bc_libs/cf_engel.bc

#!/usr/local/bin/bc -l funcs.bc cf.bc

### CF_ENGEL.BC - Engel expansion experimentation using array pass by reference

# Workhorse: Create an Engel expansion of x in pbr array en__[]
define engel_new_(mode) {
  # expects *en__[] and x to be declared elsewhere
  auto i,fc,max,q,p
  
  # error checking
  .=check_cf_max_()
  p=(mode<1&&abs(x)>=1);
  q=(scale<6);
  if(p||q){print "engel";if(mode==-1)print "fall";if(mode==0)print "alt";print "_new: "}
  if(p){print "Can't work with integer part.\n"; x-=int(x)}
  if(q){print "scale is ",scale,". Poor results likely.\n"}
  
  max=A^scale
  i=fc=0;p=1
  for(i=0;x&&p<max&&i<cf_max;i++){
    q=1/abs(x)
    if(mode==1){
      q=ceil(q) # proper engel expansion
    } else if(mode==-1){
      q=floor(q) # secondary engel expansion
    } else if(fc=!fc){q=floor(q)}else{q=ceil(q)} # tertiary engel expansion
    p*=q
    en__[i]=q*=sgn(x)
    x=x*q-1
  }
  if(p>=max){
    if(!--i)i=1
    max=A^int(scale/2)
    if(abs(en__[i])>max){
      en__[i]=0;return 0
    }else{
      en__[i]=0.0
      if(i-=2<3)return 0;
      q=en__[i]
      if(en__[--i]==q){
        while(en__[i--]==q){}
        en__[i+3]=0
        .=en__[i+2]--
      }
    }
  }else{
    en__[i]=0
  }
  return 0;
}

# Create Engel expansion of x in pbr array en__[]
# . all terms same sign as x
define engel_new(*en__[],x) {
  return x+engel_new_(1);
}

# Create secondary Engel expansion of x in pbr array en__[]
# . first term same sign as x, all other terms opposite sign
# . terms in implied Egyptian fraction sum have alternating signs
define engelfall_new(*en__[],x) {
  return x+engel_new_(-1);
}

# Create tertiary Engel expansion of x in pbr array en__[]
# . first term same sign as x, following terms alternate in sign
# . terms in implied Egyptian fraction alternate pairwise e.g. +--++--++...
define engelalt_new(*en__[],x) {
  return x+engel_new_(0);
}

# Output pbv array en__[] formatted as an Engel expansion
define engel_print(en__[]){
  auto i;
  .=check_cf_max_();
  print "{";
  if(en__[1]==0){print en__[0],"}";return 0}
  for(i=1;en__[i];i++)print en__[i-1],", ";
  print en__[i-1];
  if(scale(en__[i]))print ",...";
  print "}";return 0;
}

# Print a number as an Engel expansion
define print_as_engel(x){
  auto en[];
  .=engel_new(en[],x)+engel_print(en[])
  return x;
}
define print_as_engelfall(x){
  auto en[];
  .=engelfall_new(en[],x)+engel_print(en[])
  return x;
}
define print_as_engelalt(x){
  auto en[];
  .=engelalt_new(en[],x)+engel_print(en[])
  return x;
}

# Turn the Engel expansion in pbv array en__[] into its value
define engel_value(en__[]) {
  auto i,p,v;
  .=check_cf_max_();
  p=1;v=0;for(i=0;i<cf_max&&p*=en__[i];i++)v+=1/p
  return v;
}