Format flags for CustomFloat.

Allows user code to define custom floating-point formats. These formats are for storage only; all operations on them are performed by first implicitly extracting them to real first. After the operation is completed the result can be stored in a custom floating-point value via assignment.

// Define a 16-bit floating point values

CustomFloat!16                                x;     // Using the number of bits

CustomFloat!(10, 5)                           y;     // Using the precision and exponent width

CustomFloat!(10, 5,CustomFloatFlags.ieee)     z;     // Using the precision, exponent width and format flags

CustomFloat!(10, 5,CustomFloatFlags.ieee, 15) w;     // Using the precision, exponent width, format flags and exponent offset bias


// Use the 16-bit floats mostly like normal numbers

w = x*y - 1;
writeln(w);

// Functions calls require conversion

z = sin(+x)           + cos(+y);                     // Use uniary plus to concisely convert to a real

z = sin(x.re)         + cos(y.re);                   // Or use the .re property to convert to a real

z = sin(x.get!float)  + cos(y.get!float);            // Or use get!T

z = sin(cast(float)x) + cos(cast(float)y);           // Or use cast(T) to explicitly convert


// Define a 8-bit custom float for storing probabilities

alias CustomFloat!(4, 4, CustomFloatFlags.ieee^CustomFloatFlags.probability^CustomFloatFlags.signed ) Probability;
auto p = Probability(0.5);

Defines the fastest type to use when storing temporaries of a calculation intended to ultimately yield a result of type F (where F must be one of float, double, or real). When doing a multi-step computation, you may want to store intermediate results as  FPTemporary!F.

// Average numbers in an array

double avg(in double[] a)
{
    if (a.length == 0) return 0;
    FPTemporary!double result = 0;
    foreach (e; a) result += e;
    return result / a.length;
}

Implements the secant method for finding a root of the function fun starting from points [xn_1, x_n] (ideally close to the root). Num may be float, double, or real.

float f(float x) {
    return cos(x) - x*x*x;
}
auto x = secantMethod!(f)(0f, 1f);
assert(approxEqual(x, 0.865474));

Find a real root of a real function f(x) via bracketing.

Given a function f and a range [a..b] such that f(a) and f(b) have opposite signs, returns the value of x in the range which is closest to a root of f(x). If f(x) has more than one root in the range, one will be chosen arbitrarily. If f(x) returns NaN, NaN will be returned; otherwise, this algorithm is guaranteed to succeed.

Uses an algorithm based on TOMS748, which uses inverse cubic interpolation whenever possible, otherwise reverting to parabolic or secant interpolation. Compared to TOMS748, this implementation improves worst-case performance by a factor of more than 100, and typical performance by a factor of 2. For 80-bit reals, most problems require 8 to 15 calls to f(x) to achieve full machine precision. The worst-case performance (pathological cases) is approximately twice the number of bits.

Find root of a real function f(x) by bracketing, allowing the termination condition to be specified.

Computes Euclidean distance between input ranges a and b. The two ranges must have the same length. The three-parameter version stops computation as soon as the distance is greater than or equal to limit (this is useful to save computation if a small distance is sought).

Computes the dot product of input ranges a and b. The two ranges must have the same length. If both ranges define length, the check is done once; otherwise, it is done at each iteration.

Computes the cosine similarity of input ranges a and b. The two ranges must have the same length. If both ranges define length, the check is done once; otherwise, it is done at each iteration. If either range has all-zero elements, return 0.

Normalizes values in range by multiplying each element with a number chosen such that values sum up to sum. If elements in range sum to zero, assigns sum / range.length to all. Normalization makes sense only if all elements in range are positive.  normalize assumes that is the case without checking it.

Computes entropy of input range r in bits. This function assumes (without checking) that the values in r are all in [0, 1]. For the  entropy to be meaningful, often r should be normalized too (i.e., its values should sum to 1). The two-parameter version stops evaluating as soon as the intermediate result is greater than or equal to max.

Computes the Kullback-Leibler divergence between input ranges a and b, which is the sum ai * log(ai / bi). The base of logarithm is 2. The ranges are assumed to contain elements in [0, 1]. Usually the ranges are normalized probability distributions, but this is not required or checked by  kullbackLeiblerDivergence. If any element bi is zero and the corresponding element ai nonzero, returns infinity. (Otherwise, if ai == 0 && bi == 0, the term ai * log(ai / bi) is considered zero.) If the inputs are normalized, the result is positive.

