# Benchmarks#

Tests that will take a long time if something is wrong, but be very quick otherwise. See https://wiki.sagemath.org/symbench. The parameters chosen below are such that with pynac most of these take well less than a second, but would not even be feasible using Sage’s Maxima-based symbolics.

Problem R1

Important note. Below we do s.expand().real() because s.real() takes forever (TODO?).

```sage: f(z) = sqrt(1/3)*z^2 + i/3
sage: s = f(f(f(f(f(f(f(f(f(f(i/2))))))))))
sage: s.expand().real()
-15323490199844318074242473679071410934833494247466385771803570370858961112774390851798166656796902695599442662754502211584226105508648298600018090510170430216881977761279503642801008178271982531042720727178135881702924595044672634313417239327304576652633321095875724771887486594852083526001648217317718794685379391946143663292907934545842931411982264788766619812559999515408813796287448784343854980686798782575952258163992236113752353237705088451481168691158059505161807961082162315225057299394348203539002582692884735745377391416638540520323363224931163680324690025802009761307137504963304640835891588925883135078996398616361571065941964628043214890356454145039464055430143/160959987592246947739944859375773744043416001841910423046466880402863187009126824419781711398533250016237703449459397319370100476216445123130147322940019839927628599479294678599689928643570237983736966305423831947366332466878486992676823215303312139985015592974537721140932243906832125049776934072927576666849331956351862828567668505777388133331284248870175178634054430823171923639987569211668426477739974572402853248951261366399284257908177157179099041115431335587887276292978004143353025122721401971549897673882099546646236790739903146970578001092018346524464799146331225822142880459202800229013082033028722077703362360159827236163041299500992177627657014103138377287073792*sqrt(1/3)
```
```>>> from sage.all import *
>>> __tmp__=var("z"); f = symbolic_expression(sqrt(Integer(1)/Integer(3))*z**Integer(2) + i/Integer(3)).function(z)
>>> s = f(f(f(f(f(f(f(f(f(f(i/Integer(2)))))))))))
>>> s.expand().real()
-15323490199844318074242473679071410934833494247466385771803570370858961112774390851798166656796902695599442662754502211584226105508648298600018090510170430216881977761279503642801008178271982531042720727178135881702924595044672634313417239327304576652633321095875724771887486594852083526001648217317718794685379391946143663292907934545842931411982264788766619812559999515408813796287448784343854980686798782575952258163992236113752353237705088451481168691158059505161807961082162315225057299394348203539002582692884735745377391416638540520323363224931163680324690025802009761307137504963304640835891588925883135078996398616361571065941964628043214890356454145039464055430143/160959987592246947739944859375773744043416001841910423046466880402863187009126824419781711398533250016237703449459397319370100476216445123130147322940019839927628599479294678599689928643570237983736966305423831947366332466878486992676823215303312139985015592974537721140932243906832125049776934072927576666849331956351862828567668505777388133331284248870175178634054430823171923639987569211668426477739974572402853248951261366399284257908177157179099041115431335587887276292978004143353025122721401971549897673882099546646236790739903146970578001092018346524464799146331225822142880459202800229013082033028722077703362360159827236163041299500992177627657014103138377287073792*sqrt(1/3)
```

Problem R2:

```sage: def hermite(n,y):
....:     if n == 1: return 2*y
....:     if n == 0: return 1
....:     return expand(2*y*hermite(n-1,y) - 2*(n-1)*hermite(n-2,y))
sage: hermite(15,var('y'))
32768*y^15 - 1720320*y^13 + 33546240*y^11 - 307507200*y^9 + 1383782400*y^7 - 2905943040*y^5 + 2421619200*y^3 - 518918400*y
```
```>>> from sage.all import *
>>> def hermite(n,y):
...     if n == Integer(1): return Integer(2)*y
...     if n == Integer(0): return Integer(1)
...     return expand(Integer(2)*y*hermite(n-Integer(1),y) - Integer(2)*(n-Integer(1))*hermite(n-Integer(2),y))
>>> hermite(Integer(15),var('y'))
32768*y^15 - 1720320*y^13 + 33546240*y^11 - 307507200*y^9 + 1383782400*y^7 - 2905943040*y^5 + 2421619200*y^3 - 518918400*y
```

