DEFINITIONS
  // LibraryReals.def : access to floating point operations
  EXTERNAL_FUNCTION_RADD == REAL*REAL --> REAL;
  RADD(x,y) == 0.0;
  EXTERNAL_FUNCTION_RSUB == REAL*REAL --> REAL;
  RSUB(x,y) == 0.0;
  EXTERNAL_FUNCTION_RMUL == REAL*REAL --> REAL;
  RMUL(x,y) == 0.0;
  EXTERNAL_FUNCTION_RDIV == REAL*REAL --> REAL;
  RDIV(x,y) == 0.0;
  
  EXTERNAL_FUNCTION_RSIN == REAL --> REAL;
  RSIN(x) == 0.0;
  EXTERNAL_FUNCTION_RCOS == REAL --> REAL;
  RCOS(x) == 0.0;
  EXTERNAL_FUNCTION_RTAN == REAL --> REAL;
  RTAN(x) == 0.0;
  EXTERNAL_FUNCTION_RCOT == REAL --> REAL;
  RCOT(x) == 0.0;
  EXTERNAL_FUNCTION_RSINH == REAL --> REAL;
  RSINH(x) == 0.0;
  EXTERNAL_FUNCTION_RCOSH == REAL --> REAL;
  RCOSH(x) == 0.0;
  EXTERNAL_FUNCTION_RTANH == REAL --> REAL;
  RTANH(x) == 0.0;
  EXTERNAL_FUNCTION_RCOTH == REAL --> REAL;
  RCOTH(x) == 0.0;
  EXTERNAL_FUNCTION_RASIN == REAL --> REAL;
  RASIN(x) == 0.0;
  EXTERNAL_FUNCTION_RACOS == REAL --> REAL;
  RACOS(x) == 0.0;
  EXTERNAL_FUNCTION_RATAN == REAL --> REAL;
  RATAN(x) == 0.0;
  EXTERNAL_FUNCTION_RACOT == REAL --> REAL;
  RACOT(x) == 0.0;
  EXTERNAL_FUNCTION_RASINH == REAL --> REAL;
  RASINH(x) == 0.0;
  EXTERNAL_FUNCTION_RACOSH == REAL --> REAL;
  RACOSH(x) == 0.0;
  EXTERNAL_FUNCTION_RATANH == REAL --> REAL;
  RATANH(x) == 0.0;
  EXTERNAL_FUNCTION_RACOTH == REAL --> REAL;
  RACOTH(x) == 0.0;
  EXTERNAL_FUNCTION_RATAN2 == REAL*REAL --> REAL;
  RATAN2(x,y) == 0.0;
  EXTERNAL_FUNCTION_RADIANS == REAL --> REAL;
  RADIANS(x) == 0.0;
  EXTERNAL_FUNCTION_DEGREE == REAL --> REAL;
  DEGREE(x) == 0.0;
  
  EXTERNAL_FUNCTION_REXP == REAL --> REAL;
  REXP(x) == 0.0;
  EXTERNAL_FUNCTION_RLOGe == REAL --> REAL;
  RLOGe(x) == 0.0;
  EXTERNAL_FUNCTION_RSQRT == REAL --> REAL;
  RSQRT(x) == 0.0;
  EXTERNAL_FUNCTION_RABS == REAL --> REAL;
  RABS(x) == 0.0;
  EXTERNAL_FUNCTION_ROUND == REAL --> INTEGER;
  ROUND(x) == 0;
  EXTERNAL_FUNCTION_RSIGN == REAL --> REAL;
  RSIGN(x) == x;
  EXTERNAL_FUNCTION_RFRACTION == REAL --> REAL;
  RFRACTION(x) == x;
  EXTERNAL_FUNCTION_RINTEGER == REAL --> REAL;
  RINTEGER(x) == x;
  
  EXTERNAL_FUNCTION_RMAX == REAL*REAL --> REAL;
  RMAX(x,y) == 0.0;
  EXTERNAL_FUNCTION_RMIN == REAL*REAL --> REAL;
  RMIN(x,y) == 0.0;
  EXTERNAL_FUNCTION_RPOW == REAL*REAL --> REAL;
  RPOW(x,y) == 0.0;
  EXTERNAL_FUNCTION_RLOG == REAL*REAL --> REAL;
  RLOG(x,y) == 0.0;
  
  EXTERNAL_FUNCTION_RPI == REAL;
  RPI == 3.14;
  EXTERNAL_FUNCTION_RZERO == REAL;
  RZERO == 0.0;
  EXTERNAL_FUNCTION_RONE == REAL;
  RONE == 1.0;
  EXTERNAL_FUNCTION_REULER == REAL;
  REULER == 2.718281828459045;
  EXTERNAL_FUNCTION_REPSILON == REAL;
  REPSILON == 5.0E-324;
  EXTERNAL_FUNCTION_RMAXFLOAT == REAL;
  RMAXFLOAT == 1.0E30; // real value on 64 bit system is 1.7976931348623157E+308;

  EXTERNAL_PREDICATE_RLEQ == REAL*REAL;
  RLEQ(x,y) == btrue;
  EXTERNAL_PREDICATE_RGEQ == REAL*REAL;
  RGEQ(x,y) == btrue;
  EXTERNAL_PREDICATE_RLT == REAL*REAL;
  RLT(x,y) == btrue;
  EXTERNAL_PREDICATE_RGT == REAL*REAL;
  RGT(x,y) == btrue;
  
  EXTERNAL_PREDICATE_REQ == REAL*REAL;
  REQ(x,y) == btrue;
  EXTERNAL_PREDICATE_RNEQ == REAL*REAL;
  RNEQ(x,y) == btrue;

  EXTERNAL_FUNCTION_RMAXIMUM == POW(REAL) --> REAL;
  RMAXIMUM(set_of_reals) == 0.0;
  EXTERNAL_FUNCTION_RMINIMUM == POW(REAL) --> REAL;
  RMINIMUM(set_of_reals) == 0.0;
  
  EXTERNAL_FUNCTION_RNEXT == REAL --> REAL;
  RNEXT(nr) == nr + 0.00001;
  EXTERNAL_FUNCTION_RPREV == REAL --> REAL;
  RPREV(nr) == nr - 0.00001;

 
 // Softmax for an array of reals:
 SOFTMAX(zz) == LET zi BE zi=zz IN %ism.(ism:dom(zi)|REXP(zi(ism)) / SIGMA(jsm).(jsm:dom(zi)|REXP(zi(jsm)))) END
