Diff for "WesterBenchmarks"

Differences between revisions 5 and 6

Problem Key
→	simplify
A(...)	assume ...
S(...,x)	solve ... for x
(p.v.)	principal value

Performance Key
×	wrong answer/cannot do the problem
s sec/ms/μs	performs correctly in time s
>s sec/ms/μs	does not finish in time s
>.<,s or >.<,×	very difficult to convince system to do what you want (regardless of performance)

Problem	Maple	Mathematica	GiNaC	Maxima	Sage	Symbolics	Notes (such as code used/version etc.)
√2√3+4→1+√3
2∞−3→∞
ex−1ex/2+1→ex/2−1
A(x≥y,y≥z,z≥x);x=z?
A(x>y,y>0);2x2>2y2?
cosxcos(3x)→cos2x−3sin2x
cosxcos(3x)→2cos(2x)−1
A(x,y>0);x1/ny1/n−(xy)1/n→0
log(tan(21x+4π))−sinh−1(tan(x))→0
log2√r+1√4r+4√r+1→0
√x\|z\|√xy\|z\|2→√x√xy/→√y							Note √x=±√x
2x=0+1→2x+1=1
S(e2x+2ex+1=z,x)
S((x+1)(sin2x+1)2cos3(3x)=0,x)
M−1, where M=[[x,1],[y,ez]]
∑nk=1k3→4n2(n+1)2
∑∞k=1(1k2+1k3)→6π2+ζ(3)
∏nk=1k→n!
limn→∞(1+n1)n→e
\lim_{x\rightarrow 0}\frac{\sin x}{x} \rightarrow 1
\lim_{x\rightarrow 0}\frac{1-\cos x}{x^2} \rightarrow \frac{1}{2}
\frac{d^2}{dx^2}y(x(t)) \rightarrow \frac{d^2y}{dx^2}(\frac{dx}{dt})^2 + \frac{dy}{dx}\frac{d^2x}{dt^2}
\frac{d}{dx}(\int\frac{1}{x^3+2}dx) \rightarrow \frac{1}{x^3+2}
\int\frac{1}{a+b\cos x}dx (a < b)
\frac{d}{dx}\int\frac{1}{a+b\cos x}dx = \frac{1}{a+b\cos x}
\frac{d}{dx}\|x\| \rightarrow \frac{x}{\|x\|}
\int\|x\|dx \rightarrow \frac{x\|x\|}{2}
\int\frac{x}{\sqrt{1+x}+\sqrt{1-x}}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3}
\int\frac{\sqrt{1+x}+\sqrt{1-x}}{2}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3}
\int_{-1}^1\frac{1}{x}dx \rightarrow 0(p.v.)

