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TI-68k vs HP49G+ BenchmarkBhuvanesh Bhattbbhatt1@towson.eduJune 12, 2005Machine configurationŸ Timing for the 68k was done on my Voyage 200:AMS 2.09168 KB free RAM, empty history area, no TSRs or kernels installedMode settings: DisplayDigits ﬁ Float12, Angle ﬁ Radian, ExponentialFormat ﬁ Normal, ComplexFormat ﬁ Rectangular, PrettyPrint ﬁ On, Exact/Approx ﬁ Auto.Ÿ HP49G+:ROM revision 1.23, CAS version 4.20031005233KB + 127KB free RAMFlag settings: {#8205010FF0h, #0h, #880404402028000h, #0h}TimingsŸ Time is given in seconds, unless otherwise notedŸ A blank square means that the timing has not yet been done for that exampleŸ Like other computer algebra benchmarks, the CAS timings include evaluation time but not display timeŸ The graphics performance benchmark includes computation time as well as rendering timeŸ The display-routine performance benchmark includes conversion/printing time but not evaluation timeOther notesŸ Free add-on programs are included in this benchmarkŸ Performance comparisons are done only when both machines have that particular functionalityŸ For the most part, this benchmark is currently focused on math problems, not applications such as engineeringTI-68k vs HP49G+ Benchmark 2Ÿ Inputs are normally given in TI syntaxŸ For floating point arithmetic, approximate values of the arguments shown are usedŸ Most of the CAS timing examples were chosen without evaluating them on either calculatorŸ A blank square in the functionality ...

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TI-68k vs HP49G+ Benchmark Bhuvanesh Bhatt bbhatt1@towson.edu June 12, 2005

Machine configuration  Timing for the 68k was done on my Voyage 200: AMS 2.09 168 KB free RAM, empty history area, no TSRs or kernels installed Mode settings: DisplayDigits ® Float12, Angle ® Radian, ExponentialFormat ® Normal, ComplexFormat ® Rectangular, PrettyPrint ® On, Exact/Approx ® Auto.  HP49G+: ROM revision 1.23, CAS version 4.20031005 233KB + 127KB free RAM Flag settings: {#8205010FF0h, #0h, #880404402028000h, #0h}

Timings  Time is given in seconds, unless otherwise noted  A blank square means that the timing has not yet been done for that example  Like other computer algebra benchmarks, the CAS timings include evaluation time but not display time  The graphics performance benchmark includes computation time as well as rendering time  The display-routine performance benchmark includes conversion/printing time but not evaluation time

Other notes  Free add-on programs are included in this benchmark  Performance comparisons are done only when both machines have that particular functionality  For the most part, this benchmark is currently focused on math problems, not applications such as engineering

TI-68k vs HP49G+ Benchmark

 Inputs are normally given in TI syntax  For floating point arithmetic, approximate values of the arguments shown are used  Most of the CAS timing examples were chosen without evaluating them on either calculator  A blank square in the functionality comparison means that the calculator does not have the functionality

To do  Include ASI, numerical inverse program, and HP49G+ programs, and add references to add-ons used  This might be a good chance to make a set of regression tests (the same as or similar to the examples below)  A few examples (e.g. for polynomial factorization) may have been from Bernard Parisse; add references if needed  Consider including non-doable ("give-up") examples in CAS performance table  Add to functionality table: Special f ctions (Airy, Euler numbers, Fibonacci, Fresnel, erf, exponential integrals, Ζ H s, a L un Hurwitz zeta,       , hypergeometrics, orthogonal polynomials, cyclotomic, number theoretical)  Things to try: Poles, essential singularities, branch cuts, and discontinuous functions for symbolic and numeric definite integration. functions and discontinuous functions for optimization. Non-squarefree polynomials and large coefficients for factorization. Finite (abs, sign, min, max, when) and infinite (floor, ceiling, round, fpart, mod, intdiv) functions for integration and equation solving. Integrands of type R x, "############## × ### x ### + #### c ## . x 2 + b  Include plot settings, such as window settings (typically standard V200 settings)  Functionality comparison: Systems of nonlinear polynomial equations (check how the numerical solvers handle discontinuities) Laurent series, asymptotic series, limits, hypergeometric summation Definite integration (check singularity detection, e.g. Ù (1/(sin(x)+2),x,0,2 Π );compare nInt(1/(x+1),x,-3,3)) Graphics (e.g. 68k ComplexMap vs 49G GridMap, vector plot, vector field plot, 3D parametric curves and surfaces)  Add some semblance of a summary

Disclaimer  It is significantly harder to develop a fair benchmark for symbolics than for numerics. The examples give only a rough idea of the strengths and weaknesses. Also, I do not know many of the methods used internally by the CAS's.

