In this example we'll look at polynomial evaluation, this is not only an important use case, but it's one that number performs particularly well at because the expression templates completely eliminate all temporaries from a Horner polynomial evaluation scheme.

The following code evaluates sin(x) as a polynomial, accurate to at least 64 decimal places:

using boost::multiprecision::cpp_dec_float;
typedef boost::multiprecision::number<cpp_dec_float<64> > mp_type;

mp_type mysin(const mp_type& x)
{
  // Approximation of sin(x * pi/2) for -1 <= x <= 1, using an order 63 polynomial.
  static const std::array<mp_type, 32U> coefs =
  {{
    mp_type("+1.5707963267948966192313216916397514420985846996875529104874722961539082031431044993140174126711"), //"),
    mp_type("-0.64596409750624625365575656389794573337969351178927307696134454382929989411386887578263960484"), // ^3
    mp_type("+0.07969262624616704512050554949047802252091164235106119545663865720995702920146198554317279"), // ^5
    mp_type("-0.0046817541353186881006854639339534378594950280185010575749538605102665157913157426229824"), // ^7
    mp_type("+0.00016044118478735982187266087016347332970280754062061156858775174056686380286868007443"), // ^9
    mp_type("-3.598843235212085340458540018208389404888495232432127661083907575106196374913134E-6"), // ^11
    mp_type("+5.692172921967926811775255303592184372902829756054598109818158853197797542565E-8"), // ^13
    mp_type("-6.688035109811467232478226335783138689956270985704278659373558497256423498E-10"), // ^15
    mp_type("+6.066935731106195667101445665327140070166203261129845646380005577490472E-12"), // ^17
    mp_type("-4.377065467313742277184271313776319094862897030084226361576452003432E-14"), // ^19
    mp_type("+2.571422892860473866153865950420487369167895373255729246889168337E-16"), // ^21
    mp_type("-1.253899540535457665340073300390626396596970180355253776711660E-18"), // ^23
    mp_type("+5.15645517658028233395375998562329055050964428219501277474E-21"), // ^25
    mp_type("-1.812399312848887477410034071087545686586497030654642705E-23"), // ^27
    mp_type("+5.50728578652238583570585513920522536675023562254864E-26"), // ^29
    mp_type("-1.461148710664467988723468673933026649943084902958E-28"), // ^31
    mp_type("+3.41405297003316172502972039913417222912445427E-31"), // ^33
    mp_type("-7.07885550810745570069916712806856538290251E-34"), // ^35
    mp_type("+1.31128947968267628970845439024155655665E-36"), // ^37
    mp_type("-2.18318293181145698535113946654065918E-39"), // ^39
    mp_type("+3.28462680978498856345937578502923E-42"), // ^41
    mp_type("-4.48753699028101089490067137298E-45"), // ^43
    mp_type("+5.59219884208696457859353716E-48"), // ^45
    mp_type("-6.38214503973500471720565E-51"), // ^47
    mp_type("+6.69528558381794452556E-54"), // ^49
    mp_type("-6.47841373182350206E-57"), // ^51
    mp_type("+5.800016389666445E-60"), // ^53
    mp_type("-4.818507347289E-63"), // ^55
    mp_type("+3.724683686E-66"), // ^57
    mp_type("-2.6856479E-69"), // ^59
    mp_type("+1.81046E-72"), // ^61
    mp_type("-1.133E-75"), // ^63
  }};

  const mp_type v = x * 2 / boost::math::constants::pi<mp_type>();
  const mp_type x2 = (v * v);
  //
  // Polynomial evaluation follows, if mp_type allocates memory then
  // just one such allocation occurs - to initialize the variable "sum" -
  // and no temporaries are created at all.
  //
  const mp_type sum = (((((((((((((((((((((((((((((((     + coefs[31U]
                                                     * x2 + coefs[30U])
                                                     * x2 + coefs[29U])
                                                     * x2 + coefs[28U])
                                                     * x2 + coefs[27U])
                                                     * x2 + coefs[26U])
                                                     * x2 + coefs[25U])
                                                     * x2 + coefs[24U])
                                                     * x2 + coefs[23U])
                                                     * x2 + coefs[22U])
                                                     * x2 + coefs[21U])
                                                     * x2 + coefs[20U])
                                                     * x2 + coefs[19U])
                                                     * x2 + coefs[18U])
                                                     * x2 + coefs[17U])
                                                     * x2 + coefs[16U])
                                                     * x2 + coefs[15U])
                                                     * x2 + coefs[14U])
                                                     * x2 + coefs[13U])
                                                     * x2 + coefs[12U])
                                                     * x2 + coefs[11U])
                                                     * x2 + coefs[10U])
                                                     * x2 + coefs[9U])
                                                     * x2 + coefs[8U])
                                                     * x2 + coefs[7U])
                                                     * x2 + coefs[6U])
                                                     * x2 + coefs[5U])
                                                     * x2 + coefs[4U])
                                                     * x2 + coefs[3U])
                                                     * x2 + coefs[2U])
                                                     * x2 + coefs[1U])
                                                     * x2 + coefs[0U])
                                                     * v;

  return sum;
}

Calling the function like so:

mp_type pid4 = boost::math::constants::pi<mp_type>() / 4;
std::cout << std::setprecision(std::numeric_limits< ::mp_type>::digits10) << std::scientific;
std::cout << mysin(pid4) << std::endl;

Yields the expected output:

7.0710678118654752440084436210484903928483593768847403658833986900e-01