An efficient computation idea. Please comment

A

Aaron Fude

Hi,

In my program I evaluate long trigonometric polynomials a lot! For example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So why not
create a java snippet, compile it, load it as a class and use the compiled
version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";


I think it's a pretty cool idea. But before I implement it, I would like to
know what you think about it!

Thank you in advance!

Aaron Fude
 
P

Patrick May

Aaron Fude said:
In my program I evaluate long trigonometric polynomials a lot! For example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So
why not create a java snippet, compile it, load it as a class and
use the compiled version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";

I think it's a pretty cool idea. But before I implement it, I would
like to know what you think about it!
Click to expand...


Greenspun's Tenth Rule of Programming:
"Any sufficiently complicated C or Fortran program contains an
ad-hoc, informally-specified, bug-ridden, slow implementation of
half of Common Lisp."

You have added Fude's Lemma: "Java programs, too."

Look up defmacro in the Common Lisp Hyperspec
(http://www.lispworks.com/reference/HyperSpec, among others). Java
doesn't allow you to do this particularly well. Lisp does.

Regards,

Patrick
 
A

Aaron Fude

Patrick May said:
Aaron Fude said:
In my program I evaluate long trigonometric polynomials a lot! For
example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So
why not create a java snippet, compile it, load it as a class and
use the compiled version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";

I think it's a pretty cool idea. But before I implement it, I would
like to know what you think about it!
Click to expand...


Greenspun's Tenth Rule of Programming:
"Any sufficiently complicated C or Fortran program contains an
ad-hoc, informally-specified, bug-ridden, slow implementation of
half of Common Lisp."

You have added Fude's Lemma: "Java programs, too."

Look up defmacro in the Common Lisp Hyperspec
(http://www.lispworks.com/reference/HyperSpec, among others). Java
doesn't allow you to do this particularly well. Lisp does.

Regards,

Patrick
Click to expand...



Hi Patrck,

I myself love Scheme and agree with that quote which I heard as "Inside
every C program there's a Lisp yearning to get out."

I expect that Java won't do this well. Nevertheless, will this be worth my
effort?

Thanks.

Aaron
 
M

Mark Thornton

Aaron said:
Hi,

In my program I evaluate long trigonometric polynomials a lot! For example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So why not
create a java snippet, compile it, load it as a class and use the compiled
version. So I would have something like;
Click to expand...


A far better scheme would be to mathematically transform the calculation
to something which can be evaluated more cheaply. For example suppose we
calculate t = cos(x)

cos(0*x) = 1
cos(2*x) = 2*t*t-1
cos(3*x) = -3*t+4*t*t*t

etc

So we need only compute cos(x) (and perhaps sin(x)) and then find the
result with a series of much simpler operations (multiplication and
addition).

There are a number of different transformations which might be employed,
depending on the range of values expected for x and other
considerations. This approach can yield much more efficient evaluation
than the technique you suggest.

You might also want to review the literature on Fourier series.

Mark Thornton
 
M

Mark Thornton

Patrick said:
Aaron Fude said:
In my program I evaluate long trigonometric polynomials a lot! For example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So
why not create a java snippet, compile it, load it as a class and
use the compiled version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";

I think it's a pretty cool idea. But before I implement it, I would
like to know what you think about it!
Click to expand...



Greenspun's Tenth Rule of Programming:
"Any sufficiently complicated C or Fortran program contains an
ad-hoc, informally-specified, bug-ridden, slow implementation of
half of Common Lisp."

You have added Fude's Lemma: "Java programs, too."

Look up defmacro in the Common Lisp Hyperspec
(http://www.lispworks.com/reference/HyperSpec, among others). Java
doesn't allow you to do this particularly well. Lisp does.
Click to expand...


However in this particular case there is a much better way of evaluating
the required function. A good understanding of mathematics is what is
required.

Mark Thornton
 
A

Aaron Fude

Mark Thornton said:
Patrick said:
Aaron Fude said:
In my program I evaluate long trigonometric polynomials a lot! For
example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So
why not create a java snippet, compile it, load it as a class and
use the compiled version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";

I think it's a pretty cool idea. But before I implement it, I would
like to know what you think about it!
Click to expand...



Greenspun's Tenth Rule of Programming:
"Any sufficiently complicated C or Fortran program contains an
ad-hoc, informally-specified, bug-ridden, slow implementation of
half of Common Lisp."

You have added Fude's Lemma: "Java programs, too."

Look up defmacro in the Common Lisp Hyperspec
(http://www.lispworks.com/reference/HyperSpec, among others). Java
doesn't allow you to do this particularly well. Lisp does.
Click to expand...


However in this particular case there is a much better way of evaluating
the required function. A good understanding of mathematics is what is
required.

Mark Thornton
Click to expand...


Your comment about unwrapping cosines is undoubtedly correct. To tell you
the truth, I don't even have cosines, I have elementary polynomials. I have
contrived the cosine example so I wouldn't get responses about summing the
series in inverse order - and it backfired. (BTW, how many flops does a
cosine take? I have no idea, but I would guess about 50. Wouldn't that mean
that by the time you get to cos(50*x) it's cheaper to evaluate the cosine?)

In any case, now that we know that I have polynomial, perhaps you could
answer the posed question in the context of polynomials. Is there added
value to the compilation trick if my series has about 1000 terms and needs
to be evaluated 100k times?

