- expairseq.cpp: moved expairseq::to_rational to...
[ginac.git] / ginac / inifcns_trans.cpp
index 4473fe02aff95f365bd3928e4375ad3faf6149ec..81f92a93aca908aa6f698e79dd1a4380d5998d61 100644 (file)
@@ -4,7 +4,7 @@
  *  functions. */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
 #include "power.h"
 #include "relational.h"
 #include "symbol.h"
+#include "pseries.h"
 #include "utils.h"
 
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
 
 //////////
 // exponential function
 //////////
 
-static ex exp_evalf(ex const & x)
+static ex exp_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -50,7 +51,7 @@ static ex exp_evalf(ex const & x)
     return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
 }
 
-static ex exp_eval(ex const & x)
+static ex exp_eval(const ex & x)
 {
     // exp(0) -> 1
     if (x.is_zero()) {
@@ -80,21 +81,23 @@ static ex exp_eval(ex const & x)
     return exp(x).hold();
 }
 
-static ex exp_diff(ex const & x, unsigned diff_param)
+static ex exp_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
     // d/dx exp(x) -> exp(x)
     return exp(x);
 }
 
-REGISTER_FUNCTION(exp, exp_eval, exp_evalf, exp_diff, NULL);
+REGISTER_FUNCTION(exp, eval_func(exp_eval).
+                       evalf_func(exp_evalf).
+                       derivative_func(exp_deriv));
 
 //////////
 // natural logarithm
 //////////
 
-static ex log_evalf(ex const & x)
+static ex log_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -103,48 +106,76 @@ static ex log_evalf(ex const & x)
     return log(ex_to_numeric(x)); // -> numeric log(numeric)
 }
 
-static ex log_eval(ex const & x)
+static ex log_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
+        if (x.is_equal(_ex0()))  // log(0) -> infinity
+            throw(std::domain_error("log_eval(): log(0)"));
+        if (x.info(info_flags::real) && x.info(info_flags::negative))
+            return (log(-x)+I*Pi);
         if (x.is_equal(_ex1()))  // log(1) -> 0
             return _ex0();
-        if (x.is_equal(_ex_1())) // log(-1) -> I*Pi
-            return (I*Pi);        
         if (x.is_equal(I))       // log(I) -> Pi*I/2
             return (Pi*I*_num1_2());
         if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
             return (Pi*I*_num_1_2());
-        if (x.is_equal(_ex0()))  // log(0) -> infinity
-            throw(std::domain_error("log_eval(): log(0)"));
         // log(float)
         if (!x.info(info_flags::crational))
             return log_evalf(x);
     }
-    // log(exp(t)) -> t (for real-valued t):
+    // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
     if (is_ex_the_function(x, exp)) {
-        ex t=x.op(0);
-        if (t.info(info_flags::real))
-            return t;
+        ex t = x.op(0);
+        if (t.info(info_flags::numeric)) {
+            numeric nt = ex_to_numeric(t);
+            if (nt.is_real())
+                return t;
+        }
     }
     
     return log(x).hold();
 }
 
-static ex log_diff(ex const & x, unsigned diff_param)
+static ex log_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
-
+    GINAC_ASSERT(deriv_param==0);
+    
     // d/dx log(x) -> 1/x
-    return power(x, -1);
+    return power(x, _ex_1());
 }
 
-REGISTER_FUNCTION(log, log_eval, log_evalf, log_diff, NULL);
+static ex log_series(const ex &x, const relational &rel, int order)
+{
+    const ex x_pt = x.subs(rel);
+    if (!x_pt.info(info_flags::negative) && !x_pt.is_zero())
+        throw do_taylor();  // caught by function::series()
+    // now we either have to care for the branch cut or the branch point:
+    if (x_pt.is_zero()) {  // branch point: return a plain log(x).
+        epvector seq;
+        seq.push_back(expair(log(x), _ex0()));
+        return pseries(rel, seq);
+    } // on the branch cut:
+    const ex point = rel.rhs();
+    const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+    const symbol foo;
+    // compute the formal series:
+    ex replx = series(log(x),*s==foo,order).subs(foo==point);
+    epvector seq;
+    seq.push_back(expair(-I*csgn(x*I)*Pi,_ex0()));
+    seq.push_back(expair(Order(_ex1()),order));
+    return series(replx - I*Pi + pseries(rel, seq),rel,order);
+}
+
+REGISTER_FUNCTION(log, eval_func(log_eval).
+                       evalf_func(log_evalf).
+                       derivative_func(log_deriv).
+                       series_func(log_series));
 
