an)*ra;
        pos.y = sin(an)*ra;