Aaron Fude
 
M

Mark Thornton

Aaron said:
Patrick said:
In my program I evaluate long trigonometric polynomials a lot! For
example,

final int N = 1000;
double[] c = new double[N];
// Populate c
double x = .5, sum = 0.0;

for (int i = 0; i < N; i++)
sum += c*cos(i*x);

And this function is evaluated very many times (about 100,000). So
why not create a java snippet, compile it, load it as a class and
use the compiled version. So I would have something like;

String sum = "0"; // A String buffer, of course...
for (int i = 0; i < N; i++)
sum += "+" + c + "*cos(" + i + "*x)";
sum += ";";

I think it's a pretty cool idea. But before I implement it, I would
like to know what you think about it!


Greenspun's Tenth Rule of Programming:
"Any sufficiently complicated C or Fortran program contains an
ad-hoc, informally-specified, bug-ridden, slow implementation of
half of Common Lisp."

You have added Fude's Lemma: "Java programs, too."

Look up defmacro in the Common Lisp Hyperspec
(http://www.lispworks.com/reference/HyperSpec, among others). Java
doesn't allow you to do this particularly well. Lisp does.
Click to expand...


However in this particular case there is a much better way of evaluating
the required function. A good understanding of mathematics is what is
required.

Mark Thornton
Click to expand...



Your comment about unwrapping cosines is undoubtedly correct. To tell you
the truth, I don't even have cosines, I have elementary polynomials. I have
contrived the cosine example so I wouldn't get responses about summing the
series in inverse order - and it backfired. (BTW, how many flops does a
cosine take? I have no idea, but I would guess about 50. Wouldn't that mean
that by the time you get to cos(50*x) it's cheaper to evaluate the cosine?)
Click to expand...


No, because in practice a recurrence relation is used to compute each
successive term from previous values. Just as with evaluating
polynomials you don't compute x^n from scratch at each term. It is in
fact very similar as exp(iz) = cos(z) + i.sin(z) (where 'i' here is the
sqrt of -1). Now note that exp(i.n.z) = exp(iz)^n
In any case, now that we know that I have polynomial, perhaps you could
answer the posed question in the context of polynomials. Is there added
value to the compilation trick if my series has about 1000 terms and needs
to be evaluated 100k times?
Click to expand...

It still isn't worth generating byte code. Many JIT compilers will have
been tuned to recognise this type of loop and may do some unrolling
themselves.

More seriously you need to consider the numerical stability of a power
series with 1000 terms. There are a number of methods to 'improve' such
series, which would then usually leave you with a much smaller
polynomial to evaluate. However this a complex subject which filled many
lectures during my maths degree (25 years ago).

I suggest you find a mathematician to help economise the series, and
evaluate it in the straight forward manner.

Mark Thornton
 
A

Aaron Fude

Mark,

Here's some funny code for you which shows that compiled code may run faster
(typically by a factor of 3 on my machine).
In really, the effect will be even greater for me since the series would
have coefficients that come from some kind of a list with potentially
significant access time. Now, as you wisely noted, .999^555 is quickly
computed as .999^554*.999. That's very true, but perhaps it won't change the
fact that the compilation trick on top of your keen observation is a good
idea.

Aaron

public static void main(String[] inStr) {
int N = 0;
int M = 0;
try {
N = Math.max(1000, System.in.read()); // To prevent simple
unwrapping...
M = Math.max(100, System.in.read());
}
catch (Exception inE) { }

{
long start = System.currentTimeMillis();
double sum = 0;
for (int m = 0; m < M; m++)
for (int n = 0; n < N; n++)
sum += Math.pow(.999, n);
System.out.println(System.currentTimeMillis() - start);
}