 //////////
 // sine (trigonometric function)
 //////////
 
-static ex sin_evalf(ex const & x)
+static ex sin_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -153,40 +184,44 @@ static ex sin_evalf(ex const & x)
     return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
 }
 
-static ex sin_eval(ex const & x)
-{
-    // sin(n*Pi/6) -> { 0 | +/-1/2 | +/-sqrt(3)/2 | +/-1 }
-    ex SixExOverPi = _ex6()*x/Pi;
-    if (SixExOverPi.info(info_flags::integer)) {
-        numeric z = smod(ex_to_numeric(SixExOverPi),_num12());
-        if (z.is_equal(_num_5()))  // sin(7*Pi/6)  -> -1/2
-            return _ex_1_2();
-        if (z.is_equal(_num_4()))  // sin(8*Pi/6)  -> -sqrt(3)/2
-            return _ex_1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num_3()))  // sin(9*Pi/6)  -> -1
-            return _ex_1();
-        if (z.is_equal(_num_2()))  // sin(10*Pi/6) -> -sqrt(3)/2
-            return _ex_1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num_1()))  // sin(11*Pi/6) -> -1/2
-            return _ex_1_2();
-        if (z.is_equal(_num0()))   // sin(0)       ->  0
-            return _ex0();
-        if (z.is_equal(_num1()))   // sin(1*Pi/6)  ->  1/2
-            return _ex1_2();
-        if (z.is_equal(_num2()))   // sin(2*Pi/6)  ->  sqrt(3)/2
-            return _ex1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num3()))   // sin(3*Pi/6)  ->  1
-            return _ex1();
-        if (z.is_equal(_num4()))   // sin(4*Pi/6)  ->  sqrt(3)/2
-            return _ex1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num5()))   // sin(5*Pi/6)  ->  1/2
-            return _ex1_2();
-        if (z.is_equal(_num6()))   // sin(6*Pi/6)  ->  0
+static ex sin_eval(const ex & x)
+{
+    // sin(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z -= _num60();
+            sign = _ex_1();
+        }
+        if (z>_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+        }
+        if (z.is_equal(_num0()))  // sin(0)       -> 0
             return _ex0();
+        if (z.is_equal(_num5()))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num6()))  // sin(Pi/10)   -> sqrt(5)/4-1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+        if (z.is_equal(_num10())) // sin(Pi/6)    -> 1/2
+            return sign*_ex1_2();
+        if (z.is_equal(_num15())) // sin(Pi/4)    -> sqrt(2)/2
+            return sign*_ex1_2()*power(_ex2(),_ex1_2());
+        if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+        if (z.is_equal(_num20())) // sin(Pi/3)    -> sqrt(3)/2
+            return sign*_ex1_2()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num30())) // sin(Pi/2)    -> 1
+            return sign*_ex1();
     }
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // sin(asin(x)) -> x
         if (is_ex_the_function(x, asin))
             return t;
@@ -205,21 +240,23 @@ static ex sin_eval(ex const & x)
     return sin(x).hold();
 }
 
-static ex sin_diff(ex const & x, unsigned diff_param)
+static ex sin_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx sin(x) -> cos(x)
     return cos(x);
 }
 
-REGISTER_FUNCTION(sin, sin_eval, sin_evalf, sin_diff, NULL);
+REGISTER_FUNCTION(sin, eval_func(sin_eval).
+                       evalf_func(sin_evalf).
+                       derivative_func(sin_deriv));
 
 //////////
 // cosine (trigonometric function)
 //////////
 
-static ex cos_evalf(ex const & x)
+static ex cos_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -228,40 +265,44 @@ static ex cos_evalf(ex const & x)
     return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
 }
 