Bhuvanesh Bhatt (bbhatt1@towson.edu)

TI-68k vs HP49G+ Benchmark

Performance Problem 91356200129 + 7868039544 356530692 * 200980515627 7 477 ! 3 ! ! + !! 5 ! !! 3 ! * !! 5 ! ! ! 3 !  ! 5 ! ! I! 3 ! ! M !! 5 !! !!!!!!  13 cos H Π + ã L         400 gcd I 29 , 31 300 M 180 ! isPrime I 2 127 -1 M factor H 211 ! L factor H 4498414682539051 L factor H num1 L expand IH x + y L 99 M expand I 1  I x 6 + 1 M , x M expand H 1  poly5 L SqrFree H poly6, x L factor I x 100 -1 M factor I x 100 -1, x M factor H poly1 L  factor H poly2 L factor H ratfunc1 L solve H 3 x + 9 = 0, x L

TI HP 0.001  0.002  0.17  0.003  0.004  0.005  0.024  0.009  0.015                  11.42 7.23 0.29 0.51 34.95 32.42 5.21 7.10 5.74 3.50 72.16 No 6.47 225.3 0.88 4.09 12.32  2.05  1.09  7.26 Hang         0.06 

Comments Integer arithmetic -addition Integer arithmetic -multiplication Integer arithmetic -exponentiation Real floating point arithmetic -addition Real floating point arithmetic -multiplication Real floating point arithmetic -division Real floating point arithmetic -exponentiation Real floating point arithmetic -square root Real floating point arithmetic -cosine Real floating point arithmetic -arctangent Complex floating point arithmetic -addition Complex floating point arithmetic -multiplication Complex floating point arithmetic -division Complex floating point arithmetic -exponentiation Complex floating point arithmetic -square root Complex floating point arithmetic -absolute value Complex floating point arithmetic -natural logarithm Integer gcd Factorial Primality testing Integer factorization Integer factorization with moderately large factors Integer factorization with large factors Polynomial expansion Partial fraction expansion Partial fraction expansion Squarefree factorization Univariate factorization Univariate factorization Univariate factorization Univariate factorization with parameters Multivariate factorization Multivariate factorization Univariate polynomial solving

Bhuvanesh Bhatt (bbhatt1@towson.edu)

TI-68k vs HP49G+ Benchmark

solve I x 6 -x 4 -4 x 2 + 4 = 0, x M 1.37  cSolve I x 6 -x 4 -4 x 2 + 4 = 0, x M 1.63  cZeros I x 4 + x 3 + x 2 + x + 1, x M 9.05  cZeros I x 6 -1, x M 0.18 2.17 cZeros HH x ^ 5 -x L ^ 33, x L   solve HH x -7 L × H x -5 L × H x -3 L = 0, x L È x £ 6 0.13  cSolve H x ^ 5 + x + 1 = 0 and abs H x L = 1, x L   cSolve H x _ 5 = x _ and imag H x _ L ¹ 0, x _ L 1.32     solve I x 2 + y 2 = 1, 8 x, y <M 0.42  zeros I9 x 3 + 3 × x × y + y 3 , x + y 3 = , 8 x, y <M 7.19  cSolve I x 2 + y 2 = 1, 8 x, y <M 0.25  cZeros I9 x 3 + 3 × x × y + y 3 , x + y 3 = , 8 x, y <M 9.05  solve H polysys, 8 x, y <L   cSolve H polysys, 8 x, y <L      PolyGCD H poly7, poly8 L 0.28 0.47 PolyGCD H poly10, x ^ 5 -11 x + 9 L               â I x 71 , x, 60 M 0.30  â I 5 × x 11 + 3 × x 7 -7 × x 3 + 21 × x 2 -1, x, 5 M 0.12     â I!! x ! , x, 5 M 0.05 2.05 a * x â H ã , x, 100 L 0.82  â H cos H ln H x LL , x, 50 L 3.32  â I x sin H x L , x, 3 M 4.63  â H f H x L  g H x L , x, 4 L 11.47  limit I sin I!! x ! M , x, 0 M 0.01 0.25 limit HH 1 + 1  x L x , x, ¥ L 0.12 4.93 limit I x 12 ã -x , x, ¥ M 0.25 5.35 × limit II x 4 -6 x 3 + x 2 + 3 M  H x -1 L , x, 1 M 0.16 4.15 limit H sin H 1  x L , x, 0 L 0.04 2.31 limit I ln H x L  ! x ! ! , x, ¥ M 0.08 6.36 limit I! x !!!! + !!!! a !! -!! x ! , x, ¥ M 0.84 22.44 limit I x 22  77 x , x, ¥ M 0.72 11.01