{
long start = System.currentTimeMillis();
double sum = 0;
for (int m = 0; m < M; m++)
sum = 0 + Math.pow(.999, 0) + Math.pow(.999, 1) + Math.pow(.999,
2) + Math.pow(.999, 3) + Math.pow(.999, 4) + Math.pow(.999, 5) +
Math.pow(.999, 6) + Math.pow(.999, 7) + Math.pow(.999, 8) +
Math.pow(.999, 9) + Math.pow(.999, 10) + Math.pow(.999, 11) +
Math.pow(.999, 12) + Math.pow(.999, 13) + Math.pow(.999, 14) +
Math.pow(.999, 15) + Math.pow(.999, 16) + Math.pow(.999, 17) +
Math.pow(.999, 18) + Math.pow(.999, 19) + Math.pow(.999, 20) +
Math.pow(.999, 21) + Math.pow(.999, 22) + Math.pow(.999, 23) +
Math.pow(.999, 24) + Math.pow(.999, 25) + Math.pow(.999, 26) +
Math.pow(.999, 27) + Math.pow(.999, 28) + Math.pow(.999, 29) +
Math.pow(.999, 30) + Math.pow(.999, 31) + Math.pow(.999, 32) +
Math.pow(.999, 33) + Math.pow(.999, 34) + Math.pow(.999, 35) +
Math.pow(.999, 36) + Math.pow(.999, 37) + Math.pow(.999, 38) +
Math.pow(.999, 39) + Math.pow(.999, 40) + Math.pow(.999, 41) +
Math.pow(.999, 42) + Math.pow(.999, 43) + Math.pow(.999, 44) +
Math.pow(.999, 45) + Math.pow(.999, 46) + Math.pow(.999, 47) +
Math.pow(.999, 48) + Math.pow(.999, 49) + Math.pow(.999, 50) +
Math.pow(.999, 51) + Math.pow(.999, 52) + Math.pow(.999, 53) +
Math.pow(.999, 54) + Math.pow(.999, 55) + Math.pow(.999, 56) +
Math.pow(.999, 57) + Math.pow(.999, 58) + Math.pow(.999, 59) +
Math.pow(.999, 60) + Math.pow(.999, 61) + Math.pow(.999, 62) +
Math.pow(.999, 63) + Math.pow(.999, 64) + Math.pow(.999, 65) +
Math.pow(.999, 66) + Math.pow(.999, 67) + Math.pow(.999, 68) +
Math.pow(.999, 69) + Math.pow(.999, 70) + Math.pow(.999, 71) +
Math.pow(.999, 72) + Math.pow(.999, 73) + Math.pow(.999, 74) +
Math.pow(.999, 75) + Math.pow(.999, 76) + Math.pow(.999, 77) +
Math.pow(.999, 78) + Math.pow(.999, 79) + Math.pow(.999, 80) +
Math.pow(.999, 81) + Math.pow(.999, 82) + Math.pow(.999, 83) +
Math.pow(.999, 84) + Math.pow(.999, 85) + Math.pow(.999, 86) +
Math.pow(.999, 87) + Math.pow(.999, 88) + Math.pow(.999, 89) +
Math.pow(.999, 90) + Math.pow(.999, 91) + Math.pow(.999, 92) +
Math.pow(.999, 93) + Math.pow(.999, 94) + Math.pow(.999, 95) +
Math.pow(.999, 96) + Math.pow(.999, 97) + Math.pow(.999, 98) +
Math.pow(.999, 99) + Math.pow(.999, 100) + Math.pow(.999, 101) +
Math.pow(.999, 102) + Math.pow(.999, 103) + Math.pow(.999, 104) +
Math.pow(.999, 105) + Math.pow(.999, 106) + Math.pow(.999, 107) +
Math.pow(.999, 108) + Math.pow(.999, 109) + Math.pow(.999, 110) +
Math.pow(.999, 111) + Math.pow(.999, 112) + Math.pow(.999, 113) +
Math.pow(.999, 114) + Math.pow(.999, 115) + Math.pow(.999, 116) +
Math.pow(.999, 117) + Math.pow(.999, 118) + Math.pow(.999, 119) +
Math.pow(.999, 120) + Math.pow(.999, 121) + Math.pow(.999, 122) +
Math.pow(.999, 123) + Math.pow(.999, 124) + Math.pow(.999, 125) +
Math.pow(.999, 126) + Math.pow(.999, 127) + Math.pow(.999, 128) +
Math.pow(.999, 129) + Math.pow(.999, 130) + Math.pow(.999, 131) +
Math.pow(.999, 132) + Math.pow(.999, 133) + Math.pow(.999, 134) +
Math.pow(.999, 135) + Math.pow(.999, 136) + Math.pow(.999, 137) +
Math.pow(.999, 138) + Math.pow(.999, 139) + Math.pow(.999, 140) +
Math.pow(.999, 141) + Math.pow(.999, 142) + Math.pow(.999, 143) +
Math.pow(.999, 144) + Math.pow(.999, 145) + Math.pow(.999, 146) +
Math.pow(.999, 147) + Math.pow(.999, 148) + Math.pow(.999, 149) +
Math.pow(.999, 150) + Math.pow(.999, 151) + Math.pow(.999, 152) +
Math.pow(.999, 153) + Math.pow(.999, 154) + Math.pow(.999, 155) +
Math.pow(.999, 156) + Math.pow(.999, 157) + Math.pow(.999, 158) +
Math.pow(.999, 159) + Math.pow(.999, 160) + Math.pow(.999, 161) +
Math.pow(.999, 162) + Math.pow(.999, 163) + Math.pow(.999, 164) +
Math.pow(.999, 165) + Math.pow(.999, 166) + Math.pow(.999, 167) +
Math.pow(.999, 168) + Math.pow(.999, 169) + Math.pow(.999, 170) +
Math.pow(.999, 171) + Math.pow(.999, 172) + Math.pow(.999, 173) +
Math.pow(.999, 174) + Math.pow(.999, 175) + Math.pow(.999, 176) +
Math.pow(.999, 177) + Math.pow(.999, 178) + Math.pow(.999, 179) +
Math.pow(.999, 180) + Math.pow(.999, 181) + Math.pow(.999, 182) +
Math.pow(.999, 183) + Math.pow(.999, 184) + Math.pow(.999, 185) +
Math.pow(.999, 186) + Math.pow(.999, 187) + Math.pow(.999, 188) +
Math.pow(.999, 189) + Math.pow(.999, 190) + Math.pow(.999, 191) +
Math.pow(.999, 192) + Math.pow(.999, 193) + Math.pow(.999, 194) +
Math.pow(.999, 195) + Math.pow(.999, 196) + Math.pow(.999, 197) +
Math.pow(.999, 198) + Math.pow(.999, 199) + Math.pow(.999, 200) +
