It is achievable to write fast and precise mathematical functions. There are cutting-edge implementations in computers, phones, calculators, and game consoles. Some of them are open source, like glibc and musl, from which we can learn. I have also been working on mathematics in metallic, as a shameless plug.

It may seem that mathematical functions are hardware instructions. We usually code them in software instead. The trend is to have the hardware deal with only basic operations after decades of evolution. We can perform mathematics with only operations required in IEEE 754 and integer operations via type punning.

Target system

Instruction set

Which instructions are on the target system? C operators are probably supported. Other operations are not as available even if they are required by IEEE 754. For example, fmod rarely compiles to a single instruction. It is usually done by long division, which then translates to a series of integer operations or partial remainder instructions. This C function is also an example where operators in other programming languages can be more expensive than expected, like % in JavaScript or Python.

Programming language

I suggest programming mathematical functions in C. It is fast and precise to evaluate complicated expressions with floating-point contraction. On supported platforms, a * b + c can compile to a fused multiply–add.

// Nearest integer for a nonnegative argument
float nearest(float x)
{
    const float rectifier = 0x1p23f;
    float y = x + rectifier;
    return y - rectifier;

    // Wrong: can be optimized to (x + 0)
    return x + rectifier - rectifier;
}

Rounding

Round half to even is the default rounding in IEEE 754. The roundoff error of a series of n operations by this unbiased rounding is only in O(√n). This rounding will be the implied rounding method throughout this article unless otherwise specified.

Double rounding

The default rounding is non-associative. If we round 1.49 to the nearest tenth and then to the nearest integer, it becomes 2. Rounding to midpoints must be avoided in intermediate roundings.

Round to odd is a helpful intermediate rounding for binary numbers. It rounds irrepresentable numbers to the nearest representation with an odd significand. Only numbers with even significands can be midpoints or exact in a coarser precision. Therefore, round to odd does not interfere with subsequent roundings. Round to odd is also associative like directed roundings as it rounds all values between representations to either of them.

• When double rounding is odd, by Sylvie Boldo and Guillaume Melquiond
• GCC avoids double rounding errors with round-to-odd, by Rick Regan

Exact addition

We can obtain the exact error of addition with the default rounding. This technique is useful for storing precise intermediates to stop the propagation of error. The idea is to find s + e = a + b, where s is the rounded sum, and e is the error term. The error term is defined when s does not overflow.

Compensated summation produces precise results with preconditions. The base of the floating-point system (FLT_RADIX) can be at most 3, and logb(a) ≥ logb(b).

s = a + b;
e = a - s + b;

We can generalize this algorithm by comparison, as |a| ≥ |b| implies logb(a) ≥ logb(b). Branching is nevertheless inefficient. There is another unbranched algorithm working most of the time. This algorithm overflows only if an operand is the largest finite representation or its negative.

s = a + b;
x = a - s;
y = b - s;
e = (a + x) + (b + y);

• On the robustness of the 2Sum and Fast2Sum algorithms by Sylvie Boldo, Stef Graillat, and Jean-Michel Muller

Exact multiplication

First, a double is large enough to store the exact product of any two floats. Therefore, it is preferred to cast and multiply. This method is straightforward and fast, and double-precision multiplication is widely supported.

double multiply(float a, float b)
{
    return (double)a * b;
}

Then, we try to find s + e = ab. If the FMA instructions are available, use them. Probe this feature with FP_FAST_FMA[FL]? macros.

s = a * b;
e = fma(a, b, -s);

Next, without all the hardware witchcraft, we can still count on schoolbook multiplication. With an even number of significant digits, equally split them and obtain the higher part with a bitwise AND.

Even if the number of significant bits is odd, we can split a binary significand with the default rounding. Take IEEE 754 double-precision as an example, which has 53 significant bits. Its magic multiplier is 2²⁷ + 1, where 27 = (53 + 1) / 2.

double split(double x)
{
    double s = (0x1p27 + 1) * x;
    double c = x - s;

    return s + c;
}

The above function returns the higher half of the argument. We can extract the lower half by subtracting the higher half. Each half is guaranteed to have at most 26 significant bits. The possibility that the halves can have opposite signs recovers the seemingly lost bit.

Table maker’s dilemma

The cost of a correctly rounded transcendental function is unknown unless probed with brute force. Faithfully rounded versions are much faster and generally preferable, though they may round up or down. No matter how precise intermediate results are, they can be close to a turning point of the given rounding. For example, x is a mathematical result equal to x^* + 0.49 ulp, where x^* is an exact floating-point. An exquisite approximation gives x^* + 0.51 ulp, which is only 0.02 ulp off. Nevertheless, it becomes x ^* + 1 ulp after default rounding, which is 1 ulp off from the correctly rounded x^*.

We can correctly round an algebraic function by solving its polynomial equation at the turning point and compare the results. However, this extra cost is unwelcome if faithful rounding is enough. It is unlikely that a correctly rounded program solves a real-world problem that a faithfully rounded one does not. IEEE 754 does not require correct rounding for the cubic root. Therefore, I made cbrt faithfully rounded in metallic. Its error can be even larger in glibc and other C libraries.

