- Banned exZERO(), exONE(), exMINUSHALF() and all this from the interface.

[ginac.git] / ginac / inifcns_trans.cpp

diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp
index 40968672dcd01dfb715e9e4a08ab92f562797492..4473fe02aff95f365bd3928e4375ad3faf6149ec 100644 (file)
--- a/ginac/inifcns_trans.cpp
+++ b/ginac/inifcns_trans.cpp
@@ -29,8 +29,13 @@
 #include "constant.h"
 #include "numeric.h"
 #include "power.h"
 #include "constant.h"
 #include "numeric.h"
 #include "power.h"

+#include "relational.h"
+#include "symbol.h"
+#include "utils.h"

 
 

+#ifndef NO_GINAC_NAMESPACE

 namespace GiNaC {
 namespace GiNaC {

+#endif // ndef NO_GINAC_NAMESPACE

 
 //////////
 // exponential function
 
 //////////
 // exponential function


@@ -49,19 +54,19 @@ static ex exp_eval(ex const & x)
 {
     // exp(0) -> 1
     if (x.is_zero()) {
 {
     // exp(0) -> 1
     if (x.is_zero()) {

-        return exONE();
+        return _ex1();

     }
     // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
     }
     // exp(n*Pi*I/2) -> {+1|+I|-1|-I}

-    ex TwoExOverPiI=(2*x)/(Pi*I);
+    ex TwoExOverPiI=(_ex2()*x)/(Pi*I);

     if (TwoExOverPiI.info(info_flags::integer)) {
     if (TwoExOverPiI.info(info_flags::integer)) {

-        numeric z=mod(ex_to_numeric(TwoExOverPiI),numeric(4));
-        if (z.is_equal(numZERO()))
-            return exONE();
-        if (z.is_equal(numONE()))
+        numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
+        if (z.is_equal(_num0()))
+            return _ex1();
+        if (z.is_equal(_num1()))

             return ex(I);
             return ex(I);

-        if (z.is_equal(numTWO()))
-            return exMINUSONE();
-        if (z.is_equal(numTHREE()))
+        if (z.is_equal(_num2()))
+            return _ex_1();
+        if (z.is_equal(_num3()))

             return ex(-I);
     }
     // exp(log(x)) -> x
             return ex(-I);
     }
     // exp(log(x)) -> x


@@ -69,16 +74,17 @@ static ex exp_eval(ex const & x)
         return x.op(0);
     
     // exp(float)
         return x.op(0);
     
     // exp(float)

-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))

         return exp_evalf(x);
     
     return exp(x).hold();
         return exp_evalf(x);
     
     return exp(x).hold();

-}    
+}

 
 static ex exp_diff(ex const & x, unsigned diff_param)
 {
 
 static ex exp_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

 
 

+    // d/dx exp(x) -> exp(x)

     return exp(x);
 }
 
     return exp(x);
 }
 


@@ -100,33 +106,35 @@ static ex log_evalf(ex const & x)
 static ex log_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
 static ex log_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {

-        // log(1) -> 0
-        if (x.is_equal(exONE()))
-            return exZERO();
-        // log(-1) -> I*Pi
-        if (x.is_equal(exMINUSONE()))
-            return (I*Pi);
-        // log(I) -> Pi*I/2
-        if (x.is_equal(I))
-            return (I*Pi*numeric(1,2));
-        // log(-I) -> -Pi*I/2
-        if (x.is_equal(-I))
-            return (I*Pi*numeric(-1,2));
-        // log(0) -> throw singularity
-        if (x.is_equal(exZERO()))
+        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)
             throw(std::domain_error("log_eval(): log(0)"));
         // log(float)

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return log_evalf(x);
     }
             return log_evalf(x);
     }

+    // log(exp(t)) -> t (for real-valued t):
+    if (is_ex_the_function(x, exp)) {
+        ex t=x.op(0);
+        if (t.info(info_flags::real))
+            return t;
+    }

     
     return log(x).hold();
     
     return log(x).hold();

-}    
+}

 
 static ex log_diff(ex const & x, unsigned diff_param)
 {
 
 static ex log_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

 
 

