Improve recipe readability (GH-22685) (GH-22686)

Doc/library/random.rst

@@ -253,6 +253,8 @@ Functions for sequences
       order so that the sample is reproducible.
 
 
+.. _real-valued-distributions:
+
 Real-valued distributions
 -------------------------
 
@@ -516,52 +518,6 @@ Simulation of arrival times and service deliveries for a multiserver queue::
     print(f'Mean wait: {mean(waits):.1f}.  Stdev wait: {stdev(waits):.1f}.')
     print(f'Median wait: {median(waits):.1f}.  Max wait: {max(waits):.1f}.')
 
-Recipes
--------
-
-The default :func:`.random` returns multiples of 2⁻⁵³ in the range
-*0.0 ≤ x < 1.0*.  All such numbers are evenly spaced and exactly
-representable as Python floats.  However, many floats in that interval
-are not possible selections.  For example, ``0.05954861408025609``
-isn't an integer multiple of 2⁻⁵³.
-
-The following recipe takes a different approach.  All floats in the
-interval are possible selections.  Conceptually it works by choosing
-from evenly spaced multiples of 2⁻¹⁰⁷⁴ and then rounding down to the
-nearest representable float.
-
-For efficiency, the actual mechanics involve calling
-:func:`~math.ldexp` to construct a representable float.  The mantissa
-comes from a uniform distribution of integers in the range *2⁵² ≤
-mantissa < 2⁵³*.  The exponent comes from a geometric distribution
-where exponents smaller than *-53* occur half as often as the next
-larger exponent.
-
-::
-
-    from random import Random
-    from math import ldexp
-
-    class FullRandom(Random):
-
-        def random(self):
-            mantissa = 0x10_0000_0000_0000 | self.getrandbits(52)
-            exponent = -53
-            x = 0
-            while not x:
-                x = self.getrandbits(32)
-                exponent += x.bit_length() - 32
-            return ldexp(mantissa, exponent)
-
-All of the real valued distributions will use the new method::
-
-    >>> fr = FullRandom()
-    >>> fr.random()
-    0.05954861408025609
-    >>> fr.expovariate(0.25)
-    8.87925541791544
-
-
 .. seealso::
 
    `Statistics for Hackers <https://www.youtube.com/watch?v=Iq9DzN6mvYA>`_
@@ -583,6 +539,56 @@ All of the real valued distributions will use the new method::
    the basics of probability theory, how to write simulations, and
    how to perform data analysis using Python.
 
+
+Recipes
+-------
+
+The default :func:`.random` returns multiples of 2⁻⁵³ in the range
+*0.0 ≤ x < 1.0*.  All such numbers are evenly spaced and are exactly
+representable as Python floats.  However, many floats in that interval
+are not possible selections.  For example, ``0.05954861408025609``
+isn't an integer multiple of 2⁻⁵³.
+
+The following recipe takes a different approach.  All floats in the
+interval are possible selections.  The mantissa comes from a uniform
+distribution of integers in the range *2⁵² ≤ mantissa < 2⁵³*.  The
+exponent comes from a geometric distribution where exponents smaller
+than *-53* occur half as often as the next larger exponent.
+
+::
+
+    from random import Random
+    from math import ldexp
+
+    class FullRandom(Random):
+
+        def random(self):
+            mantissa = 0x10_0000_0000_0000 | self.getrandbits(52)
+            exponent = -53
+            x = 0
+            while not x:
+                x = self.getrandbits(32)
+                exponent += x.bit_length() - 32
+            return ldexp(mantissa, exponent)
+
+All :ref:`real valued distributions <real-valued-distributions>`
+in the class will use the new method::
+
+    >>> fr = FullRandom()
+    >>> fr.random()
+    0.05954861408025609
+    >>> fr.expovariate(0.25)
+    8.87925541791544
+
+The recipe is conceptually equivalent to an algorithm that chooses from
+all the multiples of 2⁻¹⁰⁷⁴ in the range *0.0 ≤ x < 1.0*.  All such
+numbers are evenly spaced, but most have to be rounded down to the
+nearest representable Python float.  (The value 2⁻¹⁰⁷⁴ is the smallest
+positive unnormalized float and is equal to ``math.ulp(0.0)``.)
+
+
+.. seealso::
+
    `Generating Pseudo-random Floating-Point Values
    <https://allendowney.com/research/rand/downey07randfloat.pdf>`_ a
    paper by Allen B. Downey describing ways to generate more