Math.pow(.999, 201) + Math.pow(.999, 202) + Math.pow(.999, 203) +
Math.pow(.999, 204) + Math.pow(.999, 205) + Math.pow(.999, 206) +
Math.pow(.999, 207) + Math.pow(.999, 208) + Math.pow(.999, 209) +
Math.pow(.999, 210) + Math.pow(.999, 211) + Math.pow(.999, 212) +
Math.pow(.999, 213) + Math.pow(.999, 214) + Math.pow(.999, 215) +
Math.pow(.999, 216) + Math.pow(.999, 217) + Math.pow(.999, 218) +
Math.pow(.999, 219) + Math.pow(.999, 220) + Math.pow(.999, 221) +
Math.pow(.999, 222) + Math.pow(.999, 223) + Math.pow(.999, 224) +
Math.pow(.999, 225) + Math.pow(.999, 226) + Math.pow(.999, 227) +
Math.pow(.999, 228) + Math.pow(.999, 229) + Math.pow(.999, 230) +
Math.pow(.999, 231) + Math.pow(.999, 232) + Math.pow(.999, 233) +
Math.pow(.999, 234) + Math.pow(.999, 235) + Math.pow(.999, 236) +
Math.pow(.999, 237) + Math.pow(.999, 238) + Math.pow(.999, 239) +
Math.pow(.999, 240) + Math.pow(.999, 241) + Math.pow(.999, 242) +
Math.pow(.999, 243) + Math.pow(.999, 244) + Math.pow(.999, 245) +
Math.pow(.999, 246) + Math.pow(.999, 247) + Math.pow(.999, 248) +
Math.pow(.999, 249) + Math.pow(.999, 250) + Math.pow(.999, 251) +
Math.pow(.999, 252) + Math.pow(.999, 253) + Math.pow(.999, 254) +
Math.pow(.999, 255) + Math.pow(.999, 256) + Math.pow(.999, 257) +
Math.pow(.999, 258) + Math.pow(.999, 259) + Math.pow(.999, 260) +
Math.pow(.999, 261) + Math.pow(.999, 262) + Math.pow(.999, 263) +
Math.pow(.999, 264) + Math.pow(.999, 265) + Math.pow(.999, 266) +
Math.pow(.999, 267) + Math.pow(.999, 268) + Math.pow(.999, 269) +
Math.pow(.999, 270) + Math.pow(.999, 271) + Math.pow(.999, 272) +
Math.pow(.999, 273) + Math.pow(.999, 274) + Math.pow(.999, 275) +
Math.pow(.999, 276) + Math.pow(.999, 277) + Math.pow(.999, 278) +
Math.pow(.999, 279) + Math.pow(.999, 280) + Math.pow(.999, 281) +
Math.pow(.999, 282) + Math.pow(.999, 283) + Math.pow(.999, 284) +
Math.pow(.999, 285) + Math.pow(.999, 286) + Math.pow(.999, 287) +
Math.pow(.999, 288) + Math.pow(.999, 289) + Math.pow(.999, 290) +
Math.pow(.999, 291) + Math.pow(.999, 292) + Math.pow(.999, 293) +
Math.pow(.999, 294) + Math.pow(.999, 295) + Math.pow(.999, 296) +
Math.pow(.999, 297) + Math.pow(.999, 298) + Math.pow(.999, 299) +
Math.pow(.999, 300) + Math.pow(.999, 301) + Math.pow(.999, 302) +
Math.pow(.999, 303) + Math.pow(.999, 304) + Math.pow(.999, 305) +
Math.pow(.999, 306) + Math.pow(.999, 307) + Math.pow(.999, 308) +
Math.pow(.999, 309) + Math.pow(.999, 310) + Math.pow(.999, 311) +
Math.pow(.999, 312) + Math.pow(.999, 313) + Math.pow(.999, 314) +
Math.pow(.999, 315) + Math.pow(.999, 316) + Math.pow(.999, 317) +
Math.pow(.999, 318) + Math.pow(.999, 319) + Math.pow(.999, 320) +
Math.pow(.999, 321) + Math.pow(.999, 322) + Math.pow(.999, 323) +
Math.pow(.999, 324) + Math.pow(.999, 325) + Math.pow(.999, 326) +
Math.pow(.999, 327) + Math.pow(.999, 328) + Math.pow(.999, 329) +
Math.pow(.999, 330) + Math.pow(.999, 331) + Math.pow(.999, 332) +
Math.pow(.999, 333) + Math.pow(.999, 334) + Math.pow(.999, 335) +
Math.pow(.999, 336) + Math.pow(.999, 337) + Math.pow(.999, 338) +
Math.pow(.999, 339) + Math.pow(.999, 340) + Math.pow(.999, 341) +
Math.pow(.999, 342) + Math.pow(.999, 343) + Math.pow(.999, 344) +
Math.pow(.999, 345) + Math.pow(.999, 346) + Math.pow(.999, 347) +
Math.pow(.999, 348) + Math.pow(.999, 349) + Math.pow(.999, 350) +
Math.pow(.999, 351) + Math.pow(.999, 352) + Math.pow(.999, 353) +
Math.pow(.999, 354) + Math.pow(.999, 355) + Math.pow(.999, 356) +
Math.pow(.999, 357) + Math.pow(.999, 358) + Math.pow(.999, 359) +
Math.pow(.999, 360) + Math.pow(.999, 361) + Math.pow(.999, 362) +
Math.pow(.999, 363) + Math.pow(.999, 364) + Math.pow(.999, 365) +
Math.pow(.999, 366) + Math.pow(.999, 367) + Math.pow(.999, 368) +
Math.pow(.999, 369) + Math.pow(.999, 370) + Math.pow(.999, 371) +
Math.pow(.999, 372) + Math.pow(.999, 373) + Math.pow(.999, 374) +
Math.pow(.999, 375) + Math.pow(.999, 376) + Math.pow(.999, 377) +
Math.pow(.999, 378) + Math.pow(.999, 379) + Math.pow(.999, 380) +
Math.pow(.999, 381) + Math.pow(.999, 382) + Math.pow(.999, 383) +
Math.pow(.999, 384) + Math.pow(.999, 385) + Math.pow(.999, 386) +
Math.pow(.999, 387) + Math.pow(.999, 388) + Math.pow(.999, 389) +
Math.pow(.999, 390) + Math.pow(.999, 391) + Math.pow(.999, 392) +
Math.pow(.999, 393) + Math.pow(.999, 394) + Math.pow(.999, 395) +
Math.pow(.999, 396) + Math.pow(.999, 397) + Math.pow(.999, 398) +
Math.pow(.999, 399) + Math.pow(.999, 400) + Math.pow(.999, 401) +
