derivative.cpp

/* derivative */

#include "stdafx.h"
#include "defs.h"

#define F p3
#define X p4
#define N p5

void
eval_derivative(void)
{
	int i, n;

	// evaluate 1st arg to get function F

	p1 = cdr(p1);
	push(car(p1));
	eval();

	// evaluate 2nd arg and then...

	// example	result of 2nd arg	what to do
	//
	// d(f)		nil			guess X, N = nil
	// d(f,2)	2			guess X, N = 2
	// d(f,x)	x			X = x, N = nil
	// d(f,x,2)	x			X = x, N = 2
	// d(f,x,y)	x			X = x, N = y

	p1 = cdr(p1);
	push(car(p1));
	eval();

	p2 = pop();
	if (p2 == symbol(NIL)) {
		guess();
		push(symbol(NIL));
	} else if (isnum(p2)) {
		guess();
		push(p2);
	} else {
		push(p2);
		p1 = cdr(p1);
		push(car(p1));
		eval();
	}

	N = pop();
	X = pop();
	F = pop();

	while (1) {

		// N might be a symbol instead of a number

		if (isnum(N)) {
			push(N);
			n = pop_integer();
			if (n == (int) 0x80000000)
				stop("nth derivative: check n");
		} else
			n = 1;

		push(F);

		if (n >= 0) {
			for (i = 0; i < n; i++) {
				push(X);
				derivative();
			}
		} else {
			n = -n;
			for (i = 0; i < n; i++) {
				push(X);
				integral();
			}
		}

		F = pop();

		// if N is nil then arglist is exhausted

		if (N == symbol(NIL))
			break;

		// otherwise...

		// N		arg1		what to do
		//
		// number	nil		break
		// number	number		N = arg1, continue
		// number	symbol		X = arg1, N = arg2, continue
		//
		// symbol	nil		X = N, N = nil, continue
		// symbol	number		X = N, N = arg1, continue
		// symbol	symbol		X = N, N = arg1, continue

		if (isnum(N)) {
			p1 = cdr(p1);
			push(car(p1));
			eval();
			N = pop();
			if (N == symbol(NIL))
				break;		// arglist exhausted
			if (isnum(N))
				;		// N = arg1
			else {
				X = N;		// X = arg1
				p1 = cdr(p1);
				push(car(p1));
				eval();
				N = pop();	// N = arg2
			}
		} else {
			X = N;			// X = N
			p1 = cdr(p1);
			push(car(p1));
			eval();
			N = pop();		// N = arg1
		}
	}

	push(F); // final result
}

void
derivative(void)
{
	save();
	p2 = pop();
	p1 = pop();
	if (isnum(p2))
		stop("undefined function");
	if (istensor(p1))
		if (istensor(p2))
			d_tensor_tensor();
		else
			d_tensor_scalar();
	else
		if (istensor(p2))
			d_scalar_tensor();
		else
			d_scalar_scalar();
	restore();
}

void
d_scalar_scalar(void)
{
	if (issymbol(p2))
		d_scalar_scalar_1();
	else {
		// Example: d(sin(cos(x)),cos(x))
		// Replace cos(x) <- X, find derivative, then do X <- cos(x)
		push(p1);		// sin(cos(x))
		push(p2);		// cos(x)
		push(symbol(SECRETX));	// X
		subst();		// sin(cos(x)) -> sin(X)
		push(symbol(SECRETX));	// X
		derivative();
		push(symbol(SECRETX));	// X
		push(p2);		// cos(x)
		subst();		// cos(X) -> cos(cos(x))
	}
}

void
d_scalar_scalar_1(void)
{
	// d(x,x)?

	if (equal(p1, p2)) {
		push(one);
		return;
	}

	// d(a,x)?

