Calculate (x exponent 0.19029) with low memory using lookup table?

Question

I'm writing a C program for a PIC micro-controller which needs to do a very specific exponential function. I need to calculate the following:

A = k . (1 - (p/p0)^0.19029)

k and p0 are constant, so it's all pretty simple apart from finding x^0.19029

(p/p0) ratio would always be in the range 0-1.

It works well if I add in math.h and use the power function, except that uses up all of the available 16 kB of program memory. Talk about bloatware! (Rest of program without power function = ~20% flash memory usage; add math.h and power function, =100%).

I'd like the program to do some other things as well. I was wondering if I can write a special case implementation for x^0.19029, maybe involving iteration and some kind of lookup table.

My idea is to generate a look-up table for the function x^0.19029, with perhaps 10-100 values of x in the range 0-1. The code would find a close match, then (somehow) iteratively refine it by re-scaling the lookup table values. However, this is where I get lost because my tiny brain can't visualise the maths involved.

Could this approach work?

Alternatively, I've looked at using Exp(x) and Ln(x), which can be implemented with a Taylor expansion. b^x can the be found with:

b^x = (e^(ln b))^x = e^(x.ln(b))

(See: Wikipedia - Powers via Logarithms)

This looks a bit tricky and complicated to me, though. Am I likely to get the implementation smaller then the compiler's math library, and can I simplify it for my special case (i.e. base = 0-1, exponent always 0.19029)?

Note that RAM usage is OK at the moment, but I've run low on Flash (used for code storage). Speed is not critical. Somebody has already suggested that I use a bigger micro with more flash memory, but that sounds like profligate wastefulness!

[EDIT] I was being lazy when I said "(p/p0) ratio would always be in the range 0-1". Actually it will never reach 0, and I did some calculations last night and decided that in fact a range of 0.3 - 1 would be quite adequate! This mean that some of the simpler solutions below should be suitable. Also, the "k" in the above is 44330, and I'd like the error in the final result to be less than 0.1. I guess that means an error in the (p/p0)^0.19029 needs to be less than 1/443300 or 2.256e-6

Answer 1

Use splines. The relevant part of the function is shown in the figure below. It varies approximately like the 5th root, so the problematic zone is close to p / p0 = 0. There is mathematical theory how to optimally place the knots of splines to minimize the error (see Carl de Boor: A Practical Guide to Splines). Usually one constructs the spline in B form ahead of time (using toolboxes such as Matlab's spline toolbox - also written by C. de Boor), then converts to Piecewise Polynomial representation for fast evaluation.

In C. de Boor, PGS, the function g(x) = sqrt(x + 1) is actually taken as an example (Chapter 12, Example II). This is exactly what you need here. The book comes back to this case a few times, since it is admittedly a hard problem for any interpolation scheme due to the infinite derivatives at x = -1. All software from PGS is available for free as PPPACK in netlib, and most of it is also part of SLATEC (also from netlib).

Edit (Removed)

(Multiplying by x once does not significantly help, since it only regularizes the first derivative, while all other derivatives at x = 0 are still infinite.)

Edit 2

My feeling is that optimally constructed splines (following de Boor) will be best (and fastest) for relatively low accuracy requirements. If the accuracy requirements are high (say 1e-8), one may be forced to get back to the algorithms that mathematicians have been researching for centuries. At this point, it may be best to simply download the sources of glibc and copy (provided GPL is acceptable) whatever is in

glibc-2.19/sysdeps/ieee754/dbl-64/e_pow.c

Since we don't have to include the whole math.h, there shouldn't be a problem with memory, but we will only marginally profit from having a fixed exponent.