+    // d/dx log(x) -> 1/x

     return power(x, -1);
 }
 
     return power(x, -1);
 }
 


@@ -147,33 +155,51 @@ static ex sin_evalf(ex const & x)
 
 static ex sin_eval(ex const & x)
 {
 
 static ex sin_eval(ex const & x)
 {

-    // sin(n*Pi) -> 0
-    ex xOverPi=x/Pi;
-    if (xOverPi.info(info_flags::integer))
-        return exZERO();
-    
-    // sin((2n+1)*Pi/2) -> {+|-}1
-    ex xOverPiMinusHalf=xOverPi-exHALF();
-    if (xOverPiMinusHalf.info(info_flags::even))
-        return exONE();
-    else if (xOverPiMinusHalf.info(info_flags::odd))
-        return exMINUSONE();
+    // 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
+            return _ex0();
+    }

     
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // sin(asin(x)) -> x
         if (is_ex_the_function(x, asin))
             return t;
     
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // sin(asin(x)) -> x
         if (is_ex_the_function(x, asin))
             return t;

-        // sin(acos(x)) -> (1-x^2)^(1/2)
+        // sin(acos(x)) -> sqrt(1-x^2)

         if (is_ex_the_function(x, acos))
         if (is_ex_the_function(x, acos))

-            return power(exONE()-power(t,exTWO()),exHALF());
+            return power(_ex1()-power(t,_ex2()),_ex1_2());

         // sin(atan(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))
         // sin(atan(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))

-            return t*power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()+power(t,_ex2()),_ex_1_2());

     }
     
     // sin(float) -> float
     }
     
     // sin(float) -> float

-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))

         return sin_evalf(x);
     
     return sin(x).hold();
         return sin_evalf(x);
     
     return sin(x).hold();


@@ -181,8 +207,9 @@ static ex sin_eval(ex const & x)
 
 static ex sin_diff(ex const & x, unsigned diff_param)
 {
 
 static ex sin_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

+    // d/dx sin(x) -> cos(x)

     return cos(x);
 }
 
     return cos(x);
 }
 


@@ -203,17 +230,35 @@ static ex cos_evalf(ex const & x)
 
 static ex cos_eval(ex const & x)
 {
 
 static ex cos_eval(ex const & x)
 {

-    // cos(n*Pi) -> {+|-}1
-    ex xOverPi=x/Pi;
-    if (xOverPi.info(info_flags::even))
-        return exONE();
-    else if (xOverPi.info(info_flags::odd))
-        return exMINUSONE();
-    
-    // cos((2n+1)*Pi/2) -> 0
-    ex xOverPiMinusHalf=xOverPi-exHALF();
-    if (xOverPiMinusHalf.info(info_flags::integer))
-        return exZERO();
+    // 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();
+    }

     
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
     
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);


@@ -222,14 +267,14 @@ static ex cos_eval(ex const & x)
             return t;
         // cos(asin(x)) -> (1-x^2)^(1/2)
         if (is_ex_the_function(x, asin))
             return t;
         // cos(asin(x)) -> (1-x^2)^(1/2)
         if (is_ex_the_function(x, asin))

-            return power(exONE()-power(t,exTWO()),exHALF());
+            return power(_ex1()-power(t,_ex2()),_ex1_2());

         // cos(atan(x)) -> (1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))
         // cos(atan(x)) -> (1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))

-            return power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return power(_ex1()+power(t,_ex2()),_ex_1_2());

     }
     
     // cos(float) -> float
     }
     
     // cos(float) -> float

-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))

         return cos_evalf(x);
     
     return cos(x).hold();
         return cos_evalf(x);
     
     return cos(x).hold();


@@ -237,9 +282,10 @@ static ex cos_eval(ex const & x)
 
 static ex cos_diff(ex const & x, unsigned diff_param)
 {
 
 static ex cos_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

 
 

-    return numMINUSONE()*sin(x);
+    // d/dx cos(x) -> -sin(x)
+    return _ex_1()*sin(x);

 }
 
 REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL);
 }
 
 REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL);


