The paper somehow seems to be missing the most interesting part, i.e. the optimal constructions of functions from eml in a readable format.

Here is my attempt. I think they should be optimal up to around 15 eml.nodrs, the latter might not be:

# 0

1=1

# 1

exp(x)=eml(x,1)

e-ln(x)=eml(1,x)

e=exp(1)

# 2

e-x=e-ln(exp(x))

# 3

0=e-e

ln(x)=e-(e-ln(x))

exp(x)-exp(y)=eml(x,exp(exp(y)))

# 4

id(x)=e-(e-x)

inf=e-ln(0)

x-ln(y)=eml(ln(x),y)

# 5

x-y=x-ln(exp(y))

-inf=e-ln(inf)

# 6

-ln(x)=eml(-inf,x)

ln(ln(x))=ln(ln(x))

# 7

-x=-ln(exp(x))

-1=-1

x^-1=exp(-ln(x))

ln(x)+ln(y)=e-((e-ln(x))-ln(y))

ln(x)-ln(y)=ln(x)-ln(y) # using x - ln(y)

# 8

xy=exp(ln(x)+ln(y))

x/y=exp(ln(x)-ln(y))

# 9

x + y = ln(exp(x))+ln(exp(y))

2 = 1+1

# 10

ipi = ln(-1)

# 13

-ipi=-ln(-1)

x^y = exp(ln(x)y)

# 16

1/2 = 2^-1

# 17

x/2 = x/2

x2 = x2

# 20

ln(sqrt(x)) = ln(x)/2

# 21

sqrt(x) = exp(ln(sqrt(x)))

# 25

sqrt(xy) = exp((ln(x)+ln(y))/2)

# 27

ln(i)=ln(sqrt(-1))

# 28

i = sqrt(-1)

-pi^2 = (ipi)(ipi)

# 31

pi^2 = (ipi)(-ipi)

# 37

exp(xi)=exp(xi)

# 44

exp(-xi)=exp(-(xi))

# 46

pi = (ipi)/i

# 90+x?

2cos(x)=exp(xi)+exp(-xi))

# 107+x?

cos(x) = (2cos(x))/2

# 118+x?

2sin(x)=(exp(x*i)-exp(-xi))/i # using exp(x)-exp(y)

# 145+x?

sin(x) = (2sin(x))/2

# 217+3x?

tan(x) = 2sin(x)/(2cos(x))