	if (!iscons(p1)) {
		push(zero);
		return;
	}

	if (isadd(p1)) {
		dsum();
		return;
	}

	if (car(p1) == symbol(MULTIPLY)) {
		dproduct();
		return;
	}

	if (car(p1) == symbol(POWER)) {
		dpower();
		return;
	}

	if (car(p1) == symbol(DERIVATIVE)) {
		dd();
		return;
	}

	if (car(p1) == symbol(LOG)) {
		dlog();
		return;
	}

	if (car(p1) == symbol(SIN)) {
		dsin();
		return;
	}

	if (car(p1) == symbol(COS)) {
		dcos();
		return;
	}

	if (car(p1) == symbol(TAN)) {
		dtan();
		return;
	}

	if (car(p1) == symbol(ARCSIN)) {
		darcsin();
		return;
	}

	if (car(p1) == symbol(ARCCOS)) {
		darccos();
		return;
	}

	if (car(p1) == symbol(ARCTAN)) {
		darctan();
		return;
	}

	if (car(p1) == symbol(SINH)) {
		dsinh();
		return;
	}

	if (car(p1) == symbol(COSH)) {
		dcosh();
		return;
	}

	if (car(p1) == symbol(TANH)) {
		dtanh();
		return;
	}

	if (car(p1) == symbol(ARCSINH)) {
		darcsinh();
		return;
	}

	if (car(p1) == symbol(ARCCOSH)) {
		darccosh();
		return;
	}

	if (car(p1) == symbol(ARCTANH)) {
		darctanh();
		return;
	}

	if (car(p1) == symbol(ABS)) {
		dabs();
		return;
	}

	if (car(p1) == symbol(SGN)) {
		dsgn();
		return;
	}

	if (car(p1) == symbol(HERMITE)) {
		dhermite();
		return;
	}

	if (car(p1) == symbol(ERF)) {
		derf();
		return;
	}

	if (car(p1) == symbol(ERFC)) {
		derfc();
		return;
	}

	if (car(p1) == symbol(BESSELJ)) {
		if (iszero(caddr(p1)))
			dbesselj0();
		else
			dbesseljn();
		return;
	}

	if (car(p1) == symbol(BESSELY)) {
		if (iszero(caddr(p1)))
			dbessely0();
		else
			dbesselyn();
		return;
	}

	if (car(p1) == symbol(INTEGRAL) && caddr(p1) == p2) {
		derivative_of_integral();
		return;
	}

	dfunction();
}

void
dsum(void)
{
	int h = tos;
	p1 = cdr(p1);
	while (iscons(p1)) {
		push(car(p1));
		push(p2);
		derivative();
		p1 = cdr(p1);
	}
	add_all(tos - h);
}

void
dproduct(void)
{
	int i, j, n;
	n = length(p1) - 1;
	for (i = 0; i < n; i++) {
		p3 = cdr(p1);
		for (j = 0; j < n; j++) {
			push(car(p3));
			if (i == j) {
				push(p2);
				derivative();
			}
			p3 = cdr(p3);
		}
		multiply_all(n);
	}
	add_all(n);
}

//-----------------------------------------------------------------------------
//
//	     v
//	y = u
//
//	log y = v log u
//
//	1 dy   v du           dv
//	- -- = - -- + (log u) --
//	y dx   u dx           dx
//
//	dy    v  v du           dv
//	-- = u  (- -- + (log u) --)
//	dx       u dx           dx
//
//-----------------------------------------------------------------------------

void
dpower(void)
{
	push(caddr(p1));	// v/u
	push(cadr(p1));
	divide();

	push(cadr(p1));		// du/dx
	push(p2);
	derivative();

	multiply();

	push(cadr(p1));		// log u
	logarithm();

	push(caddr(p1));	// dv/dx
	push(p2);
	derivative();

	multiply();

	add();

	push(p1);		// u^v

	multiply();
}

void
dlog(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	divide();
}

//	derivative of derivative
//
//	example: d(d(f(x,y),y),x)
//
//	p1 = d(f(x,y),y)
//
//	p2 = x
//
//	cadr(p1) = f(x,y)
//
//	caddr(p1) = y

void
dd(void)
{
	// d(f(x,y),x)

	push(cadr(p1));
	push(p2);
	derivative();

	p3 = pop();

	if (car(p3) == symbol(DERIVATIVE)) {

		// sort dx terms

		push_symbol(DERIVATIVE);
		push_symbol(DERIVATIVE);
		push(cadr(p3));

		if (lessp(caddr(p3), caddr(p1))) {
			push(caddr(p3));
			list(3);
			push(caddr(p1));
		} else {
			push(caddr(p1));
			list(3);
			push(caddr(p3));
		}

		list(3);