Problem R3:

```sage: f = sum(var('x,y,z')); a = [bool(f==f) for _ in range(100000)]
```
```>>> from sage.all import *
>>> f = sum(var('x,y,z')); a = [bool(f==f) for _ in range(Integer(100000))]
```

Problem R4:

```sage: u = [e,pi,sqrt(2)]; Tuples(u,3).cardinality()
27
```
```>>> from sage.all import *
>>> u = [e,pi,sqrt(Integer(2))]; Tuples(u,Integer(3)).cardinality()
27
```

Problem R5:

```sage: def blowup(L,n):
....:    for i in [0..n]:
....:        L.append( (L[i] + L[i+1]) * L[i+2] )
sage: L = list(var('x,y,z'))
sage: blowup(L,15)
sage: len(set(L))
19
```
```>>> from sage.all import *
>>> def blowup(L,n):
...    for i in (ellipsis_range(Integer(0),Ellipsis,n)):
...        L.append( (L[i] + L[i+Integer(1)]) * L[i+Integer(2)] )
>>> L = list(var('x,y,z'))
>>> blowup(L,Integer(15))
>>> len(set(L))
19
```

Problem R6:

```sage: sum(((x+sin(i))/x+(x-sin(i))/x) for i in range(100)).expand()
200
```
```>>> from sage.all import *
>>> sum(((x+sin(i))/x+(x-sin(i))/x) for i in range(Integer(100))).expand()
200
```

Problem R7:

```sage: f = x^24+34*x^12+45*x^3+9*x^18 +34*x^10+ 32*x^21
sage: a = [f(x=random()) for _ in range(10^4)]
```
```>>> from sage.all import *
>>> f = x**Integer(24)+Integer(34)*x**Integer(12)+Integer(45)*x**Integer(3)+Integer(9)*x**Integer(18) +Integer(34)*x**Integer(10)+ Integer(32)*x**Integer(21)
>>> a = [f(x=random()) for _ in range(Integer(10)**Integer(4))]
```

Problem R10:

```sage: v = [float(z) for z in [-pi,-pi+1/100..,pi]]
```
```>>> from sage.all import *
>>> v = [float(z) for z in (ellipsis_range(-pi,-pi+Integer(1)/Integer(100),Ellipsis,pi))]
```

Problem R11:

```sage: a = [random() + random()*I for w in [0..100]]
sage: a.sort()
```
```>>> from sage.all import *
>>> a = [random() + random()*I for w in (ellipsis_range(Integer(0),Ellipsis,Integer(100)))]
>>> a.sort()
```

Problem W3:

```sage: acos(cos(x))
arccos(cos(x))
```
```>>> from sage.all import *
>>> acos(cos(x))
arccos(cos(x))
```

PROBLEM S1:

```sage: _ = var('x,y,z')
sage: f = (x+y+z+1)^10
sage: g = expand(f*(f+1))
```
```>>> from sage.all import *
>>> _ = var('x,y,z')
>>> f = (x+y+z+Integer(1))**Integer(10)
>>> g = expand(f*(f+Integer(1)))
```

PROBLEM S2:

```sage: _ = var('x,y')
sage: a = expand((x^sin(x) + y^cos(y) - z^(x+y))^100)
```
```>>> from sage.all import *
>>> _ = var('x,y')
>>> a = expand((x**sin(x) + y**cos(y) - z**(x+y))**Integer(100))
```

PROBLEM S3:

```sage: _ = var('x,y,z')
sage: f = expand((x^y + y^z + z^x)^50)
sage: g = f.diff(x)
```
```>>> from sage.all import *
>>> _ = var('x,y,z')
>>> f = expand((x**y + y**z + z**x)**Integer(50))
>>> g = f.diff(x)
```

PROBLEM S4:

```sage: w = (sin(x)*cos(x)).series(x,400)
```
```>>> from sage.all import *
>>> w = (sin(x)*cos(x)).series(x,Integer(400))
```