Edit 3

Here is an adapted version of e_pow.c from netlib, as found by @Joni. This seems to be the grandfather of glibc's more modern implementation mentioned above. The old version has two advantages: (1) It is public domain, and (2) it uses a limited number of constants, which is beneficial if memory is a tight resource (glibc's version defines over 10000 lines of constants!). The following is completely standalone code, which calculates x^0.19029 for 0 <= x <= 1 to double precision (I tested it against Python's power function and found that at most 2 bits differed):

#define __LITTLE_ENDIAN

#ifdef __LITTLE_ENDIAN
#define __HI(x) *(1+(int*)&x)
#define __LO(x) *(int*)&x
#else
#define __HI(x) *(int*)&x
#define __LO(x) *(1+(int*)&x)
#endif

static const double
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
zero = 0.0,
one = 1.0,
two =  2.0,
two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
    /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/

double pow0p19029(double x)
{
    double y = 0.19029e+00;
    double z,ax,z_h,z_l,p_h,p_l;
    double y1,t1,t2,r,s,t,u,v,w;
    int i,j,k,n;
    int hx,hy,ix,iy;
    unsigned lx,ly;

    hx = __HI(x); lx = __LO(x);
    hy = __HI(y); ly = __LO(y);
    ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

    ax = x;
    /* special value of x */
    if(lx==0) {
        if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
            z = ax;           /*x is +-0,+-inf,+-1*/
            return z;
        }
    }

    s = one; /* s (sign of result -ve**odd) = -1 else = 1 */

    double ss,s2,s_h,s_l,t_h,t_l;
    n  = ((ix)>>20)-0x3ff;
    j  = ix&0x000fffff;
    /* determine interval */
    ix = j|0x3ff00000;      /* normalize ix */
    if(j<=0x3988E) k=0;     /* |x|<sqrt(3/2) */
    else if(j<0xBB67A) k=1; /* |x|<sqrt(3)   */
    else {k=0;n+=1;ix -= 0x00100000;}
    __HI(ax) = ix;

    /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
    u = ax-bp[k];           /* bp[0]=1.0, bp[1]=1.5 */
    v = one/(ax+bp[k]);
    ss = u*v;
    s_h = ss;
    __LO(s_h) = 0;
    /* t_h=ax+bp[k] High */
    t_h = zero;
    __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 
    t_l = ax - (t_h-bp[k]);
    s_l = v*((u-s_h*t_h)-s_h*t_l);
    /* compute log(ax) */
    s2 = ss*ss;
    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
    r += s_l*(s_h+ss);
    s2  = s_h*s_h;
    t_h = 3.0+s2+r;
    __LO(t_h) = 0;
    t_l = r-((t_h-3.0)-s2);
    /* u+v = ss*(1+...) */
    u = s_h*t_h;
    v = s_l*t_h+t_l*ss;
    /* 2/(3log2)*(ss+...) */
    p_h = u+v;
    __LO(p_h) = 0;
    p_l = v-(p_h-u);
    z_h = cp_h*p_h;         /* cp_h+cp_l = 2/(3*log2) */
    z_l = cp_l*p_h+p_l*cp+dp_l[k];
    /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
    t = (double)n;
    t1 = (((z_h+z_l)+dp_h[k])+t);
    __LO(t1) = 0;
    t2 = z_l-(((t1-t)-dp_h[k])-z_h);

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
    y1  = y;
    __LO(y1) = 0;
    p_l = (y-y1)*t1+y*t2;
    p_h = y1*t1;
    z = p_l+p_h;
    j = __HI(z);
    i = __LO(z);
    /*
     * compute 2**(p_h+p_l)
     */
    i = j&0x7fffffff;
    k = (i>>20)-0x3ff;
    n = 0;
    if(i>0x3fe00000) {            /* if |z| > 0.5, set n = [z+0.5] */
        n = j+(0x00100000>>(k+1));
        k = ((n&0x7fffffff)>>20)-0x3ff;      /* new k for n */
        t = zero;
        __HI(t) = (n&~(0x000fffff>>k));
        n = ((n&0x000fffff)|0x00100000)>>(20-k);
        if(j<0) n = -n;
        p_h -= t;
    } 
    t = p_l+p_h;
    __LO(t) = 0;
    u = t*lg2_h;
    v = (p_l-(t-p_h))*lg2+t*lg2_l;
    z = u+v;
    w = v-(z-u);
    t  = z*z;
    t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
    r  = (z*t1)/(t1-two)-(w+z*w);
    z  = one-(r-z);
    __HI(z) += (n<<20);
    return s*z;
}