	} else {
		push(p3);
		push(caddr(p1));
		derivative();
	}
}

// derivative of a generic function

void
dfunction(void)
{
	p3 = cdr(p1);	// p3 is the argument list for the function

	if (p3 == symbol(NIL) || find(p3, p2)) {
		push_symbol(DERIVATIVE);
		push(p1);
		push(p2);
		list(3);
	} else
		push(zero);
}

void
dsin(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	cosine();
	multiply();
}

void
dcos(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	sine();
	multiply();
	negate();
}

void
dtan(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	cosine();
	push_integer(-2);
	power();
	multiply();
}

void
darcsin(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push_integer(1);
	push(cadr(p1));
	push_integer(2);
	power();
	subtract();
	push_rational(-1, 2);
	power();
	multiply();
}

void
darccos(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push_integer(1);
	push(cadr(p1));
	push_integer(2);
	power();
	subtract();
	push_rational(-1, 2);
	power();
	multiply();
	negate();
}

//				Without simplify	With simplify
//
//	d(arctan(y/x),x)	-y/(x^2*(y^2/x^2+1))	-y/(x^2+y^2)
//
//	d(arctan(y/x),y)	1/(x*(y^2/x^2+1))	x/(x^2+y^2)

void
darctan(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push_integer(1);
	push(cadr(p1));
	push_integer(2);
	power();
	add();
	inverse();
	multiply();
	simplify();
}

void
dsinh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	ycosh();
	multiply();
}

void
dcosh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	ysinh();
	multiply();
}

void
dtanh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	ycosh();
	push_integer(-2);
	power();
	multiply();
}

void
darcsinh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push_integer(2);
	power();
	push_integer(1);
	add();
	push_rational(-1, 2);
	power();
	multiply();
}

void
darccosh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push_integer(2);
	power();
	push_integer(-1);
	add();
	push_rational(-1, 2);
	power();
	multiply();
}

void
darctanh(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push_integer(1);
	push(cadr(p1));
	push_integer(2);
	power();
	subtract();
	inverse();
	multiply();
}

void
dabs(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	sgn();
	multiply();
}

void
dsgn(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	dirac();
	multiply();
	push_integer(2);
	multiply();
}

void
dhermite(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push_integer(2);
	push(caddr(p1));
	multiply();
	multiply();
	push(cadr(p1));
	push(caddr(p1));
	push_integer(-1);
	add();
	hermite();
	multiply();
}

void
derf(void)
{
	push(cadr(p1));
	push_integer(2);
	power();
	push_integer(-1);
	multiply();
	exponential();
	push_symbol(PI);
	push_rational(-1,2);
	power();
	multiply();
	push_integer(2);
	multiply();
	push(cadr(p1));
	push(p2);
	derivative();
	multiply();

}

void
derfc(void)
{
	push(cadr(p1));
	push_integer(2);
	power();
	push_integer(-1);
	multiply();
	exponential();
	push_symbol(PI);
	push_rational(-1,2);
	power();
	multiply();
	push_integer(-2);
	multiply();
	push(cadr(p1));
	push(p2);
	derivative();
	multiply();

}

void
dbesselj0(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push_integer(1);
	besselj();
	multiply();
	push_integer(-1);
	multiply();
}

void
dbesseljn(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push(caddr(p1));
	push_integer(-1);
	add();
	besselj();
	push(caddr(p1));
	push_integer(-1);
	multiply();
	push(cadr(p1));
	divide();
	push(cadr(p1));
	push(caddr(p1));
	besselj();
	multiply();
	add();
	multiply();
}


void
dbessely0(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push_integer(1);
	besselj();
	multiply();
	push_integer(-1);
	multiply();
}


void
dbesselyn(void)
{
	push(cadr(p1));
	push(p2);
	derivative();
	push(cadr(p1));
	push(caddr(p1));
	push_integer(-1);
	add();
	bessely();
	push(caddr(p1));
	push_integer(-1);
	multiply();
	push(cadr(p1));
	divide();
	push(cadr(p1));
	push(caddr(p1));
	bessely();
	multiply();
	add();
	multiply();
}

void
derivative_of_integral(void)
{
	push(cadr(p1));
}

#if SELFTEST

static const char *s[] = {

	"x=quote(x)",
	"",

	"f=quote(f)",
	"",

	"g=quote(g)",
	"",

	"d(a,x)",
	"0",

	"d(x,x)",
	"1",

	"d(x^2,x)",
	"2*x",

	"d(log(x),x)",
	"1/x",

	"d(exp(x),x)",
	"exp(x)",

	"d(a^x,x)",
	"a^x*log(a)",

	"d(x^x,x)-(x^x+x^x*log(x))",
	"0",

	"d(log(x^2+5),x)-(2*x/(5+x^2))",
	"0",

	"d(d(f(x),x),y)",
	"0",

	"d(d(f(x),y),x)",
	"0",

	"d(d(f(y),x),y)",
	"0",

	"d(d(f(y),y),x)",
	"0",

	"d((x*y*z,y,x+z),(x,y,z))",
	"((y*z,x*z,x*y),(0,1,0),(1,0,1))",

	"d(x+z,(x,y,z))",
	"(1,0,1)",

	"d(cos(theta)^2,cos(theta))",
	"2*cos(theta)",

	"d(f())",
	"d(f(),x)",

	"d(x^2)",
	"2*x",

	"d(t^2)",
	"2*t",

	"d(t^2 x^2)",
	"2*t^2*x",

// trig functions

	"d(sin(x),x)-cos(x)",
	"0",

	"d(cos(x),x)+sin(x)",
	"0",

	"d(tan(x),x)-cos(x)^(-2)",
	"0",

	"d(arcsin(x),x)-1/sqrt(1-x^2)",
	"0",

	"d(arccos(x),x)+1/sqrt(1-x^2)",
	"0",

	"d(arctan(x),x)-1/(1+x^2)",
	"0",

	"d(arctan(y/x),x)",
	"-y/(x^2+y^2)",

	"d(arctan(y/x),y)",
	"x/(x^2+y^2)",

// hyp functions

	"d(sinh(x),x)-cosh(x)",
	"0",

	"d(cosh(x),x)-sinh(x)",
	"0",

	"d(tanh(x),x)-cosh(x)^(-2)",
	"0",

	"d(arcsinh(x),x)-1/sqrt(x^2+1)",
	"0",

	"d(arccosh(x),x)-1/sqrt(x^2-1)",
	"0",

	"d(arctanh(x),x)-1/(1-x^2)",
	"0",

	"d(sin(cos(x)),x)+cos(cos(x))*sin(x)",
	"0",

	"d(sin(x)^2,x)-2*sin(x)*cos(x)",
	"0",

	"d(sin(cos(x)),cos(x))-cos(cos(x))",
	"0",

	"d(abs(x),x)",
	"sgn(x)",
	
	"d(sgn(x),x)",
	"2*dirac(x)",

// generic functions

	"d(f(),x)",
	"d(f(),x)",

	"d(f(x),x)",
	"d(f(x),x)",

	"d(f(y),x)",
	"0",

	"d(g(f(x)),f(x))",
	"d(g(f(x)),f(x))",

	"d(g(f(x)),x)",
	"d(g(f(x)),x)",

// other functions

	"d(erf(x))-2*exp(-x^2)/sqrt(pi)",
	"0",

// arg lists

	"f=x^5*y^7",
	"",

	"d(f)",
	"5*x^4*y^7",

	"d(f,x)",
	"5*x^4*y^7",

	"d(f,x,0)",
	"x^5*y^7",

	"d(f,x,1)",
	"5*x^4*y^7",

	"d(f,x,2)",
	"20*x^3*y^7",

	"d(f,2)",
	"20*x^3*y^7",

	"d(f,2,y)",
	"140*x^3*y^6",

	"d(f,x,x,y,y)",
	"840*x^3*y^5",

	"f=quote(f)",
	"",
};

void
test_derivative(void)
{
	test(__FILE__, s, sizeof s / sizeof (char *));
}

#endif