Merge pull request #81 from brocksam/add-additional-cse-benchmarks

Add additional CSE benchmarks

benchmarks/cse.py

@@ -1,9 +1,29 @@
+"""Benchmarks for common subexpression elimination (CSE)."""
+
 from functools import reduce
 from operator import add
-import sympy as sp
+
+from sympy import cse, exp, sin, symbols, tan, Matrix
+
 
 def _get_args_exprs(nexprs, nterms):
-    x = sp.symbols('x:%d' % nterms)
+    """Construct expressions with a number of symbols based on chained additions
+    and subtractions.
+
+    Examples
+    ========
+
+    >>> _get_args_exprs(1, 1)
+    ((x0,), [x0])
+    >>> _get_args_exprs(1, 3)
+    ((x0, x1, x2), [x0 - x1 + x2])
+    >>> _get_args_exprs(3, 1)
+    ((x0,), [x0, -x0, x0])
+    >>> _get_args_exprs(3, 3)
+    ((x0, x1, x2), [x0 - x1 + x2, -x0 + x1 - x2, x0 - x1 + x2])
+
+    """
+    x = symbols('x:%d' % nterms)
     exprs = [
         reduce(add, [x[j]*(-1)**(i+j) for j in range(nterms)]) for i in range(nexprs)
     ]
@@ -12,13 +32,208 @@ def _get_args_exprs(nexprs, nterms):
 
 class TimeCSE:
 
-    params = ((
-        (2, 8), (3, 8), (4, 8),
-        (2, 9), (2, 10), (2, 11)
-    ),)
+    params = (((2, 8), (3, 8), (4, 8), (2, 9), (2, 10), (2, 11), ), )
 
     def setup(self, nexprs_nterms):
         self.args, self.exprs = _get_args_exprs(*nexprs_nterms)
 