-static ex cos_eval(ex const & x)
-{
-    // cos(n*Pi/6) -> { 0 | +/-1/2 | +/-sqrt(3)/2 | +/-1 }
-    ex SixExOverPi = _ex6()*x/Pi;
-    if (SixExOverPi.info(info_flags::integer)) {
-        numeric z = smod(ex_to_numeric(SixExOverPi),_num12());
-        if (z.is_equal(_num_5()))  // cos(7*Pi/6)  -> -sqrt(3)/2
-            return _ex_1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num_4()))  // cos(8*Pi/6)  -> -1/2
-            return _ex_1_2();
-        if (z.is_equal(_num_3()))  // cos(9*Pi/6)  ->  0
-            return _ex0();
-        if (z.is_equal(_num_2()))  // cos(10*Pi/6) ->  1/2
-            return _ex1_2();
-        if (z.is_equal(_num_1()))  // cos(11*Pi/6) ->  sqrt(3)/2
-            return _ex1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num0()))   // cos(0)       ->  1
-            return _ex1();
-        if (z.is_equal(_num1()))   // cos(1*Pi/6)  ->  sqrt(3)/2
-            return _ex1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num2()))   // cos(2*Pi/6)  ->  1/2
-            return _ex1_2();
-        if (z.is_equal(_num3()))   // cos(3*Pi/6)  ->  0
-            return _ex0();
-        if (z.is_equal(_num4()))   // cos(4*Pi/6)  -> -1/2
-            return _ex_1_2();
-        if (z.is_equal(_num5()))   // cos(5*Pi/6)  -> -sqrt(3)/2
-            return _ex_1_2()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num6()))   // cos(6*Pi/6)  -> -1
-            return _ex_1();
+static ex cos_eval(const ex & x)
+{
+    // cos(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z = _num120()-z;
+        }
+        if (z>=_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+            sign = _ex_1();
+        }
+        if (z.is_equal(_num0()))  // cos(0)       -> 1
+            return sign*_ex1();
+        if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
+            return sign*_ex1_2()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+        if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
+            return sign*_ex1_2()*power(_ex2(),_ex1_2());
+        if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
+            return sign*_ex1_2();
+        if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+        if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
+            return sign*_ex0();
     }
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // cos(acos(x)) -> x
         if (is_ex_the_function(x, acos))
             return t;
@@ -280,21 +321,23 @@ static ex cos_eval(ex const & x)
     return cos(x).hold();
 }
 
-static ex cos_diff(ex const & x, unsigned diff_param)
+static ex cos_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
     // d/dx cos(x) -> -sin(x)
     return _ex_1()*sin(x);
 }
 
-REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL);
+REGISTER_FUNCTION(cos, eval_func(cos_eval).
+                       evalf_func(cos_evalf).
+                       derivative_func(cos_deriv));
 
 //////////
 // tangent (trigonometric function)
 //////////
 
-static ex tan_evalf(ex const & x)
+static ex tan_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -303,28 +346,40 @@ static ex tan_evalf(ex const & x)
     return tan(ex_to_numeric(x));
 }
 
-static ex tan_eval(ex const & x)
-{
-    // tan(n*Pi/6) -> { 0 | +/-sqrt(3) | +/-sqrt(3)/2 }
-    ex SixExOverPi = _ex6()*x/Pi;
-    if (SixExOverPi.info(info_flags::integer)) {
-        numeric z = smod(ex_to_numeric(SixExOverPi),_num6());
-        if (z.is_equal(_num_2()))  // tan(4*Pi/6) -> -sqrt(3)
-            return _ex_1()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num_1()))  // tan(5*Pi/6) -> -sqrt(3)/3
-            return _ex_1_3()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num0()))   // tan(0)      ->  0
+static ex tan_eval(const ex & x)
+{
+    // tan(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z -= _num60();
+        }
+        if (z>=_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+            sign = _ex_1();
+        }
+        if (z.is_equal(_num0()))  // tan(0)       -> 0
             return _ex0();
-        if (z.is_equal(_num1()))   // tan(1*Pi/6) ->  sqrt(3)/3
-            return _ex1_3()*power(_ex3(),_ex1_2());
-        if (z.is_equal(_num2()))   // tan(2*Pi/6) ->  sqrt(3)
-            return power(_ex3(),_ex1_2());
-        if (z.is_equal(_num3()))   // tan(3*Pi/6) ->  infinity
-            throw (std::domain_error("tan_eval(): infinity"));
+        if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
+            return sign*(_ex2()-power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
+            return sign*_ex1_3()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
+            return sign*_ex1();
+        if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
+            return sign*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
+            return sign*(power(_ex3(),_ex1_2())+_ex2());
+        if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
+            throw (std::domain_error("tan_eval(): simple pole"));
     }
-        
+    
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // tan(atan(x)) -> x
         if (is_ex_the_function(x, atan))
             return t;
@@ -344,33 +399,36 @@ static ex tan_eval(ex const & x)
     return tan(x).hold();
 }
 
