Automatic differentiation (second order) on C# with Roslyn for parsing.

AutomaticDifferentiation.cs

using System;
using System.Linq.Expressions;
using System.Reflection;
using System.Threading.Tasks;
using Microsoft.CodeAnalysis.CSharp.Scripting;
using Microsoft.CodeAnalysis.Scripting;

namespace AutomaticDifferentiation
{
    public struct DualNumber
    {
        public double Value { get; }
        public double Eps { get; }
        public double Eps2 { get; }

        public DualNumber(double value, double eps = 0, double eps2 = 0)
        {
            Value = value;
            Eps = eps;
            Eps2 = eps2;
        }

        public static implicit operator DualNumber(double v) => new DualNumber(v);

        #region Arithmetic Operators

        public static DualNumber operator +(DualNumber a) => a;
        public static DualNumber operator -(DualNumber a) => new DualNumber(-a.Value, -a.Eps, -a.Eps2);

        public static DualNumber operator +(DualNumber a, DualNumber b) => new DualNumber(a.Value + b.Value, a.Eps + b.Eps, a.Eps2 + b.Eps2);
        public static DualNumber operator +(double a, DualNumber b) => new DualNumber(a + b.Value, b.Eps, b.Eps2);
        public static DualNumber operator +(DualNumber a, double b) => new DualNumber(a.Value + b, a.Eps, a.Eps2);

        public static DualNumber operator -(DualNumber a, DualNumber b) => new DualNumber(a.Value - b.Value, a.Eps - b.Eps, a.Eps2 - b.Eps2);
        public static DualNumber operator -(double a, DualNumber b) => new DualNumber(a - b.Value, -b.Eps, -b.Eps2);
        public static DualNumber operator -(DualNumber a, double b) => new DualNumber(a.Value - b, a.Eps, a.Eps2);

        public static DualNumber operator *(DualNumber a, DualNumber b)
            => new DualNumber(a.Value * b.Value, a.Eps * b.Value + b.Eps * a.Value, a.Value * b.Eps2 + a.Eps * b.Eps + a.Eps2 * b.Value);
        public static DualNumber operator *(double a, DualNumber b) => new DualNumber(a * b.Value, a * b.Eps, a * b.Eps2);
        public static DualNumber operator *(DualNumber a, double b) => new DualNumber(a.Value * b, a.Eps * b, a.Eps2 * b);

        public static DualNumber operator /(DualNumber a, DualNumber b) => a * Inverse(b);
        public static DualNumber operator /(double a, DualNumber b) => a * Inverse(b);
        public static DualNumber operator /(DualNumber a, double b) => a * (1 / b);

        #endregion

        #region Power Functions

        public static DualNumber Inverse(DualNumber a)
        {
            return new DualNumber(1 / a.Value, -a.Eps / (a.Value * a.Value), (a.Eps * a.Eps / a.Value - a.Eps2) / (a.Value * a.Value));
        }

        public static DualNumber Sqrt(DualNumber a)
        {
            var r = Math.Sqrt(a.Value);
            return new DualNumber(r, 0.5 * a.Eps / r, 0.5 * a.Eps2 / r - 0.125 * a.Eps * a.Eps / (a.Value * r));
        }

        public static DualNumber Exp(DualNumber a)
        {
            var exp = Math.Exp(a.Value);
            return new DualNumber(exp, exp * a.Eps, exp * (a.Eps2 + 0.5 * a.Eps * a.Eps));
        }

        public static DualNumber Log(DualNumber a)
        {
            return new DualNumber(Math.Log(a.Value), a.Eps / a.Value, (a.Eps2 - 0.5 * a.Eps * a.Eps / a.Value) / a.Value);
        }

        public static DualNumber Pow(DualNumber a, DualNumber b) => Exp(b * Log(a));

        public static DualNumber Pow(DualNumber a, double b)
        {
            var pow = Math.Pow(a.Value, b);
            var frac = a.Eps / a.Value;
            return new DualNumber(pow, b * pow * frac, b * pow * (a.Eps2 / a.Value + 0.5 * (b - 1) * frac * frac));
        }