@@ -259,23 +305,24 @@ static ex tan_evalf(ex const & x)
 
 static ex tan_eval(ex const & x)
 {
 
 static ex tan_eval(ex const & x)
 {

-    // tan(n*Pi/3) -> {0|3^(1/2)|-(3^(1/2))}
-    ex ThreeExOverPi=numTHREE()*x/Pi;
-    if (ThreeExOverPi.info(info_flags::integer)) {
-        numeric z=mod(ex_to_numeric(ThreeExOverPi),numeric(3));
-        if (z.is_equal(numZERO()))
-            return exZERO();
-        if (z.is_equal(numONE()))
-            return power(exTHREE(),exHALF());
-        if (z.is_equal(numTWO()))
-            return -power(exTHREE(),exHALF());
+    // 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
+            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"));

     }
     }

-    
-    // tan((2n+1)*Pi/2) -> throw
-    ex ExOverPiMinusHalf=x/Pi-exHALF();
-    if (ExOverPiMinusHalf.info(info_flags::integer))
-        throw (std::domain_error("tan_eval(): infinity"));
-    
+        

     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // tan(atan(x)) -> x
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // tan(atan(x)) -> x


@@ -283,14 +330,14 @@ static ex tan_eval(ex const & x)
             return t;
         // tan(asin(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asin))
             return t;
         // tan(asin(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asin))

-            return t*power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()-power(t,_ex2()),_ex_1_2());

         // tan(acos(x)) -> (1-x^2)^(1/2)/x
         if (is_ex_the_function(x, acos))
         // tan(acos(x)) -> (1-x^2)^(1/2)/x
         if (is_ex_the_function(x, acos))

-            return power(t,exMINUSONE())*power(exONE()-power(t,exTWO()),exHALF());
+            return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());

     }
     
     // tan(float) -> float
     }
     
     // tan(float) -> float

-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {

         return tan_evalf(x);
     }
     
         return tan_evalf(x);
     }
     


@@ -299,12 +346,25 @@ static ex tan_eval(ex const & x)
 
 static ex tan_diff(ex const & x, unsigned diff_param)
 {
 
 static ex tan_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return (1+power(tan(x),exTWO()));
+    // d/dx tan(x) -> 1+tan(x)^2;
+    return (1+power(tan(x),_ex2()));
+}
+
+static ex tan_series(ex const & x, symbol const & s, ex const & point, int order)
+{
+    // method:
+    // Taylor series where there is no pole falls back to tan_diff.
+    // On a pole simply expand sin(x)/cos(x).
+    ex xpoint = x.subs(s==point);
+    if (!(2*xpoint/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);

 }
 
 }
 

-REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, NULL);
+REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, tan_series);

 
 //////////
 // inverse sine (arc sine)
 
 //////////
 // inverse sine (arc sine)


@@ -326,19 +386,19 @@ static ex asin_eval(ex const & x)
         if (x.is_zero())
             return x;
         // asin(1/2) -> Pi/6
         if (x.is_zero())
             return x;
         // asin(1/2) -> Pi/6

-        if (x.is_equal(exHALF()))
+        if (x.is_equal(_ex1_2()))

             return numeric(1,6)*Pi;
         // asin(1) -> Pi/2
             return numeric(1,6)*Pi;
         // asin(1) -> Pi/2

-        if (x.is_equal(exONE()))
-            return numeric(1,2)*Pi;
+        if (x.is_equal(_ex1()))
+            return _num1_2()*Pi;

         // asin(-1/2) -> -Pi/6
         // asin(-1/2) -> -Pi/6

-        if (x.is_equal(exMINUSHALF()))
+        if (x.is_equal(_ex_1_2()))

             return numeric(-1,6)*Pi;
         // asin(-1) -> -Pi/2
             return numeric(-1,6)*Pi;
         // asin(-1) -> -Pi/2

-        if (x.is_equal(exMINUSONE()))
-            return numeric(-1,2)*Pi;
+        if (x.is_equal(_ex_1()))
+            return _num_1_2()*Pi;