Univariate polynomial solving -real solutions Univariate polynomial solving -complex solutions Univariate polynomial solving -complex solutions Univariate polynomial solving -complex solutions Univariate polynomial solving -complex solutions Univariate polynomial solving -constrained Univariate polynomial solving -constrained Univariate polynomial solving -constrained Univariate polynomial solving -parameters Multivariate polynomial solving -real solutions Multivariate polynomial solving -real solutions Multivariate polynomial solving -complex solutions Multivariate polynomial solving -complex solutions Polynomial system solving -real solutions Polynomial system solving -complex solutions Univariate polynomial solving H numeric L Univariate polynomial gcd Univariate polynomial gcd H partially factored input L Multivariate polynomial gcd Univariate polynomial reduction modulo a prime Polynomial reduction modulo a polynomial Univariate polynomial quotient and remainder Differentiation -polynomials Differentiation -polynomials Differentiation -rational functions Differentiation -algebraic functions H Â 3 L Differentiation -transcendental functions Differentiation -compositions of functions Differentiation -compositions of functions Differentiation -undefined functions Limit Limit Limit Limit -pole Limit -essential singularity Limit Limit Limit

Bhuvanesh Bhatt (bbhatt1@towson.edu)

TI-68k vs HP49G+ Benchmark limit IH 3 x + 5 x L 1  x , x, ¥ M taylor I x 3 + 2 x, x, 3, 2 M taylor H tan H x L , x, 10 L taylor H ln H x + 1 L , x, 20 L taylor H sin H cos H x LL , x, 8, 2 L taylor J 1  "#################### # 1 -v 2  c 2 , v, 8 N  Series J  ã  x x, 0, 4 N  x2 , Series J  x  2  + 1   a  2  , x, ¥ , 0 N Series I ã 1  x , x, ¥ , 5 M taylor IÙ 0log H z L ã sin H x L â x, z, 4, 1 M Ù H x + a L 100 â x x +  7 à JH x -1 L 26 + J  2    !  5 !!  !  N 7 N â x Ù poly9 â x Ù H sin H x L × cos n H x LL â x Ù I cos H ln H x LL  I x × ! s !! i ! n !! !H!! l !! n !! !H!! x !!L!L! ! MM â x Ù I 1  I x 8 + 1 MM â x Ù J  x  4  x  +  2   x  +  2  1  +   1  N â x Horowitz I poly3, H x 7 -x + 1 L 2 , x M à"## x ### + ########!#! x # ! # â x Ù I x 2 × I a × x 3 + b × x 2 M 1  3 M â x Ù I x 5  I 4 x 2 + 9 M 1  3 M â x Ù !!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! I x  a × x 2 + b × x + c M â x Ù I 1  I!! x ! + x 1  3 MM â x Ù I x  !! 1 !!! -!!!! x !! 4 ! ! M â x Ù I 1 -!! x ! M p -1 â x Ù j ik  "  #  # x  #  # 2 # 1   #  ## +  #  #  # 1 #  #  +  "  # x  ##  # 2  ##  #  1 # + #  #  0 # 1  #  # 0  # 0 #  #  0 #  #  # 0 #  #  # 0  #  #  y z { â x Ù H sin H a × x + b L × sin H x L 5 L â x