        #endregion

        #region Trigonometric Functions

        public static DualNumber Sin(DualNumber a)
        {
            var sin = Math.Sin(a.Value);
            var cos = Math.Cos(a.Value);
            return new DualNumber(sin, cos * a.Eps, cos * a.Eps2 - 0.5 * sin * a.Eps * a.Eps);
        }

        public static DualNumber Cos(DualNumber a)
        {
            var sin = Math.Sin(a.Value);
            var cos = Math.Cos(a.Value);
            return new DualNumber(cos, -sin * a.Eps, -sin * a.Eps2 - 0.5 * cos * a.Eps * a.Eps);
        }

        public static DualNumber Tan(DualNumber a)
        {
            var tan = Math.Tan(a.Value);
            return new DualNumber(tan, (1 + tan * tan) * a.Eps, (1 + tan * tan) * (a.Eps2 + tan * a.Eps * a.Eps));
        }

        public static DualNumber Asin(DualNumber a)
        {
            var tmp = 1 / Math.Sqrt(1 - a.Value * a.Value);
            var tmpEps = a.Eps * tmp;
            return new DualNumber(Math.Asin(a.Value), tmpEps, (a.Eps2 + 0.5 * a.Value * tmpEps * tmpEps) * tmp);
        }

        public static DualNumber Acos(DualNumber a)
        {
            var tmp = 1 / Math.Sqrt(1 - a.Value * a.Value);
            var tmpEps = a.Eps * tmp;
            return new DualNumber(Math.Acos(a.Value), -tmpEps, -(a.Eps2 + 0.5 * a.Value * tmpEps * tmpEps) * tmp);
        }

        public static DualNumber Atan(DualNumber a)
        {
            var tmp = 1 / (1 + a.Value * a.Value);
            var tmpEps = a.Eps * tmp;
            return new DualNumber(Math.Atan(a.Value), tmpEps, a.Eps2 * tmp - a.Value * tmpEps * tmpEps);
        }

        #endregion

        #region Hyperbolic Functions

        public static DualNumber Sinh(DualNumber a)
        {
            var sin = Math.Sinh(a.Value);
            var cos = Math.Cosh(a.Value);
            return new DualNumber(sin, cos * a.Eps, cos * a.Eps2 + 0.5 * sin * a.Eps * a.Eps);
        }

        public static DualNumber Cosh(DualNumber a)
        {
            var sin = Math.Sinh(a.Value);
            var cos = Math.Cosh(a.Value);
            return new DualNumber(cos, sin * a.Eps, sin * a.Eps2 + 0.5 * cos * a.Eps * a.Eps);
        }

        public static DualNumber Tanh(DualNumber a)
        {
            var tan = Math.Tanh(a.Value);
            return new DualNumber(tan, (1 - tan * tan) * a.Eps, (1 - tan * tan) * (a.Eps2 - tan * a.Eps * a.Eps));
        }

        #endregion
    }

    public class Parser
    {
        public static async Task<Func<double, double>> Compile(string text)
        {
            var options = ScriptOptions.Default.WithReferences(typeof(Expression).GetTypeInfo().Assembly);
            var expr = await CSharpScript.EvaluateAsync<Expression<Func<double, double>>>(text, options);
            return expr.Compile();
        }

        public static async Task<Func<double, (double value, double diff, double diff2)>> CompileDiff(string text)
        {
            var options = ScriptOptions.Default.WithReferences(typeof(Expression).GetTypeInfo().Assembly, typeof(DualNumber).GetTypeInfo().Assembly).WithImports(typeof(DualNumber).FullName);
            var expr = await CSharpScript.EvaluateAsync<Expression<Func<DualNumber, DualNumber>>>(text, options);
            var f = expr.Compile();
            return t => { var r = f(new DualNumber(t, 1)); return (r.Value, r.Eps, 2 * r.Eps2); };
        }
    }
}

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