-  ⇤ ← Revision 5 as of 2008-08-12 11:40:01 → 
  Size: 1330
  Editor: RobertMiller
  Comment:
+   ← Revision 6 as of 2008-08-12 12:16:36 → ⇥
  Size: 4853
  Editor: RobertMiller
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 1:
-||<-2> Problem Key ||
||  $\to$  || simplify ||
||  $A (. . .)$  || assume ... ||
+||<-2> Problem Key                    ||
||  $\to$  || simplify          ||
||  $A (. . .)$       || assume ...        ||
||  $S (. . ., x)$     || solve ... for  $x$  ||
|| (p.v.)        || principal value   ||
-Line 5:
+Line 7:
-||<-2> Performance Key ||
||  $\times$  || wrong answer/cannot do the problem ||
||  $s s e c / m s / μ s$      || performs correctly in time  $s$  ||
||  $> s s e c / m s / μ s$    || does not finish in time  $s$  ||
+||<-2> Performance Key                                           ||
||  $\times$                 || wrong answer/cannot do the problem ||
||  $s s e c / m s / μ s$        || performs correctly in time  $s$      ||
||  $> s s e c / m s / μ s$      || does not finish in time  $s$         ||
-Line 11:
+Line 13:
-|| Problem || Maple || Mathematica || GiNaC || Maxima || Sage || Symbolics || Notes (such as code used/version etc.) ||
||  $\sqrt{2 \sqrt{3} + 4} \to 1 + \sqrt{3}$  || || || || || || || ||
||  $2 \infty - 3 \to \infty$  || || || || || || || ||
||  $\frac{e^{x} - 1}{e^{x / 2} + 1} \to e^{x / 2} - 1$  || || || || || || || ||
||  $A (x \geq y, y \geq z, z \geq x); x = z ?$  || || || || || || || ||
||  $A (x > y, y > 0); 2 x^{2} > 2 y^{2} ?$  || || || || || || || ||
||  $\frac{\cos (3 x)}{\cos x} \to \cos^{2} x - 3 \sin^{2} x$  || || || || || || || ||
||  $\frac{\cos (3 x)}{\cos x} \to 2 \cos (2 x) - 1$  || || || || || || || ||
||  $A (x, y > 0); x^{1 / n} y^{1 / n} - (x y)^{1 / n} \to 0$  || || || || || || || ||
||  $\log (\tan (\frac{1}{2} x + \frac{π}{4})) - \sinh^{- 1} (\tan (x)) \to 0$  || || || || || || || ||
||  $\log \frac{2 \sqrt{r} + 1}{\sqrt{4 r + 4 \sqrt{r} + 1}} \to 0$  || || || || || || || ||
+|| Problem                                                      || Maple || Mathematica || GiNaC || Maxima || Sage || Symbolics || Notes (such as code used/version etc.) ||
||  $\sqrt{2 \sqrt{3} + 4} \to 1 + \sqrt{3}$               ||       ||             ||       ||        ||      ||           || ||
||  $2 \infty - 3 \to \infty$                              ||       ||             ||       ||        ||      ||           || ||
||  $\frac{e^{x} - 1}{e^{x / 2} + 1} \to e^{x / 2} - 1$             ||       ||             ||       ||        ||      ||           || ||
||  $A (x \geq y, y \geq z, z \geq x); x = z ?$                     ||       ||             ||       ||        ||      ||           || ||
||  $A (x > y, y > 0); 2 x^{2} > 2 y^{2} ?$                               ||       ||             ||       ||        ||      ||           || ||
||  $\frac{\cos (3 x)}{\cos x} \to \cos^{2} x - 3 \sin^{2} x$    ||       ||             ||       ||        ||      ||           || ||
||  $\frac{\cos (3 x)}{\cos x} \to 2 \cos (2 x) - 1$           ||       ||             ||       ||        ||      ||           || ||
||  $A (x, y > 0); x^{1 / n} y^{1 / n} - (x y)^{1 / n} \to 0$       ||       ||             ||       ||        ||      ||           || ||
||  $\log (\tan (\frac{1}{2} x + \frac{π}{4})) - \sinh^{- 1} (\tan (x)) \to 0$  || || ||       ||        ||      ||           || ||
||  $\log \frac{2 \sqrt{r} + 1}{\sqrt{4 r + 4 \sqrt{r} + 1}} \to 0$  || ||            ||       ||        ||      ||           || ||
||  $\frac{\sqrt{x y | z |^{2}}}{\sqrt{x} | z |} \to \frac{\sqrt{x y}}{\sqrt{x}} ↛ \sqrt{y}$  || || || || || ||       || Note  $\sqrt{x} = \pm \sqrt{x}$  ||
||  $\frac{x = 0}{2} + 1 \to \frac{x}{2} + 1 = 1$                 ||       ||             ||       ||        ||      ||           || ||
||  $S (e^{2 x} + 2 e^{x} + 1 = z, x)$                                  ||       ||             ||       ||        ||      ||           || ||
||  $S ((x + 1) (\sin^{2} x + 1)^{2} \cos^{3} (3 x) = 0, x)$                       ||       ||             ||       ||        ||      ||           || ||
||  $M^{- 1}$ , where  $M = [[x, 1], [y, e^{z}]]$                         ||       ||             ||       ||        ||      ||           || ||
||  $\sum_{k = 1}^{n} k^{3} \to \frac{n^{2} (n + 1)^{2}}{4}$           ||       ||             ||       ||        ||      ||           || ||
||  $\sum_{k = 1}^{\infty} (\frac{1}{k^{2}} + \frac{1}{k^{3}}) \to \frac{π^{2}}{6} + ζ (3)$  || || || || || || || ||
||  $\prod_{k = 1}^{n} k \to n!$                               ||       ||             ||       ||        ||      ||           || ||
||  $lim_{n \to \infty} (1 + \frac{1}{n})^{n} \to e$  ||       ||             ||       ||        ||      ||           || ||
||  $lim_{x \to 0} \frac{\sin x}{x} \to 1$         ||       ||             ||       ||        ||      ||           || ||
||  $lim_{x \to 0} \frac{1 - \cos x}{x^{2}} \to \frac{1}{2}$  || ||            ||       ||        ||      ||           || ||
||  $\frac{d^{2}}{d x^{2}} y (x (t)) \to \frac{d^{2} y}{d x^{2}} (\frac{d x}{d t})^{2} + \frac{d y}{d x} \frac{d^{2} x}{d t^{2}}$  || || || || || ||  || ||
||  $\frac{d}{d x} (\int \frac{1}{x^{3} + 2} d x) \to \frac{1}{x^{3} + 2}$  ||  ||             ||       ||        ||      ||           || ||
||  $\int \frac{1}{a + b \cos x} d x (a < b)$                           ||       ||             ||       ||        ||      ||           || ||
||  $\frac{d}{d x} \int \frac{1}{a + b \cos x} d x = \frac{1}{a + b \cos x}$ ||       ||             ||       ||        ||      ||           || ||
||  $\frac{d}{d x} | x | \to \frac{x}{| x |}$                   ||       ||             ||       ||        ||      ||           || ||
||  $\int | x | d x \to \frac{x | x |}{2}$                        ||       ||             ||       ||        ||      ||           || ||
||  $\int \frac{x}{\sqrt{1 + x} + \sqrt{1 - x}} d x \to \frac{(1 + x)^{3 / 2} + (1 - x)^{3 / 2}}{3}$  || || || ||       ||      ||           || ||
||  $\int \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} d x \to \frac{(1 + x)^{3 / 2} + (1 - x)^{3 / 2}}{3}$  || || || ||       ||      ||           || ||
||  $\int_{- 1}^{1} \frac{1}{x} d x \to 0$ (p.v.)               ||       ||             ||       ||        ||      ||           || ||