# torture test: integration

## round #3: definite integration

this test is for those machine purporting to support definite integration, numerical or otherwise. in each test a definite integral is to be performed. some of the tests have a help option which makes it easier for the machine. use of help option limits the score to 1/2 point.

rules:

• infinity can be replaced by the smallest integer, n, where f(n) underflows.
• machines that need f(x) to be defined at the limit points may have those limits adjusted by a small delta (eg 1e-10), but this counts as a help option.
• since most machines have 10 digits, five correct digits is required to score.
• manual analytic rearrangement or substitution of the formulae is not allowed. the machines have to work with the formula as stated, but non existing functions can be programmed around, eg abs(x) = sqrt(x^2) since doesnt make it any easier numerically.
• machines should not require the user to explicitly set the number of iterations for its own algorithm. selection of overall accuracy is, of course, allowed.
• calculations have to take less than 5 mins.

### 3.1: integrate(cos(ln(x)), 0, 1)

answer: 1/2

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 5.0x10^-1 420 needed 1e-10 for zero. (manual selection of 512 iterations gives 5.0x10-1 in 3:31) 0 casio fx-3900pv 5.0x10^-1 108 needed 1e-10 for zero 1/2 casio fx-5500l 0.5 45 needed 1e-10 for zero 1/2 casio fx-4800p 0.5 30 needed 1e-10 for zero 1/2 casio fx-991es 0.5 260 needed 1e-10 for zero. very slow for a modern machine. 1/2 ti89 1/2 0.5 symbolic 1 sharp el-5020 0.499990362 150 needed 1e-10 for zero, n = 256 1/2 sharp el-9900 0.500000696 8.5 default 1e-5 accuracy 1 sharp el-520w 0.5000X 300 failed to meet accuracy in all of 5 minutes - a complete donkey! 0 hp48g 0.50000 54 1 hp32sii 0.50001 81 program faster than equation. fix 5 1 hp 71b (+math rom) .500001117977 127 integral(0,1,1e-5,cos(log(ivar))). 1 hp15c 0.4999 240 vague & slow. did not meet accuracy criteria. 1/2

### 3.2: integrate(sqrt(x), 0, 2)

answer:  2*sqrt(2)^3/3 = 1.8856180831641267317

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 1.8856e00 75 correct 1 casio fx-3900pv 1.8856 25 correct 1 casio fx-5500l 1.8856 10 correct 1 casio fx-4800p 1.8856 7 correct 1 casio fx-991es 1.885618083 28 correct 1 ti89 4*sqrt(2)/3 0.5 symbolic 1 sharp el-5020 1.885598263 40 n=256 1 sharp el-9900 1.885618083 3.3 5 digits in 1.5 seconds. 1 sharp el-520w 1.885610822 22 used n=500, barely accurate enough 1 hp48g -(2*x^(3/2)/3/dx(x)|(x=0)+(2*x^(3/2)/3/dx(x)|(x=2)). ->NUM gives 1.88561808317 2 rather useless symbolic expression, but correct. 1 hp32sii 1.8856181e0 10 5 digits in 2 seconds 1 hp 71b (+math rom) 1.88561815442 2 integral(0,2,1e-5,sqr(ivar)) 1 hp15c 1.8856 20 correct 1

### 3.3: integrate(x*exp(-x), 0, inf)

answer: 1

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 1.0 240 used limit of inf = 228 1 casio fx-3900pv 1.0 58 used limit of inf = 228 1 casio fx-5500l 1. 27 used limit of inf = 228 1 casio fx-4800p 0.999 18 used limit of inf = 228. not accurate enough. did not accept int(xe-x,0,228,10). 0 casio fx-991es 1 30 used limit of inf = 228 1 ti89 1 1 symbolic 1 sharp el-5020 0.999997189 300 used limit of inf = 228, needed n=1000, only just made it in time 1 sharp el-9900 1 3 used limit inf=228 1 sharp el-520w 0.999995723 150 used limit inf=228. dreadful performance 1 hp48g 1.00000 30 use inf = 228 1 hp32sii 2.43212e-18 1 inf=1149. however x*exp(-x) < 1e-100 with inf=234 gives .99481 (not close enough). inf=228 gives .99563 (not close enough either). 0 hp71b (+math rom) .999999997617 92 integral(0,1148,1e-5,ivar*exp(-ivar)). use of built-in "inf" causes underflow warning. 1 hp15c 0.0147 13 wrong with inf = 228. the manual shows how to get this correct by splitting integral into two, but that's cheating here. 0

