Truncate a floating-point number to the leading N decimal digits

Question

Which is the most optimal way to get the n leftmost non-zero digits of a floating point number (number >= 0.0).

For example,

if n = 1:

• 0.014568 -> 0.01
• 0.246456 -> 0.2

if n = 2:

• 0.014568 -> 0.014
• 0.246456 -> 0.24

After @schil227 comment: Currently I am doing multiplications and divisions (by 10) as necessary in order to have the n digits at the decimal number field.

Answer 1

Code could use sprintf(buf, "%e",...) to do most of the heavy lifting.

There are so many corner cases that other direct code may fail, sprintf() is likely to be, at least, as good solid reference solution.

This code prints the double to DBL_DECIMAL_DIG places to insure there is no rounding in digits that would make a difference. Then it zeros out various digits depending on n.

See @Mark Dickinson comment for reasons to use a greater value than DBL_DECIMAL_DIG. Perhaps on the order of DBL_DECIMAL_DIG*2. As mentioned above, there are many corner cases.

#include <float.h>
#include <math.h>
#include <stdio.h>

double foo(double x, int n) {
  if (!isfinite(x)) {
    return x;
  }
  printf("%g\n", x);
  char buf[DBL_DECIMAL_DIG + 11];
  sprintf(buf, "%+.*e", DBL_DECIMAL_DIG, x);
  //puts(buf);
  assert(n >= 1 && n <= DBL_DECIMAL_DIG + 1);
  memset(buf + 2 + n, '0', DBL_DECIMAL_DIG - n + 1);
  //puts(buf);
  char *endptr;
  x = strtod(buf, &endptr);
  printf("%g\n", x);
  return x;
}

int main() {
 foo(0.014568, 1);
 foo(0.246456, 1);
 foo(0.014568, 2);
 foo(0.246456, 2);
 return 0;
}

Output

0.014568
0.01
0.246456
0.2
0.014568
0.014
0.246456
0.24

This answer assumes OP does not want a rounded answer. Re: 0.246456 -> 0.24

Answer 2

If you want the result as a string, you should probably print to a string with extra precision, then chop that off yourself. (See @chux's answer for details on how much extra precision you need for IEEE 64-bit double to avoid rounding up from a string of 9s, since you want truncation but all the usual to-string functions round to nearest.)

If you want a double result, then are you sure you really want this? Rounding / truncating early in the middle of a calculation usually just worsens the accuracy of the final result. Of course, there are uses in real algorithms for floor/ceil, trunc, and nearbyint, and this is just a scaled version of trunc.

---

If you just want a double, you can get fairly good results without ever going to a string. Use ndigits and floor(log10(fabs(x))) to work out a scale factor, then truncate the scaled value to an integer, then scale back.

Tested and working (with and without -ffast-math). See the asm on the Godbolt compiler explorer. This might run reasonably efficiently, especially with -ffast-math -msse4.1 (so floor and trunc can inline to roundsd).

If you care about speed, look into replacing pow() with something that takes advantage of the fact that the exponent is a small integer. I'm not sure how fast library pow() implementations are in that case. GNU C __builtin_powi(x, n) trades accuracy for speed, for integer exponents, doing a multiplication tree, which is less accurate than what pow() does.

#include <float.h>
#include <math.h>
#include <stdio.h>

double truncate_n_digits(double x, int digits)
{
    if (x==0 || !isfinite(x))
        return x;   // good idea stolen from Chux's answer :)

    double l10 = log10(fabs(x));
    double scale = pow(10.,  floor(l10) + (1 - digits));  // floor rounds towards -Inf
    double scaled = x / scale;
    double scaletrunc = trunc(scaled);  // trunc rounds towards zero
    double truncated = scaletrunc * scale;

#if 1    // debugging code
    printf("%2d %24.14g =>\t%24.14g\t scale=%g, scaled=%.30g\n", digits, x, truncated, scale, scaled);
    // print with more accuracy to reveal the real behaviour
    printf("   %24.20g =>\t%24.20g\n", x, truncated);
#endif

    return truncated;
}

test cases:

int main() {
 truncate_n_digits(0.014568, 1);
 truncate_n_digits(0.246456, 1);
 truncate_n_digits(0.014568, 2);
 truncate_n_digits(-0.246456, 2);
 truncate_n_digits(1234567, 2);
 truncate_n_digits(99999999999, 6);
 truncate_n_digits(-99999999999, 6);
 truncate_n_digits(99999, 10);
 truncate_n_digits(-0.0000000001234567, 3);
 truncate_n_digits(1000, 6);
 truncate_n_digits(0.001, 6);
 truncate_n_digits(1e-312, 2);  // denormal, and not exactly representable: 9.999...e-313
 truncate_n_digits(nextafter(1e-312, INFINITY), 2);  // denormal, just above 1.00000e-312
 return 0;
}

each result shown twice: first with only %.14g so rounding gives the string we want, then again with %.20g to show enough places to reveal the realities of floating point math. Most numbers are not exactly-representable, so even with perfect rounding it's impossible to return a double exactly represents the truncated decimal string. (Integers up to about the size of the mantissa are exactly representable, and so are fractions where the denominator is a power of 2.)

