See also [:SymbolicBenchmarks: this other page].

TableOfContents

The "Real World" Symbolic Benchmark Suite

The conditions for something to be listed here: (a) it must be resemble an actual computation somebody actually wanted to do in Sage, and (b) the question must be precisely formulated with Sage code that uses the Sage symbolics in a straightforward way (i.e., don't cleverly use number fields). Do not post any "synthetic" benchmarks. This page is supposed to be about nailing down exactly why people consider the sage symbolics at present "so slow as to be completely useless for anything but fast float".

Just to emphasize, some of these seem silly but they all come up when REAL USERS use Sage. For synthetic benchmarks, see the second section below.

Problem R1

SETUP: Define a function f(z) = \sqrt{1/3}\cdot z^2 + i/3. COMPUTATION: Compute the real part of f(f(f(...(f(i/2))...) iterated 10 times.

# setup
def f(z): return sqrt(1/3)*z^2 + i/3
# computation
real(f(f(f(f(f(f(f(f(f(f(i/2)))))))))))
//
-15323490199844318074242473679071410934833494247466385771803570370858961112774390851798166656796902695599442662754502211584226105508648298600018090510170430216881977761279503642801008178271982531042720727178135881702924595044672634313417239327304576652633321095875724771887486594852083526001648217317718794685379391946143663292907934545842931411982264788766619812559999515408813796287448784343854980686798782575952258163992236113752353237705088451481168691158059505161807961082162315225057299394348203539002582692884735745377391416638540520323363224931163680324690025802009761307137504963304640835891588925883135078996398616361571065941964628043214890356454145039464055430143/(160959987592246947739944859375773744043416001841910423046466880402863187009126824419781711398533250016237703449459397319370100476216445123130147322940019839927628599479294678599689928643570237983736966305423831947366332466878486992676823215303312139985015592974537721140932243906832125049776934072927576666849331956351862828567668505777388133331284248870175178634054430823171923639987569211668426477739974572402853248951261366399284257908177157179099041115431335587887276292978004143353025122721401971549897673882099546646236790739903146970578001092018346524464799146331225822142880459202800229013082033028722077703362360159827236163041299500992177627657014103138377287073792*sqrt(3))
Time: CPU 0.11 s, Wall: 0.34 s

Problem R2

def hermite(n,y):
  if n == 1:
      return 2*y
  if n == 0:
      return 1
  return 2*y*hermite(n-1,y) - 2*(n-1)*hermite(n-2,y)

def phi(n,y):
  return 1/(sqrt(2^n*factorial(n))*pi^(1/4))*exp(-y^2/2)*hermite(n,y)

time a = phi(25,4)
//
Time: CPU 0.59 s, Wall: 0.60 s

Problem R3

sage: var('x,y,z')
sage: f = x+y+z
sage: time for _ in range(10): a = bool(f==f)
//
CPU time: 0.09 s,  Wall time: 0.52 s

Problem R4

sage: u=[e,pi,sqrt(2)]
sage: time Tuples(u,3).count()
//
27
Time: CPU 0.23 s, Wall: 1.55 s

For comparison, see what happens with integers.

sage: u=[1,2,3]
sage: time Tuples(u,3).count()
27
Time: CPU 0.00 s, Wall: 0.00 s

Problem R5

def blowup(L,n):
    for i in [0..n]:
        L.append( (L[i] + L[i+1]) * L[i+2] )

(x,y,z)=var('x,y,z')
L = [x,y,z]
blowup(L,8)
time L=uniq(L)
//
Time: CPU 0.17 s, Wall: 0.68 s

R.<x,y,z> = QQ[]
L = [x,y,z]
blowup(L,8)
time L=uniq(L)
//
Time: CPU 0.08 s, Wall: 0.08 s

Problem R6

sage: time sum(((x+sin(i))/x+(x-sin(i))/x).rational_simplify() for i in xrange(100))
///
200
CPU time: 1.39 s,  Wall time: 8.65 s

The Synthetic Symbolic Benchmark Suite

Here is where synthetic benchmarks go. These are made up because you abstract think they are good benchmarks. They don't have to come up in real world problems.

Problem S1

We use only 7, since Sage's current symbolics are SO slow at this.

sage: var('x,y,z')
sage: f = (x+y+z+1)^7
sage: time g = expand(f*(f+1))
///
CPU time: 0.14 s,  Wall time: 2.76 s