         // asin(float) -> float
         // asin(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return asin_evalf(x);
     }
     
             return asin_evalf(x);
     }
     


@@ -347,9 +407,10 @@ static ex asin_eval(ex const & x)
 
 static ex asin_diff(ex const & x, unsigned diff_param)
 {
 
 static ex asin_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return power(1-power(x,exTWO()),exMINUSHALF());
+    // 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, asin_eval, asin_evalf, asin_diff, NULL);


@@ -371,22 +432,22 @@ static ex acos_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
         // acos(1) -> 0
 {
     if (x.info(info_flags::numeric)) {
         // acos(1) -> 0

-        if (x.is_equal(exONE()))
-            return exZERO();
+        if (x.is_equal(_ex1()))
+            return _ex0();

         // acos(1/2) -> Pi/3
         // acos(1/2) -> Pi/3

-        if (x.is_equal(exHALF()))
+        if (x.is_equal(_ex1_2()))

             return numeric(1,3)*Pi;
         // acos(0) -> Pi/2
         if (x.is_zero())
             return numeric(1,2)*Pi;
         // acos(-1/2) -> 2/3*Pi
             return numeric(1,3)*Pi;
         // acos(0) -> Pi/2
         if (x.is_zero())
             return numeric(1,2)*Pi;
         // acos(-1/2) -> 2/3*Pi

-        if (x.is_equal(exMINUSHALF()))
+        if (x.is_equal(_ex_1_2()))

             return numeric(2,3)*Pi;
         // acos(-1) -> Pi
             return numeric(2,3)*Pi;
         // acos(-1) -> Pi

-        if (x.is_equal(exMINUSONE()))
+        if (x.is_equal(_ex_1()))

             return Pi;
         // acos(float) -> float
             return Pi;
         // acos(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return acos_evalf(x);
     }
     
             return acos_evalf(x);
     }
     


@@ -395,9 +456,10 @@ static ex acos_eval(ex const & x)
 
 static ex acos_diff(ex const & x, unsigned diff_param)
 {
 
 static ex acos_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return numMINUSONE()*power(1-power(x,exTWO()),exMINUSHALF());
+    // 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, acos_eval, acos_evalf, acos_diff, NULL);


@@ -419,10 +481,10 @@ static ex atan_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
         // atan(0) -> 0
 {
     if (x.info(info_flags::numeric)) {
         // atan(0) -> 0

-        if (x.is_equal(exZERO()))
-            return exZERO();
+        if (x.is_equal(_ex0()))
+            return _ex0();

         // atan(float) -> float
         // atan(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return atan_evalf(x);
     }
     
             return atan_evalf(x);
     }
     


@@ -431,7 +493,7 @@ static ex atan_eval(ex const & x)
 
 static ex atan_diff(ex const & x, unsigned diff_param)
 {
 
 static ex atan_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

 
     return power(1+x*x, -1);
 }
 
     return power(1+x*x, -1);
 }


@@ -454,8 +516,8 @@ static ex atan2_evalf(ex const & y, ex const & x)
 
 static ex atan2_eval(ex const & y, ex const & x)
 {
 
 static ex atan2_eval(ex const & y, ex const & x)
 {

-    if (y.info(info_flags::numeric) && !y.info(info_flags::rational) &&
-        x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
+    if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
+        x.info(info_flags::numeric) && !x.info(info_flags::crational)) {

         return atan2_evalf(y,x);
     }
     
         return atan2_evalf(y,x);
     }
     


@@ -464,14 +526,14 @@ static ex atan2_eval(ex const & y, ex const & x)
 
 static ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
 {
 
 static ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param<2);
-
+    GINAC_ASSERT(diff_param<2);
+    

     if (diff_param==0) {
         // d/dy atan(y,x)
     if (diff_param==0) {
         // d/dy atan(y,x)

-        return power(x*(1+y*y/(x*x)),-1);
+        return x*pow(pow(x,2)+pow(y,2),-1);

     }
     // d/dx atan(y,x)
     }
     // d/dx atan(y,x)

-    return -y*power(x*x+y*y,-1);
+    return -y*pow(pow(x,2)+pow(y,2),-1);

 }
 
 REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL);
 }
 
 REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL);


@@ -492,14 +554,16 @@ static ex sinh_evalf(ex const & x)
 static ex sinh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
 static ex sinh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {

-        // sinh(0) -> 0
-        if (x.is_zero())
-            return exZERO();
-        // sinh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // sinh(0) -> 0
+            return _ex0();        
+        if (!x.info(info_flags::crational))  // sinh(float) -> float

             return sinh_evalf(x);
     }
     
             return sinh_evalf(x);
     }
     

+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
+        return I*sin(x/I);
+    

     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // sinh(asinh(x)) -> x
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // sinh(asinh(x)) -> x


@@ -507,10 +571,10 @@ static ex sinh_eval(ex const & x)
             return t;
         // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
         if (is_ex_the_function(x, acosh))
             return t;
         // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
         if (is_ex_the_function(x, acosh))

-            return power(t-exONE(),exHALF())*power(t+exONE(),exHALF());
+            return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());

         // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))
         // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))

-            return t*power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()-power(t,_ex2()),_ex_1_2());

     }
     
     return sinh(x).hold();
     }
     
     return sinh(x).hold();


@@ -518,8 +582,9 @@ static ex sinh_eval(ex const & x)
 
 static ex sinh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex sinh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

+    // d/dx sinh(x) -> cosh(x)

     return cosh(x);
 }
 
     return cosh(x);
 }
 


@@ -541,14 +606,16 @@ static ex cosh_evalf(ex const & x)
 static ex cosh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
 static ex cosh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {

-        // cosh(0) -> 1
-        if (x.is_zero())
-            return exONE();
-        // cosh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // cosh(0) -> 1
+            return _ex1();
+        if (!x.info(info_flags::crational))  // cosh(float) -> float

             return cosh_evalf(x);
     }
     
             return cosh_evalf(x);
     }
     

+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
+        return cos(x/I);
+    

     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // cosh(acosh(x)) -> x
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // cosh(acosh(x)) -> x


@@ -556,10 +623,10 @@ static ex cosh_eval(ex const & x)
             return t;
         // cosh(asinh(x)) -> (1+x^2)^(1/2)
         if (is_ex_the_function(x, asinh))
             return t;
         // cosh(asinh(x)) -> (1+x^2)^(1/2)
         if (is_ex_the_function(x, asinh))

-            return power(exONE()+power(t,exTWO()),exHALF());
+            return power(_ex1()+power(t,_ex2()),_ex1_2());

         // cosh(atanh(x)) -> (1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))
         // cosh(atanh(x)) -> (1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))

-            return power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return power(_ex1()-power(t,_ex2()),_ex_1_2());

     }
     
     return cosh(x).hold();
     }
     
     return cosh(x).hold();


@@ -567,8 +634,9 @@ static ex cosh_eval(ex const & x)
 
 static ex cosh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex cosh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

+    // d/dx cosh(x) -> sinh(x)

     return sinh(x);
 }
 
     return sinh(x);
 }
 


@@ -590,14 +658,16 @@ static ex tanh_evalf(ex const & x)
 static ex tanh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {
 static ex tanh_eval(ex const & x)
 {
     if (x.info(info_flags::numeric)) {

-        // tanh(0) -> 0
-        if (x.is_zero())
-            return exZERO();
-        // tanh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // tanh(0) -> 0
+            return _ex0();
+        if (!x.info(info_flags::crational))  // tanh(float) -> float

             return tanh_evalf(x);
     }
     
             return tanh_evalf(x);
     }
     

+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
+        return I*tan(x/I);
+    

     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // tanh(atanh(x)) -> x
     if (is_ex_exactly_of_type(x, function)) {
         ex t=x.op(0);
         // tanh(atanh(x)) -> x


@@ -605,10 +675,10 @@ static ex tanh_eval(ex const & x)
             return t;
         // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asinh))
             return t;
         // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asinh))

-            return t*power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()+power(t,_ex2()),_ex_1_2());

         // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
         if (is_ex_the_function(x, acosh))
         // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
         if (is_ex_the_function(x, acosh))

-            return power(t-exONE(),exHALF())*power(t+exONE(),exHALF())*power(t,exMINUSONE());
+            return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());

     }
     
     return tanh(x).hold();
     }
     
     return tanh(x).hold();