0.37 59.51 Limit 0.07  Taylor series -polynomials H about 2 to order 3 L 13.00  Taylor series H about zero to order 10 L 0.63  Taylor series 12.47  Taylor series 9.98  Taylor series   Taylor series 0.49 1.94 Laurent series H about zero to order 4 L 1.96 1.01 Asymptotic series 0.20 1.20 Asymptotic series   Series of definite integrals 0.027 236.9 Indefinite integration -factored polynomials 0.027 115.8 Indefinite integration -partially factored polynomials 3.05  Indefinite integration -high degree polynomials 0.38 No Indefinite integration -f H x L × f ' H x L 0.14 2.80 Indefinite integration -f ' H x L  f H x L 19.26 No Indefinite integration -rational functions 1.46 1.78 Indefinite integration -rational functions  2.09 Indefinite integration -rational functions  Hang Indefinite integration -algebraic functions 9.05 No Indefinite integration -algebraic functions 0.87 23.49 Indefinite integration -algebraic functions 1.04 15.53 Indefinite integration -algebraic functions 0.53 5.99 Indefinite integration -algebraic functions 0.23 No Indefinite integration -algebraic functions 1.02 No Indefinite integration -algebraic functions H G & R-based L 0.23 7.09 Indefinite integration -algebraic functions H Â 4 L 3.16 33.83 Indefinite integration -transcendental functions

Bhuvanesh Bhatt (bbhatt1@towson.edu)

TI-68k vs HP49G+ Benchmark Ù I sin H x L 12 × cos H x L 29 M â x 11.2 7.15 Ù H ã x × sin 2 H x LM â x 0.13 2.34 Ù I  ã  3  ã  x   x  -  1  N â x 1.31 3.69 Ù I ln H x L 4  x m M â x 1.03 No Ù sin H log H x LL â x   Ù H x × a x × cos H x LL â x 0.94 8.61 Ù I x × sinh -1 H a  x LM â x 0.62 No Ù I tan -1 I  xx  --  3  5  M â x 1.10 3.56 ! ! ################## à J    !! 5  !  3 !    x  2   -1 N "# -3 x 2 + 4 ## x #### + #### 2 #######!# 3 ### â x 7.89 12.44 !!!!!!!!!!!! ! Ù tan H x L â x Hang 5.40 Ù J f H x L g H x L × J  f  '  H  fx   LH  ×  xg  L  H   x   L + ln H f H x LL × g ' H x LNN â x 1.00     0.29 Hang Ù aa -+ 11 H  x  - a  1  L  9   9   9  â x Ù 1 ¥  z  1  n  â z É n > 1      rr !!!!!!!!!!!!! ! Ù -x 2 -r 2 â x 0.84 4.45 Ù 01 I 1 -!! x ! M p -1 â x Ë p > 0   a x × ã x â x 0.32 5.35 Ù -¥ Ù 0 Π  4 H tan 5 H x LL â x 62.47 4.70 Π  s  i  n   3  H  x   L  â x 3.63 17.93 Ù 0 p + q cos H x L Ù 0 ¥ t 2 × ã -s * t â t É s > 0 1.47  Ù 22 H abs H cos H z LLL â z 12.43 3.60 -   Ù 0 ¥ H f H t L  t L â t 0.16 2.56 Ù -Μ¥ I x × ã H x -Μ L Σ  H 2 Σ LM â x É Σ > 0   Ù 0 ¥ J x × ã -x2 N â x 0.43 1.52 Ù ¥ HH 1x +  +  xa   LL  pp  -+    11  â x Ë a > 0    0 Ù 0 ¥ sech H a × x L â x É a > 0   Ù -¥¥  x  4  x  +  2   x  +  2  1  +    1  â x 2.94 6.91 CPVInt I 1  I 5 x 4 + 6 x M , x, -1, 1 M 5.87 3.67

Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions Indefinite integration -transcendental functions H Â 1 L Indefinite integration -mixed functions Indefinite integration -mixed functions Indefinite integration -undefined functions Definite integration -polynomials Definite integration -rational functions Definite integration -rational functions Definite integration -rational functions Definite integration -algebraic functions Definite integration -algebraic functions H Â 5 L Definite integration -transcendental functions Definite integration -transcendental functions H Â 5 L Definite integration -transcendental functions H Â 5 L Definite integration -transcendental functions Definite integration -piecewise functions Definite integration -piecewise functions Definite integration -undefined functions Definite integration -other improper integrals Definite integration -other improper integrals Definite integration -other improper integrals H Â 5 L Definite integration -other improper integrals H Â 5 L Definite integration -contour-type improper integrals Definite integration -principal value integrals

Bhuvanesh Bhatt (bbhatt1@towson.edu)

TI-68k vs HP49G+ Benchmark

CPVInt H tan H x L , x, Π  4, 3 Π  4 L CPVInt H 1  H x × ln H x LL , x, 1  2, 2 L Ù -11 Ù -11 1 â x â y 4 z -y2 1 â x â y â z Ù 04 Ù 02 * !! z ! Ù 0 "############### # Ù 01 Ù 0x I x 2 + y 2 M â y â x nInt I x 2 -2 x + 3, x, 0, 1 M nInt I 1  I x 4 + x 2 + 9  10 M , x, -1, 1 M nInt H sec H tan H x LL , x, 0, 1 L nInt H ln H x L , x, -1, 1 L nInt H cos H x L , x, -10, 10 L nInt H when H x ³ 0 and x < 0.3, 0, 1 L , x, 0, 1 L nInt H min H sin H x L , cos H x LL , x, -3 Π  2, 3 Π  2 L ! ! nInt I 1 !! x !¤ , x, -1, 1 M nInt I 1  x 8 , x, -Π , Π M nInt J  1  x  - sc  i   no  H s  H x  x  LL  , x, 0, Π  2 N nInt I!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! M H x + 1 L × H x -1 L × ln H x L , x, 0, 1 nInt I x 3 × ln I¡I x 2 -1 M I x 2 -2 M¥M , x, 0, 3 M nInt I sin H x L  I x 100 + 1 M , x, 1, ¥ M H x -20 L 2 i 1 -      y    !!  !  !!  !!  × ã 2 * 102 , x, nInt j k 10 ! 2 Π 104, 136 { z nInt I 1  !! x ! , x, 0, ¥ M nInt I ln H x L  I 10 × x 2 + 1 M , x, 0, ¥ M 3  2 nInt I nInt I abs H x -y L , x, 0, 1 M , y, 0, 1 M Laplace H t 7 × sin H t L , t, s L 1 ILaplace J        , s, t N 1 + s2 + s4  deSolve I y ' = x × y 2 , x, y M deSolve H y ' -x × y = 1, x, y L deSolve I x 2 × y '' + a × x × y ' = ln H x L , x, y M deSolve H y × y ' = 1, x, y L deSolve I x 5 × y ' + y 2 -4 x 4 × y + x 8 = 0, x, y M deSolve H ode1, x, y L deSolve I y '' + y × H y ' L 3 = 0, x, y M deSolve I 8 × y '' + 9 × H y ' L 4 = 0, x, y M ! deSolve I y '' = a × H!! y !! ' !!L! 2 !!!! + !!!! 1 ! , x, y M deSolve H y '' = ã y , x, y L

4.38 5.60 0.02 0.76 0.11 1.20             7.51                

3.83 3.16 0.65 9.25 1.78 ?   No 13.8  Hang                       

Definite integration -principal value integrals Definite integration -principal value integrals Definite integration -multidimensional Definite integration -multidimensional Definite integration -multidimensional Numeric integration -polynomials Numeric integration -rational functions Numeric integration -finite intervals Numeric integration -finite intervals Numeric integration -even integrands Numeric integration -piecewise integrands Numeric integration -piecewise integrands Numeric integration -singularity Numeric integration -singularity Numeric integration -singularity at endpoint H Â 5 L Numeric integration -singularity at endpoint Numeric integration -singularity Numeric integration -oscillatory integrands Numeric integration -Gaussians Numeric integration -other improper integrals Numeric integration -other improper integrals Numeric integration -multidimensional Laplace transform Inverse Laplace transform Linear constant coefficient ODE Linear first-order variable coefficient ODE Linear first-order variable coefficient ODE Linear second-order variable coefficient ODE Nonlinear first-order ODE Nonlinear first-order ODE H Riccati-type L Nonlinear first-order ODE H Kamke 1.505 L Nonlinear second-order ODE Nonlinear second-order ODE Nonlinear second-order ODE Nonlinear second-order ODE

Bhuvanesh Bhatt (bbhatt1@towson.edu)