 1                 0.014568 =>                      0.01         scale=0.01, scaled=1.45679999999999987281285029894
    0.014567999999999999353 =>   0.010000000000000000208
 1                 0.246456 =>                       0.2         scale=0.1, scaled=2.46456000000000008398615136684
      0.2464560000000000084 =>     0.2000000000000000111
 2                 0.014568 =>                     0.014         scale=0.001, scaled=14.5679999999999996163069226895
    0.014567999999999999353 =>   0.014000000000000000291
 2                -0.246456 =>                     -0.24         scale=0.01, scaled=-24.6456000000000017280399333686
     -0.2464560000000000084 =>   -0.23999999999999999112
 3               1234.56789 =>                      1230         scale=10, scaled=123.456789000000000555701262783
       1234.567890000000034 =>                      1230
 6               1234.56789 =>                   1234.56         scale=0.01, scaled=123456.789000000004307366907597
       1234.567890000000034 =>     1234.5599999999999454
 6              99999999999 =>               99999900000         scale=100000, scaled=999999.999990000040270388126373
                99999999999 =>               99999900000
 6             -99999999999 =>              -99999900000         scale=100000, scaled=-999999.999990000040270388126373
               -99999999999 =>              -99999900000
10                    99999 =>                     99999         scale=1e-05, scaled=9999900000
                      99999 =>     99999.000000000014552
 3            -1.234567e-10 =>                 -1.23e-10         scale=1e-12, scaled=-123.456699999999983674570103176
   -1.234566999999999879e-10 => -1.2299999999999998884e-10
 6                     1000 =>                      1000         scale=0.01, scaled=100000
                       1000 =>                      1000
 6                    0.001 =>                     0.001         scale=1e-08, scaled=100000
   0.0010000000000000000208 =>  0.0010000000000000000208
 2     9.9999999999847e-313 =>      9.9999999996388e-313         scale=1e-314, scaled=100.000000003458453079474566039
   9.9999999999846534143e-313 =>        9.9999999996388074622e-313
 2     1.0000000000034e-312 =>      9.0000000001196e-313         scale=1e-313, scaled=9.9999999999011865980946822674
   1.0000000000034059979e-312 =>        9.0000000001195857973e-31

Since the result you want will often not be exactly representable, (and because of other rounding errors) the resulting double will sometimes be below the result you want, so printing it with full precision might give 1.19999999 instead of 1.20000011. You might want to use nextafter(result, copysign(INFINITY, original)) to get a result that's more likely to have a higher magnitude than what you want.

Of course, that could just make things worse in some cases. But since we truncate towards zero, most often we get a result that's just below (in magnitude) the unrepresentable exact value.

Answer 3

Ok, another one like @Peter Cordes but more generic.

/** Return \c digits semantic digis of number \c x.
    \tparam T Type of number \c x can be floating point or integral.
    \param x The number.
    \param digits The requested number of semantic digits of number \c x.
    \return The number with only \c digits semantic digits of number \c x. */
template<typename T>
requires(std::integral<T> || std::floating_point<T>)
T roundn(T x, unsigned int digits)
{
    if (!x || !std::isfinite(x)) return x;
    typedef std::conditional_t<std::floating_point<T>, T, double> Tp;
    Tp mul = pow(10, floor(digits - log10(abs(x))));
    Tp y = round(x * mul) / mul;
    if constexpr (std::floating_point<T>) return y;
    else return round(y);
}

int main()
{
    cout << setprecision(100);
    cout << roundn(123.456789, 1) << "\n";
    cout << roundn(123.456789, 2) << "\n";
    cout << roundn(123.456789, 3) << "\n";
    cout << roundn(123.456789, 4) << "\n";
    cout << roundn(123.456789, 5) << "\n";
    cout << roundn(-123.456789, 1) << "\n";
    cout << roundn(-123.456789, 2) << "\n";
    cout << roundn(-123.456789, 3) << "\n";
    cout << roundn(-123.456789, 4) << "\n";
    cout << roundn(-123.456789, 5) << "\n";
    cout << roundn(-123.456789, 15) << "\n";
    cout << roundn(123456, 1) << "\n";
    cout << roundn(123456, 2) << "\n";
    cout << roundn(123456, 3) << "\n";
    cout << roundn(123456, 10) << "\n";
    cout << roundn(-123456, 1) << "\n";
    cout << roundn(-123456, 2) << "\n";
    cout << roundn(-123456, 3) << "\n";
    cout << roundn(-123456, 10) << "\n";
    cout << roundn(0.0123456789, 1) << "\n";
    cout << roundn(0.0123456789, 2) << "\n";
    cout << roundn(-0.0123456789, 1) << "\n";
    cout << roundn(-0.0123456789, 2) << "\n";
    return 0;
}

It returns

99.9999999999999857891452847979962825775146484375
120
123
123.5
123.4599999999999937472239253111183643341064453125
-99.9999999999999857891452847979962825775146484375
-120
-123
-123.5
-123.4599999999999937472239253111183643341064453125
-123.4567890000000005557012627832591533660888671875
100000
120000
123000
123456
-100000
-120000
-123000
-123456
0.01000000000000000020816681711721685132943093776702880859375
0.0120000000000000002498001805406602215953171253204345703125
-0.01000000000000000020816681711721685132943093776702880859375
-0.0120000000000000002498001805406602215953171253204345703125