     def time_cse(self, nexprs_nterms):
-        sp.cse(self.exprs)
+        cse(self.exprs)
+
+
+class GriewankBabyExampleCSE:
+    """Simple test function with multiple repeated subexpressions.
+
+    Function is the "baby example" taken from Griewank, A., & Walther, A.
+    (2008). Evaluating derivatives: principles and techniques of algorithmic
+    differentiation. SIAM.
+
+    The baby example function expressed as a tree structure is as follows:
+
+                           (MUL)
+                      _______|______
+                     /              \
+                  (SUB)            (SUB)
+                 ___|___           __|__
+                /       \         /     \
+              (ADD)    (EXP)   (DIV)   (EXP)
+             ___|___     |      _|_      |
+            /       \    |     /   \     |
+          (SIN)   (DIV) (x2) (x1) (x2)  (x2)
+            |      _|_
+            |     /   \
+          (DIV) (x1) (x2)
+           _|_
+          /   \
+        (x1) (x2)
+
+    It involves 1 ADD, 3 DIV, 2 EXP, 1 MUL, 1 SIN, and 2 SUB operations.
+
+    When CSE is conducted on the expression, the following intermediate terms
+    are introduced:
+
+    @0: EXP[x2]
+    @1: DIV[x1, x2]
+    @2: SUB[@1, @0]
+    @3: SIN[@1]
+    @4: ADD[@2, @3]
+    @5: MUL[@2, @4]
+
+    @0, @3, and @4 are operands in only one operation each so can be collapsed.
+    @1 and @2 are operands in two operations each so are required common
+    subexpressions. @1 will become the first common subexpression (x0, x1 / x2)
+    and @2 will become the second common subexpression (x3, x0 - exp(x2)). @6 is
+    the expression itself so can used directly there resulting in the
+    substituted expression x3 * (x3 + sin(x0)).
+
+    This example highlights an important optimisation around operation ordering
+    in CSE. The @4 operation could instead be constructed as
+    ADD[@1, SUB[@3, @0]]. This is suboptimal CSE as it would require the
+    introduction of an additional unnecessary operation.
+
+    """
+
+    def setup(self):
+        """Create the required symbols (x1, x2), the expression (y), and its
+        Jacobian (G).
+
+        G is the 1x2 matrix that contains the derivatives of y with respect to
+        x1 and x2 as the two columns respectively.
+        
+        """
+        x1, x2 = symbols("x1, x2")
+        x = Matrix([x1, x2])
+        self.y = (sin(x1 / x2) + (x1 / x2) - exp(x2)) * ((x1 / x2) - exp(x2))
+        self.G = self.y.diff(x)
+
+    def test_function_cse(self):
+        """Expected result from CSE on the baby example."""
+        x0 = sym.Symbol("x0")
+        x1 = sym.Symbol("x1")
+        x2 = sym.Symbol("x2")
+        x3 = sym.Symbol("x3")
+
+        cse = [
+            (x0, x1 / x2),
+            (x3, x0 - exp(x2)),
+        ]
+        expr = [
+            x3 * (x3 + sin(x0)),
+        ]
+
+        assert cse(self.y) == (cse, expr)
+
+    def time_function_cse(self):
+        """Time CSE on the baby example."""
+        cse(self.y)
+
+    def time_jacobian_cse(self):
+        """Time CSE on the baby example's Jacobian."""
+        cse(self.G)
+
+    def time_combined_cse(self):
+        """Time simultaneous CSE on the baby example and its Jacobian."""
+        cse([self.y, self.G])
+
+
+class GriewankLighthouseExampleCSE:
+    """Simple matrix test function with multiple repeated subexpressions.
+
+    Function is the "lighthouse example" taken from Griewank, A., & Walther, A.
+    (2008). Evaluating derivatives: principles and techniques of algorithmic
+    differentiation. SIAM.
+
+    The lighthouse example function expressed as a tree structure is as follows:
+
+                               (MATRIX)
+                        __________|___________
+                       /                      \    
+                     (MUL)                  (MUL)
+                     __|__                   _|_              
+                    /     \                 /   \              
+                  (MUL) (gamma)          (DIV) (nu)      
+                   _|_               ______|_____          
+                  /   \             /            \             
+               (DIV) (nu)         (TAN)        (SUB)             
+           ______|_____             |          __|__              
+          /            \            |         /     \               
+        (TAN)        (SUB)        (MUL)   (gamma) (TAN)                
+          |          __|__         _|_              |                
+          |         /     \       /   \             |                 
+        (MUL)   (gamma) (TAN) (omega) (t)         (MUL)                
+         _|_              |                        _|_                
+        /   \             |                       /   \            
+    (omega) (t)         (MUL)                 (omega) (t)            
+                         _|_             
+                        /   \             
+                    (omega) (t)          
+
+    It involves 2 DIV, 7 MUL, 2 SUB, and 4 TAN operations. Additionally, one
+    matrix population is required.
+
+    When CSE is conducted on the expression, the following intermediate terms
+    are introduced:
+
+    @0: MUL[omega, t]
+    @1: TAN[@0]
+    @2: SUB[gamma, @1]
+    @3: DIV[@1, @2]
+    @4: MUL[nu, @3]
+    @5: MUL[gamma, @4]
+
+    @0, @2, and @3 are operands in only one operation each so can be collapsed.
+    @1 is an operand in four operations and @4 is both a required expression and
+    an operand in an expression so both are required common subexpressions. @1
+    will become the first common subexpression (x0, tan(omega * t)) and @4 will
+    become the second common subexpression (x1, nu * x0 / (gamma - x0)). @5 is
+    one of the required expressions in the matrix so can used directly there
+    resulting in the substituted matrix expression Matrix([x1, gamma * x1]).
+
+    """
+
+    def setup(self):
+        """Create the required symbols (nu, gamma, omega, t), the matrix of
+        expressions (y), and its Jacobian (G).
+
+        G is the 2x4 matrix that contains the derivatives of y with respect to
+        nu, gamma, omega, and t as the four columns respectively.
+        
+        """
+        nu, gamma, omega, t = symbols("nu, gamma, omega, t")
+        x = Matrix([nu, gamma, omega, t])
+        self.y = Matrix([
+            nu * tan(omega * t) / (gamma - tan(omega * t)),
+            nu * gamma * tan(omega * t) / (gamma - tan(omega * t)),
+        ])
+        self.G = self.y.jacobian(x)
+
+    def test_function_cse(self):
+        """Expected result from CSE on the lighthouse example."""
+        nu = sym.Symbol("nu")
+        gamma = sym.Symbol("gamma")
+        omega = sym.Symbol("omega")
+        t = sym.Symbol("t")
+        x0 = sym.Symbol("x0")
+        x1 = sym.Symbol("x1")
+
+        cse = [
+            (x0, tan(omega * t)),
+            (x1, nu * x0 / (gamma - x0)),
+        ]
+        expr = [
+            Matrix([x1, gamma * x1]),
+        ]
+
+        assert cse(self.y) == (cse, expr)
+
+    def time_function_cse(self):
+        """Time CSE on the lighthouse example."""
+        cse(self.y)
+
+    def time_jacobian_cse(self):
+        """Time CSE on the lighthouse example's Jacobian."""
+        cse(self.G)
+
+    def time_combined_cse(self):
+        """Time simultaneous CSE on the lighthouse example and its Jacobian."""
+        cse([self.y, self.G])