# This worksheet contains the Maple commands from Chapter 5 of
# Introduction to Scientific Programming by Joseph L. Zachary.
# 
# We set the mantissa length to 4.
# 
> Digits := 4;
# 
# (5.3)  When we try computing the distance to the horizon using the
# straightforward approach and four digits of mantissa, we get zero as
# an answer.  
# 
> sqrt((2.09e7 + 66)^2 - 2.09e7^2);
# 
# (5.4)  This shows what went wrong with example (5.3).  With only four
# digits of mantissa, adding 66 feet to the radius of the earth has no
# effect.
# 
> 2.09e7 + 66;
# 
# (5.7)  When we try computing the distance to the horizon using the
# improved approach and four digits of mantissa, we get an answer that
# is correct to four digits.
# 
> sqrt(2 * 2.09e7 * 66 + 66^2);
# 
# For the remainder of the worksheet we set the mantissa size back to
# 10.
# 
> Digits := 10;
# 
# (5.8)  This repetition of the calculation from (5.7) using a hill size
# of 100 feet helps to motivate the utility of programmer-defined
# functions.
# 
> sqrt(2 * 2.09e7 * 100 + 100^2);
# 
# (5.9)  We define a function called horizon that encapsulates the types
# of calculations done in (5.7) and (5.8).
# 
> horizon := (R, h) -> sqrt(2*R*h + h^2);
# 
# (5.10)  We use horizon to compute the distance to the horizon from the
# top of a 100 foot hill.  The answer is the same as in (5.8).
# 
> horizon(2.09e7, 100);
# 
# (5.12)  We use horizon to compute the distance to the horizon from
# Kill Devil Hill using interval arithmetic.  This is the maximum
# possible distance ...
# 
> horizon(2.095e7, 66.5);
# 
# (5.13) ... and this is the minimum possible distance.
# 
> horizon(2.085e7, 65.5);
# 
# (5.15)  We create a programmer-defined function "horizonFast" to
# calculate the distance to the horizon using an algebraically suspect
# but computationally acceptable approximation.
# 
> horizonFast := (R, h) -> sqrt(2*R*h);
# 
# (5.16)  We use horizon to compute the distance to the horizon from
# Kill Devil Hill at high tide ...
# 
> horizon(2.09e7 + 5, 66 - 5);
# 
# (5.17) ... and at low tide.
# 
> horizon(2.09e7 - 5, 66 + 5);
# 
# (5.18)  We compute the distance to the horizon from Kill Devil Hill
# when our eyepoint is three feet above the summit.
# 
> horizon(2.09e7, 69);
>