C# Damping for Godot et al

Damp.cs

//SPDX-License-Identifier: MIT

using System.Runtime.CompilerServices;
using Godot;
using Vector3 = System.Numerics.Vector3;

namespace Jovian.Math
{
    public static class Damp
    {
        public static void Smooth(ref double x, ref double v, double x_goal, double v_goal, double lambda, double dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = v_goal;
                return;
            }

            var g = x_goal;
            var q = v_goal;
            var d = lambda_to_damping(lambda);
            var c = g + d * q / (d * d / 4.0);
            var y = d / 2.0;
            var j0 = x - c;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + c;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref float x, ref float v, float x_goal, float v_goal, float lambda, float dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = v_goal;
                return;
            }

            var g = x_goal;
            var q = v_goal;
            var d = lambda_to_damping(lambda);
            var c = g + d * q / (d * d / 4.0f);
            var y = d / 2.0f;
            var j0 = x - c;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + c;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref Vector3 x, ref Vector3 v, Vector3 x_goal, Vector3 v_goal, float lambda, float dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = v_goal;
                return;
            }

            var g = x_goal;
            var q = v_goal;
            var d = lambda_to_damping(lambda);
            var c = g + d * q / (d * d / 4.0f);
            var y = d / 2.0f;
            var j0 = x - c;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + c;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref Godot.Vector3 x, ref Godot.Vector3 v, Godot.Vector3 x_goal, Godot.Vector3 v_goal, float lambda, float dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = v_goal;
                return;
            }

            var g = x_goal;
            var q = v_goal;
            var d = lambda_to_damping(lambda);
            var c = g + d * q / (d * d / 4.0f);
            var y = d / 2.0f;
            var j0 = x - c;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + c;
            v = ey_dt * (v - j1 * y * dt);
        }
        
        public static void Smooth(ref Vector3D x, ref Vector3D v, Vector3D x_goal, Vector3D v_goal, double lambda, double dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = v_goal;
                return;
            }

            var g = x_goal;
            var q = v_goal;
            var d = lambda_to_damping(lambda);
            var c = g + q * d / (d * d / 4.0);
            var y = d / 2.0;
            var j0 = x - c;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + c;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref double x, ref double v, double x_goal, double lambda, double dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = 0;
                return;
            }

            var y = lambda_to_damping(lambda) / 2.0;
            var j0 = x - x_goal;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref float x, ref float v, float x_goal, float lambda, float dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = 0;
                return;
            }

            var y = lambda_to_damping(lambda) / 2.0f;
            var j0 = x - x_goal;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }


        public static void Smooth(ref Vector3 x, ref Vector3 v, Vector3 x_goal, float lambda, float dt)
        {
            if (lambda <= 0)
            {
                x = x_goal;
                v = Vector3.Zero;
                return;
            }

            var y = lambda_to_damping(lambda) / 2.0f;
            var j0 = x - x_goal;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref Godot.Vector3 x, ref Godot.Vector3 v, Godot.Vector3 x_goal, float lambda, float dt)
        {
            var y = lambda_to_damping(lambda) / 2.0f;
            var j0 = x - x_goal;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref Vector3D x, ref Vector3D v, Vector3D x_goal, double lambda, double dt)
        {
            var y = lambda_to_damping(lambda) / 2.0;
            var j0 = x - x_goal;
            var j1 = v + j0 * y;
            var ey_dt = fast_neg_exp(y * dt);

            x = ey_dt * (j0 + j1 * dt) + x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }

        public static void Smooth(ref Godot.Quaternion x, ref Godot.Vector3 v, Quaternion x_goal, float lambda, float dt)
        {
            var y = lambda_to_damping(lambda) / 2.0f;

            //Compensate for double cover of quaternions: q and -q represent the same rotation
            var dot = x.Dot(x_goal);
            var multi = dot > 0f ? 1f : -1f;
            x_goal *= multi;
            
            var j0 = Qx.ToScaledAngleAxis(x * x_goal.Inverse());
            var j1 = v + j0 * y;

            var ey_dt = fast_neg_exp(y * dt);
            x = Qx.FromScaledAngleAxis(ey_dt * (j0 + j1 * dt)) * x_goal;
            v = ey_dt * (v - j1 * y * dt);
        }
        
        private const double Ln2 = 0.6931471805599453094172321214581765680755001343602552541206800094d;
        private const float Ln2F = 0.693147180559945f;
        
        #region Helper Maths

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static float lambda_to_damping(float lambda)
        {
            return 4.0f * Ln2F / lambda;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static double lambda_to_damping(double lambda)
        {
            return 4.0 * Ln2 / lambda;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static float fast_neg_exp(float x)
        {
            return 1.0f / (1.0f + x + 0.48f * x * x + 0.235f * x * x * x);
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static double fast_neg_exp(double x)
        {
            return 1.0 / (1.0 + x + 0.48 * x * x + 0.235 * x * x * x);
            // Better for x element [0.5..1]: (our case is most often x < 0.5, usually 0.15 to 0.05-ish)
            // return 1.0 / (1.0 + 1.006722 * x + 0.453568 * x * x + 0.254559 * x * x * x);
        }
        #endregion
    }
}

/*
MIT License

Copyright (c) 2023 tiger.blue

Modified and ported to Godot and System.Numerics, with float/double precision.
Changed core use case, which is SmoothDamp / Critical with and without goal velocity, 
for float, double, Vector2, Vector3, Vector3D, and Quaternions

Copyright (c) 2021 Daniel Holden, Meta

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*/