Clearly, 50+ years of research have gone into this, so it's probably very hard to do any better. (One has to appreciate that there are 0 loops, only 2 divisions, and only 6 if statements in the whole algorithm!) The reason for this is, again, the behavior at x = 0, where all derivatives diverge, which makes it extremely hard to keep the error under control: I once had a spline representation with 18 knots that was good up to x = 1e-4, with absolute and relative errors < 5e-4 everywhere, but going to x = 1e-5 ruined everything again.

So, unless the requirement to go arbitrarily close to zero is relaxed, I recommend using the adapted version of e_pow.c given above.

Edit 4

Now that we know that the domain 0.3 <= x <= 1 is sufficient, and that we have very low accuracy requirements, Edit 3 is clearly overkill. As @MvG has demonstrated, the function is so well behaved that a polynomial of degree 7 is sufficient to satisfy the accuracy requirements, which can be considered a single spline segment. @MvG's solution minimizes the integral error, which already looks very good.

The question arises as to how much better we can still do? It would be interesting to find the polynomial of a given degree that minimizes the maximum error in the interval of interest. The answer is the minimax polynomial, which can be found using Remez' algorithm, which is implemented in the Boost library. I like @MvG's idea to clamp the value at x = 1 to 1, which I will do as well. Here is minimax.cpp:

#include <ostream>
#define TARG_PREC 64
#define WORK_PREC (TARG_PREC*2)

#include <boost/multiprecision/cpp_dec_float.hpp>
typedef boost::multiprecision::number<boost::multiprecision::cpp_dec_float<WORK_PREC> > dtype;
using boost::math::pow;

#include <boost/math/tools/remez.hpp>
boost::shared_ptr<boost::math::tools::remez_minimax<dtype> > p_remez;

dtype f(const dtype& x) {
    static const dtype one(1), y(0.19029);
    return one - pow(one - x, y);
}

void out(const char *descr, const dtype& x, const char *sep="") {
    std::cout << descr << boost::math::tools::real_cast<double>(x) << sep << std::endl;
}

int main() {
    dtype a(0), b(0.7);   // range to optimise over
    bool rel_error(false), pin(true);
    int orderN(7), orderD(0), skew(0), brake(50);

    int prec = 2 + (TARG_PREC * 3010LL)/10000;
    std::cout << std::scientific << std::setprecision(prec);

    p_remez.reset(new boost::math::tools::remez_minimax<dtype>(
        &f, orderN, orderD, a, b, pin, rel_error, skew, WORK_PREC));
    out("Max error in interpolated form: ", p_remez->max_error()); 

    p_remez->set_brake(brake);

    unsigned i, count(50);
    for (i = 0; i < count; ++i) {
        std::cout << "Stepping..." << std::endl;
        dtype r = p_remez->iterate();
        out("Maximum Deviation Found:                     ", p_remez->max_error());
        out("Expected Error Term:                         ", p_remez->error_term());
        out("Maximum Relative Change in Control Points:   ", r);
    }

    boost::math::tools::polynomial<dtype> n = p_remez->numerator();
    for(i = n.size(); i--; ) {
        out("", n[i], ",");
    }
}

Since all parts of boost that we use are header-only, simply build with:

c++ -O3 -I<path/to/boost/headers> minimax.cpp -o minimax

We finally get the coefficients, which are after multiplication by 44330:

24538.3409, -42811.1497, 34300.7501, -11284.1276, 4564.5847, 3186.7541, 8442.5236, 0.

The following error plot demonstrates that this is really the best possible degree-7 polynomial approximation, since all extrema are of equal magnitude (0.06659):

Should the requirements ever change (while still keeping well away from 0!), the C++ program above can be simply adapted to spit out the new optimal polynomial approximation.