-static ex tan_diff(ex const & x, unsigned diff_param)
+static ex tan_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx tan(x) -> 1+tan(x)^2;
-    return (1+power(tan(x),_ex2()));
+    return (_ex1()+power(tan(x),_ex2()));
 }
 
-static ex tan_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex tan_series(const ex &x, const relational &rel, int order)
 {
     // method:
-    // Taylor series where there is no pole falls back to tan_diff.
+    // Taylor series where there is no pole falls back to tan_deriv.
     // On a pole simply expand sin(x)/cos(x).
-    ex xpoint = x.subs(s==point);
-    if (!(2*xpoint/Pi).info(info_flags::odd))
+    const ex x_pt = x.subs(rel);
+    if (!(2*x_pt/Pi).info(info_flags::odd))
         throw do_taylor();  // caught by function::series()
     // if we got here we have to care for a simple pole
-    return (sin(x)/cos(x)).series(s, point, order+2);
+    return (sin(x)/cos(x)).series(rel, order+2);
 }
 
-REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, tan_series);
+REGISTER_FUNCTION(tan, eval_func(tan_eval).
+                       evalf_func(tan_evalf).
+                       derivative_func(tan_deriv).
+                       series_func(tan_series));
 
 //////////
 // inverse sine (arc sine)
 //////////
 
-static ex asin_evalf(ex const & x)
+static ex asin_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -379,7 +437,7 @@ static ex asin_evalf(ex const & x)
     return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
 }
 
-static ex asin_eval(ex const & x)
+static ex asin_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // asin(0) -> 0
@@ -405,21 +463,23 @@ static ex asin_eval(ex const & x)
     return asin(x).hold();
 }
 
-static ex asin_diff(ex const & x, unsigned diff_param)
+static ex asin_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx asin(x) -> 1/sqrt(1-x^2)
     return power(1-power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(asin, asin_eval, asin_evalf, asin_diff, NULL);
+REGISTER_FUNCTION(asin, eval_func(asin_eval).
+                        evalf_func(asin_evalf).
+                        derivative_func(asin_deriv));
 
 //////////
 // inverse cosine (arc cosine)
 //////////
 
-static ex acos_evalf(ex const & x)
+static ex acos_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -428,7 +488,7 @@ static ex acos_evalf(ex const & x)
     return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
 }
 
-static ex acos_eval(ex const & x)
+static ex acos_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // acos(1) -> 0
@@ -436,10 +496,10 @@ static ex acos_eval(ex const & x)
             return _ex0();
         // acos(1/2) -> Pi/3
         if (x.is_equal(_ex1_2()))
-            return numeric(1,3)*Pi;
+            return _ex1_3()*Pi;
         // acos(0) -> Pi/2
         if (x.is_zero())
-            return numeric(1,2)*Pi;
+            return _ex1_2()*Pi;
         // acos(-1/2) -> 2/3*Pi
         if (x.is_equal(_ex_1_2()))
             return numeric(2,3)*Pi;
@@ -454,21 +514,23 @@ static ex acos_eval(ex const & x)
     return acos(x).hold();
 }
 
-static ex acos_diff(ex const & x, unsigned diff_param)
+static ex acos_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx acos(x) -> -1/sqrt(1-x^2)
     return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(acos, acos_eval, acos_evalf, acos_diff, NULL);
+REGISTER_FUNCTION(acos, eval_func(acos_eval).
+                        evalf_func(acos_evalf).
+                        derivative_func(acos_deriv));
 
 //////////
 // inverse tangent (arc tangent)
 //////////
 
-static ex atan_evalf(ex const & x)
+static ex atan_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -477,7 +539,7 @@ static ex atan_evalf(ex const & x)
     return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
 }
 
-static ex atan_eval(ex const & x)
+static ex atan_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // atan(0) -> 0
@@ -491,20 +553,23 @@ static ex atan_eval(ex const & x)
     return atan(x).hold();
 }    
 
-static ex atan_diff(ex const & x, unsigned diff_param)
+static ex atan_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
-    return power(1+x*x, -1);
+    // d/dx atan(x) -> 1/(1+x^2)
+    return power(_ex1()+power(x,_ex2()), _ex_1());
 }
 
