See also [:SymbolicBenchmarks: this other page].

The 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".

Problem 1

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 2

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 3

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 4

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