Answer 2

Instead of a lookup table, I'd use a polynomial approximation:

1 - x^0.19029 ≈ - 1073365.91783x¹⁵ + 8354695.40833x¹⁴ - 29422576.6529x¹³ + 61993794.537x¹² - 87079891.4988x¹¹ + 86005723.842x¹⁰ - 61389954.7459x⁹ + 32053170.1149x⁸ - 12253383.4372x⁷ + 3399819.97536x⁶ - 672003.142815x⁵ + 91817.6782072x⁴ - 8299.75873768x³ + 469.530204564x² - 16.6572179869x + 0.722044145701

Or in code:

double f(double x) {
  double fx;
  fx =      - 1073365.91783;
  fx = fx*x + 8354695.40833;
  fx = fx*x - 29422576.6529;
  fx = fx*x + 61993794.537;
  fx = fx*x - 87079891.4988;
  fx = fx*x + 86005723.842;
  fx = fx*x - 61389954.7459;
  fx = fx*x + 32053170.1149;
  fx = fx*x - 12253383.4372;
  fx = fx*x + 3399819.97536;
  fx = fx*x - 672003.142815;
  fx = fx*x + 91817.6782072;
  fx = fx*x - 8299.75873768;
  fx = fx*x + 469.530204564;
  fx = fx*x - 16.6572179869;
  fx = fx*x + 0.722044145701;
  return fx;
}

I computed this in sage using the least squares approach:

f(x) = 1-x^(19029/100000) # your function
d = 16                    # number of terms, i.e. degree + 1
A = matrix(d, d, lambda r, c: integrate(x^r*x^c, (x, 0, 1)))
b = vector([integrate(x^r*f(x), (x, 0, 1)) for r in range(d)])
A.solve_right(b).change_ring(RDF)

Here is a plot of the error this will entail:

Blue is the error from my 16 term polynomial, while red is the error you'd get from piecewise linear interpolation with 16 equidistant values. As you can see, both errors are quite small for most parts of the range, but will become really huge close to x=0. I actually clipped the plot there. If you can somehow narrow the range of possible values, you could use that as the domain for the integration, and obtain an even better fit for the relevant range. At the cost of worse fit outside, of course. You could also increase the number of terms to obtain a closer fit, although that might also lead to higher oscillations.

I guess you can also combine this approach with the one Stefan posted: use his to split the domain into several parts, then use mine to find a close low degree polynomial for each part.

Update

Since you updated the specification of your question, with regard to both the domain and the error, here is a minimal solution to fit those requirements:

44330(1 - x^0.19029) ≈ + 23024.9160933(1-x)⁷ - 39408.6473636(1-x)⁶ + 31379.9086193(1-x)⁵ - 10098.7031260(1-x)⁴ + 4339.44098317(1-x)³ + 3202.85705860(1-x)² + 8442.42528906(1-x)

double f(double x) {
  double fx, x1 = 1. - x;
  fx =       + 23024.9160933;
  fx = fx*x1 - 39408.6473636;
  fx = fx*x1 + 31379.9086193;
  fx = fx*x1 - 10098.7031260;
  fx = fx*x1 + 4339.44098317;
  fx = fx*x1 + 3202.85705860;
  fx = fx*x1 + 8442.42528906;
  fx = fx*x1;
  return fx;
}

I integrated x from 0.293 to 1 or equivalently 1 - x from 0 to 0.707 to keep the worst oscillations outside the relevant domain. I also omitted the constant term, to ensure an exact result at x=1. The maximal error for the range [0.3, 1] now occurs at x=0.3260 and amounts to 0.0972 < 0.1. Here is an error plot, which of course has bigger absolute errors than the one above due to the scale factor k=44330 which has been included here.

I can also state that the first three derivatives of the function will have constant sign over the range in question, so the function is monotonic, convex, and in general pretty well-behaved.