@@ -616,12 +686,25 @@ static ex tanh_eval(ex const & x)
 
 static ex tanh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex tanh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return exONE()-power(tanh(x),exTWO());
+    // 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)
+{
+    // method:
+    // Taylor series where there is no pole falls back to tanh_diff.
+    // On a pole simply expand sinh(x)/cosh(x).
+    ex xpoint = x.subs(s==point);
+    if (!(2*I*xpoint/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);

 }
 
 }
 

-REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, NULL);
+REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, tanh_series);

 
 //////////
 // inverse hyperbolic sine (trigonometric function)
 
 //////////
 // inverse hyperbolic sine (trigonometric function)


@@ -641,9 +724,9 @@ static ex asinh_eval(ex const & x)
     if (x.info(info_flags::numeric)) {
         // asinh(0) -> 0
         if (x.is_zero())
     if (x.info(info_flags::numeric)) {
         // asinh(0) -> 0
         if (x.is_zero())

-            return exZERO();
+            return _ex0();

         // asinh(float) -> float
         // asinh(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return asinh_evalf(x);
     }
     
             return asinh_evalf(x);
     }
     


@@ -652,9 +735,10 @@ static ex asinh_eval(ex const & x)
 
 static ex asinh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex asinh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return power(1+power(x,exTWO()),exMINUSHALF());
+    // d/dx asinh(x) -> 1/sqrt(1+x^2)
+    return power(1+power(x,_ex2()),_ex_1_2());

 }
 
 REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL);
 }
 
 REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL);


@@ -679,13 +763,13 @@ static ex acosh_eval(ex const & x)
         if (x.is_zero())
             return Pi*I*numeric(1,2);
         // acosh(1) -> 0
         if (x.is_zero())
             return Pi*I*numeric(1,2);
         // acosh(1) -> 0

-        if (x.is_equal(exONE()))
-            return exZERO();
+        if (x.is_equal(_ex1()))
+            return _ex0();

         // acosh(-1) -> Pi*I
         // acosh(-1) -> Pi*I

-        if (x.is_equal(exMINUSONE()))
+        if (x.is_equal(_ex_1()))

             return Pi*I;
         // acosh(float) -> float
             return Pi*I;
         // acosh(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return acosh_evalf(x);
     }
     
             return acosh_evalf(x);
     }
     


@@ -694,9 +778,10 @@ static ex acosh_eval(ex const & x)
 
 static ex acosh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex acosh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return power(x-1,exMINUSHALF())*power(x+1,exMINUSHALF());
+    // 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, acosh_eval, acosh_evalf, acosh_diff, NULL);


@@ -719,12 +804,12 @@ static ex atanh_eval(ex const & x)
     if (x.info(info_flags::numeric)) {
         // atanh(0) -> 0
         if (x.is_zero())
     if (x.info(info_flags::numeric)) {
         // atanh(0) -> 0
         if (x.is_zero())

-            return exZERO();
+            return _ex0();

         // atanh({+|-}1) -> throw
         // atanh({+|-}1) -> throw

-        if (x.is_equal(exONE()) || x.is_equal(exONE()))
+        if (x.is_equal(_ex1()) || x.is_equal(_ex1()))

             throw (std::domain_error("atanh_eval(): infinity"));
         // atanh(float) -> float
             throw (std::domain_error("atanh_eval(): infinity"));
         // atanh(float) -> float

-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))

             return atanh_evalf(x);
     }
     
             return atanh_evalf(x);
     }
     


@@ -733,11 +818,14 @@ static ex atanh_eval(ex const & x)
 
 static ex atanh_diff(ex const & x, unsigned diff_param)
 {
 
 static ex atanh_diff(ex const & x, unsigned diff_param)
 {

-    ASSERT(diff_param==0);
+    GINAC_ASSERT(diff_param==0);

     
     

-    return power(exONE()-power(x,exTWO()),exMINUSONE());
+    // 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, atanh_eval, atanh_evalf, atanh_diff, NULL);
 

+#ifndef NO_GINAC_NAMESPACE

 } // namespace GiNaC
 } // namespace GiNaC

+#endif // ndef NO_GINAC_NAMESPACE