Math.pow(.999, 402) + Math.pow(.999, 403) + Math.pow(.999, 404) +
Math.pow(.999, 405) + Math.pow(.999, 406) + Math.pow(.999, 407) +
Math.pow(.999, 408) + Math.pow(.999, 409) + Math.pow(.999, 410) +
Math.pow(.999, 411) + Math.pow(.999, 412) + Math.pow(.999, 413) +
Math.pow(.999, 414) + Math.pow(.999, 415) + Math.pow(.999, 416) +
Math.pow(.999, 417) + Math.pow(.999, 418) + Math.pow(.999, 419) +
Math.pow(.999, 420) + Math.pow(.999, 421) + Math.pow(.999, 422) +
Math.pow(.999, 423) + Math.pow(.999, 424) + Math.pow(.999, 425) +
Math.pow(.999, 426) + Math.pow(.999, 427) + Math.pow(.999, 428) +
Math.pow(.999, 429) + Math.pow(.999, 430) + Math.pow(.999, 431) +
Math.pow(.999, 432) + Math.pow(.999, 433) + Math.pow(.999, 434) +
Math.pow(.999, 435) + Math.pow(.999, 436) + Math.pow(.999, 437) +
Math.pow(.999, 438) + Math.pow(.999, 439) + Math.pow(.999, 440) +
Math.pow(.999, 441) + Math.pow(.999, 442) + Math.pow(.999, 443) +
Math.pow(.999, 444) + Math.pow(.999, 445) + Math.pow(.999, 446) +
Math.pow(.999, 447) + Math.pow(.999, 448) + Math.pow(.999, 449) +
Math.pow(.999, 450) + Math.pow(.999, 451) + Math.pow(.999, 452) +
Math.pow(.999, 453) + Math.pow(.999, 454) + Math.pow(.999, 455) +
Math.pow(.999, 456) + Math.pow(.999, 457) + Math.pow(.999, 458) +
Math.pow(.999, 459) + Math.pow(.999, 460) + Math.pow(.999, 461) +
Math.pow(.999, 462) + Math.pow(.999, 463) + Math.pow(.999, 464) +
Math.pow(.999, 465) + Math.pow(.999, 466) + Math.pow(.999, 467) +
Math.pow(.999, 468) + Math.pow(.999, 469) + Math.pow(.999, 470) +
Math.pow(.999, 471) + Math.pow(.999, 472) + Math.pow(.999, 473) +
Math.pow(.999, 474) + Math.pow(.999, 475) + Math.pow(.999, 476) +
Math.pow(.999, 477) + Math.pow(.999, 478) + Math.pow(.999, 479) +
Math.pow(.999, 480) + Math.pow(.999, 481) + Math.pow(.999, 482) +
Math.pow(.999, 483) + Math.pow(.999, 484) + Math.pow(.999, 485) +
Math.pow(.999, 486) + Math.pow(.999, 487) + Math.pow(.999, 488) +
Math.pow(.999, 489) + Math.pow(.999, 490) + Math.pow(.999, 491) +
Math.pow(.999, 492) + Math.pow(.999, 493) + Math.pow(.999, 494) +
Math.pow(.999, 495) + Math.pow(.999, 496) + Math.pow(.999, 497) +
Math.pow(.999, 498) + Math.pow(.999, 499) + Math.pow(.999, 500) +
Math.pow(.999, 501) + Math.pow(.999, 502) + Math.pow(.999, 503) +
Math.pow(.999, 504) + Math.pow(.999, 505) + Math.pow(.999, 506) +
Math.pow(.999, 507) + Math.pow(.999, 508) + Math.pow(.999, 509) +
Math.pow(.999, 510) + Math.pow(.999, 511) + Math.pow(.999, 512) +
Math.pow(.999, 513) + Math.pow(.999, 514) + Math.pow(.999, 515) +
Math.pow(.999, 516) + Math.pow(.999, 517) + Math.pow(.999, 518) +
Math.pow(.999, 519) + Math.pow(.999, 520) + Math.pow(.999, 521) +
Math.pow(.999, 522) + Math.pow(.999, 523) + Math.pow(.999, 524) +
Math.pow(.999, 525) + Math.pow(.999, 526) + Math.pow(.999, 527) +
Math.pow(.999, 528) + Math.pow(.999, 529) + Math.pow(.999, 530) +
Math.pow(.999, 531) + Math.pow(.999, 532) + Math.pow(.999, 533) +
Math.pow(.999, 534) + Math.pow(.999, 535) + Math.pow(.999, 536) +
Math.pow(.999, 537) + Math.pow(.999, 538) + Math.pow(.999, 539) +
Math.pow(.999, 540) + Math.pow(.999, 541) + Math.pow(.999, 542) +
Math.pow(.999, 543) + Math.pow(.999, 544) + Math.pow(.999, 545) +
Math.pow(.999, 546) + Math.pow(.999, 547) + Math.pow(.999, 548) +
Math.pow(.999, 549) + Math.pow(.999, 550) + Math.pow(.999, 551) +
Math.pow(.999, 552) + Math.pow(.999, 553) + Math.pow(.999, 554) +
Math.pow(.999, 555) + Math.pow(.999, 556) + Math.pow(.999, 557) +
Math.pow(.999, 558) + Math.pow(.999, 559) + Math.pow(.999, 560) +
Math.pow(.999, 561) + Math.pow(.999, 562) + Math.pow(.999, 563) +
Math.pow(.999, 564) + Math.pow(.999, 565) + Math.pow(.999, 566) +
Math.pow(.999, 567) + Math.pow(.999, 568) + Math.pow(.999, 569) +
Math.pow(.999, 570) + Math.pow(.999, 571) + Math.pow(.999, 572) +
Math.pow(.999, 573) + Math.pow(.999, 574) + Math.pow(.999, 575) +
Math.pow(.999, 576) + Math.pow(.999, 577) + Math.pow(.999, 578) +
Math.pow(.999, 579) + Math.pow(.999, 580) + Math.pow(.999, 581) +
Math.pow(.999, 582) + Math.pow(.999, 583) + Math.pow(.999, 584) +
Math.pow(.999, 585) + Math.pow(.999, 586) + Math.pow(.999, 587) +
Math.pow(.999, 588) + Math.pow(.999, 589) + Math.pow(.999, 590) +
Math.pow(.999, 591) + Math.pow(.999, 592) + Math.pow(.999, 593) +
Math.pow(.999, 594) + Math.pow(.999, 595) + Math.pow(.999, 596) +
Math.pow(.999, 597) + Math.pow(.999, 598) + Math.pow(.999, 599) +
Math.pow(.999, 600) + Math.pow(.999, 601) + Math.pow(.999, 602) +