• Errors in math functions (glibc). Their cbrtf rounds faithfully, while cbrt can incur an error of 3 ulp.

Approximation

Eventually, we break down mathematical functions to basic arithmetics. This section covers how to turn mathematics into source code.

Argument reduction

Sometimes we can shrink the domain to a short interval with an identity. For example, to compute exp for a binary format, we can divide its argument x by ln 2.

\begin{align*} x &= n \ln 2 + r \\ \exp x &= 2^n \exp r \end{align*}

Where n is an integer, bit twiddling takes care of multiplication by 2ⁿ. If we pick n as an integer nearest to x, we simultaneously restrict r into [-0.5 ln 2, 0.5 ln 2].

Approximation to exp r is fast because [-0.5 ln 2, 0.5 ln 2] is a short interval. We approximate exp r with few terms to achieve the desired precision.

It is also precise because r is small. Computations involving small numbers are accurate. Floating-points are dense near the origin since they are essentially scientific notation. In IEEE 754 binary formats, there is the same number of representations in (0, 1.5) and in (1.5, ∞). Therefore, it is wise to shift the domain close to 0.

Transformations

Most mathematical functions we compute are well-behaved. We can save computations by taking advantage of continuity, differentiability, or symmetry.

When a function f passes through and is monotone at the origin, divide it by the identity function and approximate the quotient g instead.

\begin{align*} f(0) &= 0 \\ f(x) &= x g(x) \end{align*}

This transformation explicitly omits the constant term and reduces the overall relative error. The value of g(0) can be any finite number. We define g as a continuous extension for rigor.

\[ g(x) = \begin{cases} \displaystyle \frac{f(x)}{x} & \mbox{if } x \ne 0 \\ \displaystyle \lim_{t \to 0} \frac{f(t)}{t} & \mbox{if } x = 0 \end{cases} \]

Given an approximant ĝ of g, the overall absolute error x |ĝ − g| tends to 0 when x also approaches 0. This transformation enables approximating g without a weight function and simplifies calculation.

When f is an even function, view it as another function g of a squared variable.

\begin{align*} f(x) &= f (-x) \\ f(x) &= g \left( x^2 \right) \\ g(x) &= f \left( \sqrt x \right) \end{align*}

This transformation explicitly omits odd terms and halves the degree of the approximant.

An odd function is a combination of the above two. It is a product of the identity function and an even function.

\begin{align*} f(x) &= -f (-x) \\ f(x) &= x g \left( x^2 \right) \\ g(x) &= \frac{f \left( \sqrt x \right)}{\sqrt x} \end{align*}

The value of g(0) does not affect the approximation of g as it creates no hole in the domain. In practice, set the lower bound to a tiny positive number like 2^-200, and everything is fine.

Remez algorithm

Remez exchange algorithm is an iterative minimax algorithm. It minimizes the error of a rational approximation of a function. The best explanation of this algorithm I found is from the Boost libraries. I recommend Remez.jl, a public module in the Julia language. It works out of the box after installation.

For example, the following snippet finds a cubic approximation of cos(√·) in [0, (π / 4)²] with minimax absolute errors. The last argument is 0 because we want a polynomial, whose denominator is a constant (of degree 0) if regarded as a rational function.

import Remez

N, D, E, X = Remez.ratfn_minimax(x -> cos(√x), (0, (big(π) / 4)^2), 3, 0)

The variables N, D, E, X are the numerator, the denominator, the maximum error, and coordinates of the extrema, respectively. In this case, we are interested in N and E only. If we run the snippet in the REPL, it is straightforward to inspect variables.

julia> N
4-element Array{BigFloat,1}:
  0.9999999724233229210670051040057597041917874465747537951681676248240168483719746
 -0.4999985669584884771720232450657038606385147149244782395789475085368551172067715
  0.04165502688425152443762347668780274316867072837392713367475023020736799395672903
 -0.001358590851011329858521158876238716265345398772374942259275377959127201806930143

julia> E
2.757667707893299489599424029580821255342524620485822487536621483700643103529491e-08

The resulting coefficients are in ascending order. For example, the first element of N is the constant term.

Polynomial evaluation

The best polynomial evaluation method depends on the system. For example, pipelining influences execution time. Luckily, there are well-known evaluation schemes that provide decent performance and reasonable errors.

Horner’s scheme produces the fewest operations. It is the fastest if its argument is already a vector. It is also usually the most accurate method for a polynomial approximant of a well-conditioned function. However, its dependency chain is also the longest. It underuses the pipeline because all operations except one depend on another. Hence, it is less than ideal on single-threaded systems.

On the other hand, Estrin’s scheme tries to be as parallel as possible. It groups terms in a binary fashion to achieve the shallowest dependency tree at the expense of O(log(n)) extra squaring ops.

There are also other evaluation schemes with different pros and cons. Benchmark to find the most suited one if their difference is critical.