Numeric

Table of Contents

• Numbers
  • Trigonometry
  • Checks
• Interpolation
• Arithmetic Interval
• EFloat

Numerical classes provide all sort of number operations and functions. The hermes::Constants, for example, offer you common constants:

#include <hermes/numeric/numeric.h>
// Here are some constants provided by hermes
// and how you can access them
hermes::Constants::pi;
hermes::Constants::pi_over_four;
hermes::Constants::machine_epsilon;
// and more

Numbers

The hermes::Numbers namespace, included by hermes/numeric/numeric.h, provides lots of functions to work with floating-point, integer and binary number representations. Most functions are template functions. Here is an incomplete list of what you can find there, for more functions, please check the documentation of hermes::Numbers.

• Type representation limits ```cpp hermes::Numbers::lowest<T>(); hermes::Numbers::greatest<T>(); ```
• Functions

  hermes::Numbers::fract(x);
  hermes::Numbers::clamp(x, l, u);
  hermes::Numbers::sqr(x);
  hermes::Numbers::safe_sqrt(x);
  hermes::Numbers::cube(x);
  hermes::Numbers::sign(x);
  hermes::Numbers::max({-1, 2, 4, 10});
  hermes::Numbers::pow<N>(x);

• Binary representations

  hermes::Numbers::countHexDigits(x);
  hermes::Numbers::floatExponent(x);
  hermes::Numbers::floatSignificand(x);
  hermes::Numbers::floatSignBit(x);
  hermes::Numbers::floatToBits(x);
  hermes::Numbers::bitsToFloat(u);
  hermes::Numbers::interleaveBits(x, y, z);

• Rounding and precision

  hermes::Numbers::nextFloatUp(x);
  hermes::Numbers::nextFloatDown(x);
  hermes::Numbers::ceilToInt(x);
  hermes::Numbers::floorToInt(x);
  hermes::Numbers::roundToInt(x);
  // all four arithmetic operations with rounding 
  // to next float up to machine precision
  hermes::Numbers::subRoundUp(x, y);
  ...
  hermes::Numbers::divRoundDonw(x, y);
  hermes::Numbers::sqrtRoundDown(x, y);
  hermes::Numbers::gamma(n);

Trigonometry

A few trigonometric functions are listed in hermes::Trigonometry:

#include <hermes/numeric/numeric.h>
// Examples of trigonometric functions
hermes::Trigonometry::radians2degrees(hermes::Constants::two_pi);
hermes::Trigonometry::safe_acos(-2);

Checks

You can make some floating-number comparisons as well using hermes::Check:

#include <hermes/numeric/numeric.h>
float x, y;
hermes::Check::is_zero(x);
hermes::Check::is_nan(x);
hermes::Check::is_equal(x, y, 1e-8);
hermes::Check::is_between(0.1, x, y);
hermes::Check::is_between_closed(0.1, x, y);

Interpolation

Check hermes::interpolation namespace for interpolation functions.

#include <hermes/numeric/interpolation.h>
 
hermes::interpolation::smooth(a, b);
hermes::interpolation::smoothStep(a, b, v);
hermes::interpolation::sharpen(a, b);
hermes::interpolation::linearStep(v, a, b);
hermes::interpolation::mix(v, a, b);
hermes::interpolation::lerp(t, a, b);
// also bilerp and trilerp
hermes::interpolation::nearest(t, a, b);
hermes::monotonicCubicInterpolation(...);

Arithmetic Interval

hermes::Interval represents a numeric interval that supports interval arithmetic. It is very useful when you want to represent your values considering the uncertainty that comes from the limitations of the computer representations of floating-point numbers. This way, hermes::Interval represents a closed interval [low, high] that guarantees to enclose your real numerical value, by providing all sorts of arithmetic operations of intervals.

#include <hermes/numeric/interval.h>
 
int main() {
  // suppose two intervals
  hermes::Interval<f32> a(-1, 1), (0, 2);
  // you can do common arithmetic (+,-,*,/)
  auto c = a * b; // [-2,2]
  // also
  c.sqr();  // [0, 4]
  c.sqrt(); // [nan, 1.41421]
return 0;
}

You can also make some queries:

hermes::Interval<i32> a(-1,1);
a.contains(0); // true
a.center();    // 0
a.radius();    // 1
a.width();     // 2
a.isExact();   // false ([1,1] is exact)

EFloat

For certain calculations, you may want to keep track of floating-point operation errors accumulated over time. The hermes::EFloat class uses the hermes::Interval to carry error bounds during floating-point operations.

#include <hermes/numeric/e_float.h>
 
int main() {
  // EFloats start as exact intervals
  // in this case, [3,3] and [-1,-1]
  hermes::EFloat a(3.f), b(-1.f);
  // suppose you do math with them
  auto c = a * 3 + a * b - 4 * b + a / b;
  // the resulting quantity will contain errors
  // that can be checked this way:
  c.absoluteError(); // 1.14441e-05
  c.upperBound();    // 7.00001
  c.lowerBound();    // 6.99999
  return 0;
}