Math.pow(.999, 603) + Math.pow(.999, 604) + Math.pow(.999, 605) +
Math.pow(.999, 606) + Math.pow(.999, 607) + Math.pow(.999, 608) +
Math.pow(.999, 609) + Math.pow(.999, 610) + Math.pow(.999, 611) +
Math.pow(.999, 612) + Math.pow(.999, 613) + Math.pow(.999, 614) +
Math.pow(.999, 615) + Math.pow(.999, 616) + Math.pow(.999, 617) +
Math.pow(.999, 618) + Math.pow(.999, 619) + Math.pow(.999, 620) +
Math.pow(.999, 621) + Math.pow(.999, 622) + Math.pow(.999, 623) +
Math.pow(.999, 624) + Math.pow(.999, 625) + Math.pow(.999, 626) +
Math.pow(.999, 627) + Math.pow(.999, 628) + Math.pow(.999, 629) +
Math.pow(.999, 630) + Math.pow(.999, 631) + Math.pow(.999, 632) +
Math.pow(.999, 633) + Math.pow(.999, 634) + Math.pow(.999, 635) +
Math.pow(.999, 636) + Math.pow(.999, 637) + Math.pow(.999, 638) +
Math.pow(.999, 639) + Math.pow(.999, 640) + Math.pow(.999, 641) +
Math.pow(.999, 642) + Math.pow(.999, 643) + Math.pow(.999, 644) +
Math.pow(.999, 645) + Math.pow(.999, 646) + Math.pow(.999, 647) +
Math.pow(.999, 648) + Math.pow(.999, 649) + Math.pow(.999, 650) +
Math.pow(.999, 651) + Math.pow(.999, 652) + Math.pow(.999, 653) +
Math.pow(.999, 654) + Math.pow(.999, 655) + Math.pow(.999, 656) +
Math.pow(.999, 657) + Math.pow(.999, 658) + Math.pow(.999, 659) +
Math.pow(.999, 660) + Math.pow(.999, 661) + Math.pow(.999, 662) +
Math.pow(.999, 663) + Math.pow(.999, 664) + Math.pow(.999, 665) +
Math.pow(.999, 666) + Math.pow(.999, 667) + Math.pow(.999, 668) +
Math.pow(.999, 669) + Math.pow(.999, 670) + Math.pow(.999, 671) +
Math.pow(.999, 672) + Math.pow(.999, 673) + Math.pow(.999, 674) +
Math.pow(.999, 675) + Math.pow(.999, 676) + Math.pow(.999, 677) +
Math.pow(.999, 678) + Math.pow(.999, 679) + Math.pow(.999, 680) +
Math.pow(.999, 681) + Math.pow(.999, 682) + Math.pow(.999, 683) +
Math.pow(.999, 684) + Math.pow(.999, 685) + Math.pow(.999, 686) +
Math.pow(.999, 687) + Math.pow(.999, 688) + Math.pow(.999, 689) +
Math.pow(.999, 690) + Math.pow(.999, 691) + Math.pow(.999, 692) +
Math.pow(.999, 693) + Math.pow(.999, 694) + Math.pow(.999, 695) +
Math.pow(.999, 696) + Math.pow(.999, 697) + Math.pow(.999, 698) +
Math.pow(.999, 699) + Math.pow(.999, 700) + Math.pow(.999, 701) +
Math.pow(.999, 702) + Math.pow(.999, 703) + Math.pow(.999, 704) +
Math.pow(.999, 705) + Math.pow(.999, 706) + Math.pow(.999, 707) +
Math.pow(.999, 708) + Math.pow(.999, 709) + Math.pow(.999, 710) +
Math.pow(.999, 711) + Math.pow(.999, 712) + Math.pow(.999, 713) +
Math.pow(.999, 714) + Math.pow(.999, 715) + Math.pow(.999, 716) +
Math.pow(.999, 717) + Math.pow(.999, 718) + Math.pow(.999, 719) +
Math.pow(.999, 720) + Math.pow(.999, 721) + Math.pow(.999, 722) +
Math.pow(.999, 723) + Math.pow(.999, 724) + Math.pow(.999, 725) +
Math.pow(.999, 726) + Math.pow(.999, 727) + Math.pow(.999, 728) +
Math.pow(.999, 729) + Math.pow(.999, 730) + Math.pow(.999, 731) +
Math.pow(.999, 732) + Math.pow(.999, 733) + Math.pow(.999, 734) +
Math.pow(.999, 735) + Math.pow(.999, 736) + Math.pow(.999, 737) +
Math.pow(.999, 738) + Math.pow(.999, 739) + Math.pow(.999, 740) +
Math.pow(.999, 741) + Math.pow(.999, 742) + Math.pow(.999, 743) +
Math.pow(.999, 744) + Math.pow(.999, 745) + Math.pow(.999, 746) +
Math.pow(.999, 747) + Math.pow(.999, 748) + Math.pow(.999, 749) +
Math.pow(.999, 750) + Math.pow(.999, 751) + Math.pow(.999, 752) +
Math.pow(.999, 753) + Math.pow(.999, 754) + Math.pow(.999, 755) +
Math.pow(.999, 756) + Math.pow(.999, 757) + Math.pow(.999, 758) +
Math.pow(.999, 759) + Math.pow(.999, 760) + Math.pow(.999, 761) +
Math.pow(.999, 762) + Math.pow(.999, 763) + Math.pow(.999, 764) +
Math.pow(.999, 765) + Math.pow(.999, 766) + Math.pow(.999, 767) +
Math.pow(.999, 768) + Math.pow(.999, 769) + Math.pow(.999, 770) +
Math.pow(.999, 771) + Math.pow(.999, 772) + Math.pow(.999, 773) +
Math.pow(.999, 774) + Math.pow(.999, 775) + Math.pow(.999, 776) +
Math.pow(.999, 777) + Math.pow(.999, 778) + Math.pow(.999, 779) +
Math.pow(.999, 780) + Math.pow(.999, 781) + Math.pow(.999, 782) +
Math.pow(.999, 783) + Math.pow(.999, 784) + Math.pow(.999, 785) +
Math.pow(.999, 786) + Math.pow(.999, 787) + Math.pow(.999, 788) +
Math.pow(.999, 789) + Math.pow(.999, 790) + Math.pow(.999, 791) +
Math.pow(.999, 792) + Math.pow(.999, 793) + Math.pow(.999, 794) +
Math.pow(.999, 795) + Math.pow(.999, 796) + Math.pow(.999, 797) +
Math.pow(.999, 798) + Math.pow(.999, 799) + Math.pow(.999, 800) +
