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

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

Problem R4

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

For comparison, see what happens with integers.

u = [1,2,3]
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

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

Problem R7

var('x')
f = x^24+34*x^12+45*x^3+9*x^18 +34*x^10+ 32*x^21
time a = [f(random()) for _ in xrange(10^4)]
///
Time: CPU 11.92 s, Wall: 12.73 s

Problem R8

def right(f,a,b,n):
   Deltax = (b-a)/n; c=a; est=0
   for i in range(n):
       c += Deltax
       est += f(c)
   return est*Deltax

var('x')
time right(x^2,0,5,10^4)
///
Time: CPU 1.53 s, Wall: 1.57 s
66676667/1600000

Problem R9

var('x,y')
time factor(x^20 - pi^5*y^20)
///
-(pi*y^4 - x^4)*(pi^4*y^16 + pi^3*x^4*y^12 + pi^2*x^8*y^8 + pi*x^12*y^4 + x^16)
Time: CPU 0.02 s, Wall: 7.43 s

Note that singular takes .05 seconds to do x^{20} - w^5 y^{20}, and one could use that.

Sage Gets the Answer All Wrong

Examples where Sage is just blatantly wrong (often because of Maxima).

Problem W1

var('a b c')
eqn1 = a - exp((-pi*b)/sqrt(1-b)) == 0
eqn2 = c - atan(2*b*sqrt(1/(sqrt(4*b^4+1) - 2*b^2)))==0
solve([eqn1,eqn2,a==1/8],b,c,a) 
///
[]

WRONG and LAME! This is because of a bug in Maxima.

Problem W2

This is related R10. It's quick, but the output is WRONG, since if you replace pi by 2^{1/3} above you will get a

var('x,y')
time factor(x^20 - (2^(1/3))^5*y^20)
///
-(2*2^(2/3)*y^20 - x^20)
Time: CPU 0.02 s, Wall: 0.12 s

Observe that

w = (2^(1/3))
expand(-(w*y^4 - x^4)*(w^4*y^16 + w^3*x^4*y^12 + w^2*x^8*y^8 + w*x^12*y^4 + x^16))
///
x^20 - 2*2^(2/3)*y^20

This is because of a bug in Maxima.

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