There is a lot of excellent information in this book.However, it is very dense and the reader should beware that many of the programs in the book don't work. I surmise that they are more intended to show the algorithm rather than the actual Haskell implementation.In Haskell it is particularly important to pay attention to the details, which is why I give this book only three stars.Here is an example of a program in the book that does not work.On page 82 the author Richard Bird provides this sample implementation of the floor function:floor x = searchFrom 0where searchFrom = decrease . upper . lowerlower = until (<=x) decreaseupper = until (>x) increasedecrease n = n - 1increase n = n + 1The problem with that implementation is that it does not return an Integer value; rather, it returns a decimal (Double) value. Here's an example:floor (-3.4) -- returns (-4.0)That is wrong. The specification for floor says that it "maps a value of type Float to a value of type Integer." Clearly it is not producing an Integer result.I will now explain how to revise floor so that it returns an Integer value.The heart of the problem is with the until function.Here is how Richard Bird implements it:until :: (a -> Bool) -> (a -> a) -> a -> auntil p f y = if p y then y else until p f (f y)It takes three arguments, p, f, and y. Look at the signature of that function. The type of the third argument, y, dictates the type of the other arguments and the type of the result--whatever type y is, the other arguments must be the same type and the result must be the same type.Function until is first invoked by function lower:lower = until (<=x) decrease 0Here are the three arguments provided to function until:(1) p is the partial function (<=x), where x is the input to the floor function. Suppose x is this value: x = (-3.4)(2) f is the function decrease(3) y is 0Now you may argue, "Hey, the third argument, y, is 0 and that's an Integer, so clearly the result of function until will be an Integer." However, that is not correct. The type of 0 is ambiguous, it could be an Integer or it could be a Double. This ambiguity can be seen by checking its type using WinGHCi::type 00 :: Num a => aThe class Num is high up in Haskell's type hierarchy and it represents any number (Integer, Double, etc.). Thus, we cannot determine the type of function until's result just by examining its third argument. The other arguments will determine whether the 0 is an Integer or a Double. Let's examine the first argument:p is the partial function (<=x), where x = (-3.4)p compares "some value" against (-3.4). Let's check the type of (-3.4) using WinGHCi::type (-3.4)(-3.4) :: Fractional a => aThe datatype Double falls within the class Fractional. So p compares "some value" against a Double.Fundamental rule of Haskell: you cannot compare anInteger against a Double, you can only compare aDouble against a Double.Recall that p compares "some value" against (-3.4). What is that "some value"? If we examine the body of function until we see that it is the third argument, y, and we know that y = 0. Ah, now we know how Haskell will treat 0: since the 0 is being compared against a Double value Haskell will treat the 0 as a Double. Okay, now that we know the type of y we can plug it into the type signature for function until:until :: (a -> Bool) -> (a -> a) -> Double -> aAll a's must be of the same type, so the other a's must also be Double:until :: (Double -> Bool) -> (Double -> Double) -> Double -> DoubleTherefore function until will return a Double value. For example:until (<=x) decrease 0 -- returns (0.0)The output of function until is assigned to function lower:lower = until (<=x) decreaseSo the result of function lower is a Double value.The output of lower is then input to upper and the Double datatype propagates through the entire chain of composed functions:decrease . upper . lowerThe result of function floor is therefore a Double value.Okay, so how do we get floor to return an Integer value? The key is to prevent p in function until from casting the type of y to a Double. Recall function lower:lower = until (<=x) decrease 0Notice that p is this partial function:(<=x)We must modify p to express, "Hey, compare x against an Integer value that has been cast to a Double value." That is implemented using a Lambda (anonymous) function:(\a -> (realToFrac a) <= x)Read that as: For whatever value, a, is provided convert it to a Fractional (Double) value and then compare it to x.Wow!So we are telling Haskell that the 0 is not to be treated as a Fractional (Double) value, it is to be treated as a Real (Integer) value.At last, we can implement function floor and it will return the desired Integer result:floor x = searchFrom 0where searchFrom = decrease . upper . lowerlower = until (\a -> (realToFrac a) <= x) decreaseupper = until (\a -> (realToFrac a) > x) increasedecrease n = n - 1increase n = n + 1Notice that wherever function until is called (in lower and upper), the first argument, p, is a Lambda (anonymous) function that takes its argument, a, and casts it from a Real (Integer) value to a Fractional (Double) value. Here are a couple examples of using this revised floor function:floor (-3.4) -- returns (-4)floor 3.4 -- returns 3Notice that floor now returns an Integer value, which is what we want.Here is the signature for floor:floor :: (Fractional a, Ord a, Real c) => a -> cRead as: Invoke function floor with a Fractional (Double) value and it will return a Real (Integer) value.On page 83 Richard Bird shows a second version of function floor that uses a binary search:floor x = searchFrom (-1, 1)where searchFrom = fst . middle . cross(lower, upper)lower = until (<= x) doubleupper = until (> x) doublemiddle = until done improvedone (m, n) = (m + 1 == n)improve (m, n) = if p <= x then (p, n) else (m, p)where p = (m + n) div 2That has multiple problems. First, it is syntactically not a well-formed Haskell program because the div operator (on the last line) must have back ticks ( ` ) surrounding it:where p = (m + n) `div` 2Second, the functions lower and upper invoke function until. The first argument to until must be a Lambda function as described above:lower = until (\m -> (realToFrac m) <= x) doubleupper = until (\n -> (realToFrac n) > x) doubleThird, the function improve compares p (an Integer) against x (a Double), so p must be cast to a Fractional (Double) value:improve (m, n) = if (realToFrac p) <= x then (p, n) else (m, p)where p = (m + n) div 2With those three changes the function works as desired:floor x = searchFrom (-1, 1)where searchFrom = fst . middle . cross(lower, upper)lower = until (\m -> (realToFrac m) <= x) doubleupper = until (\n -> (realToFrac n) > x) doublemiddle = until done improvedone (m, n) = (m + 1 == n)improve (m, n) = if (realToFrac p) <= x then (p, n) else (m, p)where p = (m + n) `div` 2Here are a couple examples of using the revised floor function:floor (-3.4) -- returns (-4)floor 3.4 -- returns 3Notice that floor now returns an Integer value, which is what we want.Here is the signature for floor:floor :: (Fractional a, Ord a, Real c) => a -> cRead as: Invoke function floor with a Fractional (Double) value and it will return a Real (Integer) value.