Computes the Jensen-Shannon divergence between a and b, which is the sum (ai * log(2 * ai / (ai + bi)) + bi * log(2 * bi / (ai + bi))) / 2. The base of logarithm is 2. The ranges are assumed to contain elements in [0, 1]. Usually the ranges are normalized probability distributions, but this is not required or checked by  jensenShannonDivergence. If the inputs are normalized, the result is bounded within [0, 1]. The three-parameter version stops evaluations as soon as the intermediate result is greater than or equal to limit.

The so-called "all-lengths gap-weighted string kernel" computes a similarity measure between s and t based on all of their common subsequences of all lengths. Gapped subsequences are also included.

To understand what  gapWeightedSimilarity(s, t, lambda) computes, consider first the case lambda = 1 and the strings s = ["Hello", "brave", "new", "world"] and t = ["Hello", "new", "world"]. In that case,  gapWeightedSimilarity counts the following matches:

1. three matches of length 1, namely "Hello", "new", and "world";
2. three matches of length 2, namely ("Hello", "new"), ("Hello", "world"), and ("new", "world");
3. one match of length 3, namely ("Hello", "new", "world").

string[] s = ["Hello", "brave", "new", "world"];
string[] t = ["Hello", "new", "world"];
assert(gapWeightedSimilarity(s, t, 1) == 7);

string[] s = ["Hello", "brave", "new", "world"];
string[] t = ["Hello", "new", "world"];
assert(gapWeightedSimilarity(s, t, 0) == 4);

string[] s = ["Hello", "brave", "new", "world"];
string[] t = ["Hello", "new", "world"];
assert(gapWeightedSimilarity(s, t, 0.5) == 4 + 0.5 * 2 + 0.125);

The similarity per gapWeightedSimilarity has an issue in that it grows with the lengths of the two strings, even though the strings are not actually very similar. For example, the range ["Hello", "world"] is increasingly similar with the range ["Hello", "world", "world", "world",...] as more instances of "world" are appended. To prevent that,  gapWeightedSimilarityNormalized computes a normalized version of the similarity that is computed as gapWeightedSimilarity(s, t, lambda) / sqrt(gapWeightedSimilarity(s, t, lambda) * gapWeightedSimilarity(s, t, lambda)). The function  gapWeightedSimilarityNormalized (a so-called normalized kernel) is bounded in [0, 1], reaches 0 only for ranges that don't match in any position, and 1 only for identical ranges.

string[] s = ["Hello", "brave", "new", "world"];
string[] t = ["Hello", "new", "world"];
assert(gapWeightedSimilarity(s, s, 1) == 15);
assert(gapWeightedSimilarity(t, t, 1) == 7);
assert(gapWeightedSimilarity(s, t, 1) == 7);
assert(gapWeightedSimilarityNormalized(s, t, 1) == 7. / sqrt(15. * 7));

Similar to gapWeightedSimilarity, just works in an incremental manner by first revealing the matches of length 1, then gapped matches of length 2, and so on. The memory requirement is Ο(s.length * t.length). The time complexity is Ο(s.length * t.length) time for computing each step. Continuing on the previous example:

string[] s = ["Hello", "brave", "new", "world"];
string[] t = ["Hello", "new", "world"];
auto simIter = gapWeightedSimilarityIncremental(s, t, 1);
assert(simIter.front == 3); // three 1-length matches

simIter.popFront();
assert(simIter.front == 3); // three 2-length matches

simIter.popFront();
assert(simIter.front == 1); // one 3-length match

simIter.popFront();
assert(simIter.empty);     // no more match

Computes the greatest common divisor of a and b by using Euler's algorithm.

A class for performing fast Fourier transforms of power of two sizes. This class encapsulates a large amount of state that is reusable when performing multiple FFTs of sizes smaller than or equal to that specified in the constructor. This results in substantial speedups when performing multiple FFTs with a known maximum size. However, a free function API is provided for convenience if you need to perform a one-off FFT.

Convenience functions that create an Fft object, run the FFT or inverse FFT and return the result. Useful for one-off FFTs.