-REGISTER_FUNCTION(atan, atan_eval, atan_evalf, atan_diff, NULL);
+REGISTER_FUNCTION(atan, eval_func(atan_eval).
+                        evalf_func(atan_evalf).
+                        derivative_func(atan_deriv));
 
 //////////
 // inverse tangent (atan2(y,x))
 //////////
 
-static ex atan2_evalf(ex const & y, ex const & x)
+static ex atan2_evalf(const ex & y, const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(y,numeric)
@@ -514,7 +579,7 @@ static ex atan2_evalf(ex const & y, ex const & x)
     return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
 }
 
-static ex atan2_eval(ex const & y, ex const & x)
+static ex atan2_eval(const ex & y, const ex & x)
 {
     if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
         x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
@@ -524,25 +589,27 @@ static ex atan2_eval(ex const & y, ex const & x)
     return atan2(y,x).hold();
 }    
 
-static ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
+static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param<2);
+    GINAC_ASSERT(deriv_param<2);
     
-    if (diff_param==0) {
+    if (deriv_param==0) {
         // d/dy atan(y,x)
-        return x*pow(pow(x,2)+pow(y,2),-1);
+        return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
     }
     // d/dx atan(y,x)
-    return -y*pow(pow(x,2)+pow(y,2),-1);
+    return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 }
 
-REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL);
+REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
+                         evalf_func(atan2_evalf).
+                         derivative_func(atan2_deriv));
 
 //////////
 // hyperbolic sine (trigonometric function)
 //////////
 
-static ex sinh_evalf(ex const & x)
+static ex sinh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -551,7 +618,7 @@ static ex sinh_evalf(ex const & x)
     return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
 }
 
-static ex sinh_eval(ex const & x)
+static ex sinh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         if (x.is_zero())  // sinh(0) -> 0
@@ -565,7 +632,7 @@ static ex sinh_eval(ex const & x)
         return I*sin(x/I);
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // sinh(asinh(x)) -> x
         if (is_ex_the_function(x, asinh))
             return t;
@@ -580,21 +647,23 @@ static ex sinh_eval(ex const & x)
     return sinh(x).hold();
 }
 
-static ex sinh_diff(ex const & x, unsigned diff_param)
+static ex sinh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx sinh(x) -> cosh(x)
     return cosh(x);
 }
 
-REGISTER_FUNCTION(sinh, sinh_eval, sinh_evalf, sinh_diff, NULL);
+REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
+                        evalf_func(sinh_evalf).
+                        derivative_func(sinh_deriv));
 
 //////////
 // hyperbolic cosine (trigonometric function)
 //////////
 
-static ex cosh_evalf(ex const & x)
+static ex cosh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -603,7 +672,7 @@ static ex cosh_evalf(ex const & x)
     return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
 }
 
-static ex cosh_eval(ex const & x)
+static ex cosh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         if (x.is_zero())  // cosh(0) -> 1
@@ -617,7 +686,7 @@ static ex cosh_eval(ex const & x)
         return cos(x/I);
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // cosh(acosh(x)) -> x
         if (is_ex_the_function(x, acosh))
             return t;
@@ -632,21 +701,24 @@ static ex cosh_eval(ex const & x)
     return cosh(x).hold();
 }
 
-static ex cosh_diff(ex const & x, unsigned diff_param)
+static ex cosh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx cosh(x) -> sinh(x)
     return sinh(x);
 }
 
-REGISTER_FUNCTION(cosh, cosh_eval, cosh_evalf, cosh_diff, NULL);
+REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
+                        evalf_func(cosh_evalf).
+                        derivative_func(cosh_deriv));
+
 
 //////////
 // hyperbolic tangent (trigonometric function)
 //////////
 
-static ex tanh_evalf(ex const & x)
+static ex tanh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -655,7 +727,7 @@ static ex tanh_evalf(ex const & x)
     return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
 }
 
-static ex tanh_eval(ex const & x)
+static ex tanh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         if (x.is_zero())  // tanh(0) -> 0
@@ -669,7 +741,7 @@ static ex tanh_eval(ex const & x)
         return I*tan(x/I);
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // tanh(atanh(x)) -> x
         if (is_ex_the_function(x, atanh))
             return t;
@@ -684,33 +756,36 @@ static ex tanh_eval(ex const & x)
     return tanh(x).hold();
 }
 