### 3.4: integrate(exp(-x)/x, 1, inf)

answer: 0.219383934

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 2.2e-1 240 used limit of inf = 228, but not accurate enough 0 casio fx-3900pv 2.2e-1 58 used limit of inf = 228, but not accurate enough 0 casio fx-5500l 0.22 29 used limit of inf = 228, but not accurate enough 0 casio fx-4800p 0.22 19 used limit of inf = 228, but not accurate enough 0 casio fx-991es 0.2193839344 37 used limit of inf = 228. good accuracy 1 ti89 0.219383934406 7 >=10 digits 1 sharp el-5020 0.219392 305 used limit of inf = 228, but not accurate enough even after n = 900 for 5 mins 0 sharp el-9900 0.219383934 3.2 1 sharp el-520w 0.21939 use inf=228 hp48g 0.219383934395 177 use inf=228 1 hp32sii 1.34793e-21 1 use inf=1142. inf=228 gives 0.00015 also wrong. 0 hp71b (+math rom) .219383941631 95 integral(1,1142,1e-5,exp(-ivar)/ivar). gives a few underflows, use of "inf" causes too many underflows. 1 hp15c 0.0001 15 wrong 0

### 3.5: integrate(sqrt(abs(x-1)), 0, 2)

answer: 4/3

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 1.3333e00 180 used abs(x) = sqrt(x^2) 1 casio fx-3900pv 1.3333 96 used abs(x) = sqrt(x^2) 1 casio fx-5500l 1.3333 22 used abs(x) = sqrt(x^2) 1 casio fx-4800p 1.33329 10 used abs function. barely accurate enough 1 casio fx-991es 1.333333333 58 used abs function. good accuracy, but a bit slow 1 ti89 4*sqrt(2)/3 0.5 symbolic understands abs. 1 sharp el-5020 4 used abs(x) = sqrt(x^2-2*x+1), binding of x^2 key was weird. cant get it to do this one. 0 sharp el-9900 1.333333333 7.4 using 1e-9 accuracy 1 hp48g 1.33333 152 symbolic doesn't handle abs function. cant use sqrt(x^2) trick symbolically. full (std) accuracy overran 5 mins. 1 hp32sii 1.33333 190 fix 5 1 hp71b (+math rom) 1.3333305072 250 integral(0,2,1e-5,sqr(abs(ivar-1))).  a bit slower than i expected. 1 hp15c N/A 300 still running after 5 mins. 0

### 3.6: integrate(exp(-x^2), -inf, inf)

answer: sqrt(pi) = 1.7724538509055160273

 calculator displayed result time (s) notes score casio fx-3500p, fx-180p 1.772453851e00 75 needed inf=16 1 casio fx-3900pv 1.772453851 28 needed inf=16 1 casio fx-5500l 1.7725 8 needed inf=16 1 casio fx-4800p 1.772453851 9 needed inf=16. good accuracy 1 casio fx-991es 1.772453851 48 needed inf=16. good accuracy 1 ti89 1.77245385091 5 numerical. unfortunately no symbolic sqrt(pi) 1 sharp el-5020 0.586 140 needed inf=16. cant get it to work 0 sharp el-9900 1.772453851 3 1 hp48g 1.77245385091 80 use inf=16 1 hp32sii 1.77245385092 230 use inf=34. fix 5in 59s 1 hp 71b (+math rom) 1.77245385295 60 integral(-32,32,1e-5,exp(-ivar^2)) 1 hp15c N/A 300 still running after 5 mins. this example is also in the manual and is solved by formula substitution. 0

 calculator overall score casio fx-3500p, fx-180p 67% casio fx-3900pv 75% casio fx-5500l 75% casio fx-4800p 58% casio fx-991es 92% ti89 100% sharp el-5020 42% sharp el-9900 100% hp48g 100% hp32sii 67% hp71b (+math rom) 100% hp15c 25%