Math.pow(.999, 801) + Math.pow(.999, 802) + Math.pow(.999, 803) +
Math.pow(.999, 804) + Math.pow(.999, 805) + Math.pow(.999, 806) +
Math.pow(.999, 807) + Math.pow(.999, 808) + Math.pow(.999, 809) +
Math.pow(.999, 810) + Math.pow(.999, 811) + Math.pow(.999, 812) +
Math.pow(.999, 813) + Math.pow(.999, 814) + Math.pow(.999, 815) +
Math.pow(.999, 816) + Math.pow(.999, 817) + Math.pow(.999, 818) +
Math.pow(.999, 819) + Math.pow(.999, 820) + Math.pow(.999, 821) +
Math.pow(.999, 822) + Math.pow(.999, 823) + Math.pow(.999, 824) +
Math.pow(.999, 825) + Math.pow(.999, 826) + Math.pow(.999, 827) +
Math.pow(.999, 828) + Math.pow(.999, 829) + Math.pow(.999, 830) +
Math.pow(.999, 831) + Math.pow(.999, 832) + Math.pow(.999, 833) +
Math.pow(.999, 834) + Math.pow(.999, 835) + Math.pow(.999, 836) +
Math.pow(.999, 837) + Math.pow(.999, 838) + Math.pow(.999, 839) +
Math.pow(.999, 840) + Math.pow(.999, 841) + Math.pow(.999, 842) +
Math.pow(.999, 843) + Math.pow(.999, 844) + Math.pow(.999, 845) +
Math.pow(.999, 846) + Math.pow(.999, 847) + Math.pow(.999, 848) +
Math.pow(.999, 849) + Math.pow(.999, 850) + Math.pow(.999, 851) +
Math.pow(.999, 852) + Math.pow(.999, 853) + Math.pow(.999, 854) +
Math.pow(.999, 855) + Math.pow(.999, 856) + Math.pow(.999, 857) +
Math.pow(.999, 858) + Math.pow(.999, 859) + Math.pow(.999, 860) +
Math.pow(.999, 861) + Math.pow(.999, 862) + Math.pow(.999, 863) +
Math.pow(.999, 864) + Math.pow(.999, 865) + Math.pow(.999, 866) +
Math.pow(.999, 867) + Math.pow(.999, 868) + Math.pow(.999, 869) +
Math.pow(.999, 870) + Math.pow(.999, 871) + Math.pow(.999, 872) +
Math.pow(.999, 873) + Math.pow(.999, 874) + Math.pow(.999, 875) +
Math.pow(.999, 876) + Math.pow(.999, 877) + Math.pow(.999, 878) +
Math.pow(.999, 879) + Math.pow(.999, 880) + Math.pow(.999, 881) +
Math.pow(.999, 882) + Math.pow(.999, 883) + Math.pow(.999, 884) +
Math.pow(.999, 885) + Math.pow(.999, 886) + Math.pow(.999, 887) +
Math.pow(.999, 888) + Math.pow(.999, 889) + Math.pow(.999, 890) +
Math.pow(.999, 891) + Math.pow(.999, 892) + Math.pow(.999, 893) +
Math.pow(.999, 894) + Math.pow(.999, 895) + Math.pow(.999, 896) +
Math.pow(.999, 897) + Math.pow(.999, 898) + Math.pow(.999, 899) +
Math.pow(.999, 900) + Math.pow(.999, 901) + Math.pow(.999, 902) +
Math.pow(.999, 903) + Math.pow(.999, 904) + Math.pow(.999, 905) +
Math.pow(.999, 906) + Math.pow(.999, 907) + Math.pow(.999, 908) +
Math.pow(.999, 909) + Math.pow(.999, 910) + Math.pow(.999, 911) +
Math.pow(.999, 912) + Math.pow(.999, 913) + Math.pow(.999, 914) +
Math.pow(.999, 915) + Math.pow(.999, 916) + Math.pow(.999, 917) +
Math.pow(.999, 918) + Math.pow(.999, 919) + Math.pow(.999, 920) +
Math.pow(.999, 921) + Math.pow(.999, 922) + Math.pow(.999, 923) +
Math.pow(.999, 924) + Math.pow(.999, 925) + Math.pow(.999, 926) +
Math.pow(.999, 927) + Math.pow(.999, 928) + Math.pow(.999, 929) +
Math.pow(.999, 930) + Math.pow(.999, 931) + Math.pow(.999, 932) +
Math.pow(.999, 933) + Math.pow(.999, 934) + Math.pow(.999, 935) +
Math.pow(.999, 936) + Math.pow(.999, 937) + Math.pow(.999, 938) +
Math.pow(.999, 939) + Math.pow(.999, 940) + Math.pow(.999, 941) +
Math.pow(.999, 942) + Math.pow(.999, 943) + Math.pow(.999, 944) +
Math.pow(.999, 945) + Math.pow(.999, 946) + Math.pow(.999, 947) +
Math.pow(.999, 948) + Math.pow(.999, 949) + Math.pow(.999, 950) +
Math.pow(.999, 951) + Math.pow(.999, 952) + Math.pow(.999, 953) +
Math.pow(.999, 954) + Math.pow(.999, 955) + Math.pow(.999, 956) +
Math.pow(.999, 957) + Math.pow(.999, 958) + Math.pow(.999, 959) +
Math.pow(.999, 960) + Math.pow(.999, 961) + Math.pow(.999, 962) +
Math.pow(.999, 963) + Math.pow(.999, 964) + Math.pow(.999, 965) +
Math.pow(.999, 966) + Math.pow(.999, 967) + Math.pow(.999, 968) +
Math.pow(.999, 969) + Math.pow(.999, 970) + Math.pow(.999, 971) +
Math.pow(.999, 972) + Math.pow(.999, 973) + Math.pow(.999, 974) +
Math.pow(.999, 975) + Math.pow(.999, 976) + Math.pow(.999, 977) +
Math.pow(.999, 978) + Math.pow(.999, 979) + Math.pow(.999, 980) +
Math.pow(.999, 981) + Math.pow(.999, 982) + Math.pow(.999, 983) +
Math.pow(.999, 984) + Math.pow(.999, 985) + Math.pow(.999, 986) +
Math.pow(.999, 987) + Math.pow(.999, 988) + Math.pow(.999, 989) +
Math.pow(.999, 990) + Math.pow(.999, 991) + Math.pow(.999, 992) +
Math.pow(.999, 993) + Math.pow(.999, 994) + Math.pow(.999, 995) +
Math.pow(.999, 996) + Math.pow(.999, 997) + Math.pow(.999, 998) +
Math.pow(.999, 999);
System.out.println(System.currentTimeMillis() - start);
}

// StringBuffer s = new StringBuffer(" sum = 0");
// for (int n = 0; n < N; n++)
// s.append(" + Math.pow(.999, " + n + ")");
// s.append(";");
// System.out.println(s);
}
 
A

Aaron Fude

For the record, the computational efficiency you are suggesting speeds up
the code by a factor of 100. All I'm saying, why not speed it up by a factor
of 2 or 3 on top of that. (I think it would be even better than a factor of
2 or 3 on top of that since it's obvious that most of the time is spent
computing powers and not performing the loop. Once computing the powers is
removed and only looping remains, unwinding and compiling should make a big
difference.)
 
B

bugbear

Aaron Fude wrote:

I think a book on numerical analysis would serve
you better than one on compiler design.

BugBear
 
M

Mark Thornton

Aaron said:
For the record, the computational efficiency you are suggesting speeds up
the code by a factor of 100. All I'm saying, why not speed it up by a factor
of 2 or 3 on top of that.
Click to expand...

You would be better off spending the time ensuring that the calculation
is numerically stable. As I indicated, a better understanding of
numerical analysis may well allow a far larger improvement in speed.

Mark Thornton
 

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments. After that, you can post your question and our members will help you out.

Ask a Question

Similar Threads

Custom matrix multiplication produces different results to glm 0
Can anyone help me whats wrong with this 2
Trouble with prediction code, for the life of me I can't figure out why it isnt running properly. Help would be appreciated. 0
How to keep count of right answer and wrong answers in C++? 0
Comparison of Integer and Pointer (that's supposed to be an Integer). Where did I go wrong? 0
Crossword 14
please debug this generic program 4
Weird Behavior with Rays in C and OpenGL 4

Members online

No members online now.
Total: 22 (members: 0, guests: 22)
Robots: 162

Forum statistics

Threads
473,774
Messages
2,569,596
Members
45,130
Latest member
MitchellTe

Latest Threads

Top