-static ex tanh_diff(ex const & x, unsigned diff_param)
+static ex tanh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx tanh(x) -> 1-tanh(x)^2
     return _ex1()-power(tanh(x),_ex2());
 }
 
-static ex tanh_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex tanh_series(const ex &x, const relational &rel, int order)
 {
     // method:
-    // Taylor series where there is no pole falls back to tanh_diff.
+    // Taylor series where there is no pole falls back to tanh_deriv.
     // On a pole simply expand sinh(x)/cosh(x).
-    ex xpoint = x.subs(s==point);
-    if (!(2*I*xpoint/Pi).info(info_flags::odd))
+    const ex x_pt = x.subs(rel);
+    if (!(2*I*x_pt/Pi).info(info_flags::odd))
         throw do_taylor();  // caught by function::series()
     // if we got here we have to care for a simple pole
-    return (sinh(x)/cosh(x)).series(s, point, order+2);
+    return (sinh(x)/cosh(x)).series(rel, order+2);
 }
 
-REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, tanh_series);
+REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
+                        evalf_func(tanh_evalf).
+                        derivative_func(tanh_deriv).
+                        series_func(tanh_series));
 
 //////////
 // inverse hyperbolic sine (trigonometric function)
 //////////
 
-static ex asinh_evalf(ex const & x)
+static ex asinh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -719,7 +794,7 @@ static ex asinh_evalf(ex const & x)
     return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
 }
 
-static ex asinh_eval(ex const & x)
+static ex asinh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // asinh(0) -> 0
@@ -733,21 +808,23 @@ static ex asinh_eval(ex const & x)
     return asinh(x).hold();
 }
 
-static ex asinh_diff(ex const & x, unsigned diff_param)
+static ex asinh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx asinh(x) -> 1/sqrt(1+x^2)
-    return power(1+power(x,_ex2()),_ex_1_2());
+    return power(_ex1()+power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL);
+REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
+                         evalf_func(asinh_evalf).
+                         derivative_func(asinh_deriv));
 
 //////////
 // inverse hyperbolic cosine (trigonometric function)
 //////////
 
-static ex acosh_evalf(ex const & x)
+static ex acosh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -756,7 +833,7 @@ static ex acosh_evalf(ex const & x)
     return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
 }
 
-static ex acosh_eval(ex const & x)
+static ex acosh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // acosh(0) -> Pi*I/2
@@ -776,21 +853,23 @@ static ex acosh_eval(ex const & x)
     return acosh(x).hold();
 }
 
-static ex acosh_diff(ex const & x, unsigned diff_param)
+static ex acosh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
     return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
 }
 
-REGISTER_FUNCTION(acosh, acosh_eval, acosh_evalf, acosh_diff, NULL);
+REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
+                         evalf_func(acosh_evalf).
+                         derivative_func(acosh_deriv));
 
 //////////
 // inverse hyperbolic tangent (trigonometric function)
 //////////
 
-static ex atanh_evalf(ex const & x)
+static ex atanh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -799,15 +878,15 @@ static ex atanh_evalf(ex const & x)
     return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
 }
 
-static ex atanh_eval(ex const & x)
+static ex atanh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // atanh(0) -> 0
         if (x.is_zero())
             return _ex0();
         // atanh({+|-}1) -> throw
-        if (x.is_equal(_ex1()) || x.is_equal(_ex1()))
-            throw (std::domain_error("atanh_eval(): infinity"));
+        if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
+            throw (std::domain_error("atanh_eval(): logarithmic pole"));
         // atanh(float) -> float
         if (!x.info(info_flags::crational))
             return atanh_evalf(x);
@@ -816,16 +895,18 @@ static ex atanh_eval(ex const & x)
     return atanh(x).hold();
 }
 
-static ex atanh_diff(ex const & x, unsigned diff_param)
+static ex atanh_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
     // d/dx atanh(x) -> 1/(1-x^2)
     return power(_ex1()-power(x,_ex2()),_ex_1());
 }
 
-REGISTER_FUNCTION(atanh, atanh_eval, atanh_evalf, atanh_diff, NULL);
+REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
+                         evalf_func(atanh_evalf).
+                         derivative_func(atanh_deriv));
 
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC