Version 10 (modified by sam, 9 years ago) (diff)

typo

Trigonometric functions

Our research notes about implementation of fast trigonometric functions.

Other implementations

Minimax polynomial for sin(x)

Absolute error

Suppose we want to approximate sin(x) on [-π/2; π/2] with a polynomial P(x) such that the absolute error is never more than E:

\[\max_{x \in [-\pi/2, \pi/2]}{\big\vert\sin(x) - P(x)\big\vert} = E\]

We know sin(x) is an odd function, so instead we look for a polynomial Q(x) such that P(x) = xQ(x²), and we reduce the range to positive values:

\[\max_{x \in [0, \pi/2]}{\big\vert\sin(x) - xQ(x^2)\big\vert} = E\]

Substitute y for x²:

\[\max_{y \in [0, \pi^2/4]}{\big\lvert\sin(\sqrt{y}) - \sqrt{y}Q(y)\big\rvert} = E\]

Divide through by √y:

\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]

If we want to force the asymptotic behaviour at x=0, we substitute Q(y) with 1+yR(y):

\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - 1 - yR(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]

Divide through by y:

\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\dfrac{1}{|y\sqrt{y}|}}} = E\]

We then use the following code:

static real myfun(real const &y)
{
    real x = sqrt(y);
    return (sin(x) - x) / (x * y);
}

static real myerr(real const &y)
{
    return re(y * sqrt(y));
}

RemezSolver<6> solver;
solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 40);

These are the resulting R, Q and P:

\begin{eqnarray*}
R(y) & = & a_0 + a_1 y + a_2 y^2 + a_3 y^3 + a_4 y^4 + a_5 y^5 + a_6 y^6 \\
Q(y) & = & 1 + a_0 y + a_1 y^2 + a_2 y^3 + a_3 y^4 + a_4 y^5 + a_5 y^6 + a_6 y^7 \\
P(x) & = & x + a_0 x^3 + a_1 x^5 + a_2 x^7 + a_3 x^9 + a_4 x^{11} + a_5 x^{13} + a_6 x^{15} \\
\end{eqnarray*}

With the following coefficients:

a0 = -1.666666666666580938362041558393413542600e-1;
a1 = +8.333333333262715528278347301093116699226e-3;
a2 = -1.984126982005911547055378498482154233331e-4;
a3 = +2.755731607338689059680115311170244593349e-6;
a4 = -2.505185130214293461336864464029272975945e-8;
a5 = +1.604729591825977276746033955401272849354e-10;
a6 = -7.364589573262279656101943192883347174447e-13;

E = 1.098969630370672683831702893969063712485e-16;

Relative error

Searching for relative error instead:

\[\max_{x \in [-\pi/2, \pi/2]}{\dfrac{\big\vert\sin(x) - P(x)\big\vert}{|\sin(x)|}} = E\]

Using the same method as for absolute error, we get:

\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(y)}{y\sqrt{y}}\bigg\rvert}} = E\]
static real myfun(real const &y)
{
    real x = sqrt(y);
    return (sin(x) - x) / (x * y);
}

static real myerr(real const &y)
{
    real x = sqrt(y);
    return sin(x) / (x * y);
}

RemezSolver<6> solver;
solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 40);
a0 = -1.666666666666618655330686951220600109093e-1;
a1 = +8.333333333285509269011390197274270193963e-3;
a2 = -1.984126982504390943378441670654599796999e-4;
a3 = +2.755731659890484005079622148869365590255e-6;
a4 = -2.505188017067512158000464673183918592305e-8;
a5 = +1.604809231079007402834515514014751626424e-10;
a6 = -7.373308642081174610234470417310065337523e-13;

E = 1.536616934979294924070319645582179175464e-16;

Another example

Trying to reproduce Mike Acton's experimental result:

\[\max_{x \in [-1, 1]}{\dfrac{\big\vert\sin(\frac\pi{2}x) - P(x)\big\vert}{|\sin(\frac\pi{2}x)|}} = E\]

Using P(x) = xQ(x²):

\[\max_{y \in [0, 1]}{\dfrac{\bigg\lvert\dfrac{\sin(\frac\pi{2}\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(\frac\pi{2}\sqrt(y))}{\sqrt{y}}\bigg\rvert}} = E\]
static real myfun(real const &y)
{
    real x = sqrt(y);
    return sin(real::R_PI_2 * x) / x;
}

RemezSolver<4> solver;
solver.Run(real::R_1 >> 400, real::R_1, myfun, myfun, 40);
a0 = +1.570796318452974170937444182514099057668;
a1 = -6.459637106518004316895285560913086847441e-1;
a2 = +7.968967893119537554369879233657335437244e-2;
a3 = -4.673766402124326801818077428151491285886e-3;
a4 = +1.514849803879821510688050931907848092814e-4;

E = 5.310632770140865101487747414605249755889e-9;