Audience: LEWG, SG6; WG14
S. Davis Herring <>
Los Alamos National Laboratory
February 9, 2018

# Introduction

The simple problem of computing a value between two other values is surprisingly subtle in general. This paper proposes library functions to compute the midpoint between two integer, floating-point, or pointer values, as well as a more general routine that interpolates (or extrapolates) between two floating-point values.

These utilities are a pure library extension. With the exception of the pointer versions, they would be valuable and straightforward to implement in C (perhaps via type-generic macros); it would be appropriate to consult with WG14 about including them in the C standard library.

# Midpoint

## Integers

Computing the (integer) midpoint of two integer values via

``(a+b)/2``

can cause overflow for signed or unsigned integers as well as for floating-point values. Java’s binary search implementation had this integer overflow bug for nearly a decade, and Mozilla had the same issue in its JavaScript implementation.

The standard alternative

``a+(b-a)/2``

works for unsigned integers (even if `b<a`). On a typical two’s complement implementation where conversion from unsigned to signed preserves bit patterns, the library can provide the simple implementation

``````Integer midpoint(Integer a, Integer b) {
using U = make_unsigned_t<Integer>;
return Integer(U(a)+(U(b)-U(a))/2);
}``````

that works for signed or unsigned `Integer`. Note that when `b==a+1` or `b==a-1` (without overflow), the result is a because of truncating division. This can be exploited to round half-integers up or down by supplying `a>=b` or `a<=b` respectively. (The simple `(a+b)/2` always truncates half-integers towards 0, yielding `min(a,b)` when they differ by 1.)

## Pointers

When a binary search over an array is implemented using pointers, the array size must not exceed `PTRDIFF_MAX` to avoid undefined behavior ([expr.add]/5). The library can also provide a function template

``````template<class T>
T* midpoint(T *a, T *b);``````

which is straightforward on common architectures but, it seems, cannot be implemented portably and efficiently in C++. As with integers, when the midpoint lies between two pointer values the one closer to `a` is chosen; for the usual case of `a<b`, this is compatible with the usual half-open ranges by selecting `a` when `[a,b)` is `[a,a+1)`.

## Floating-point types

Each of the midpoint formulas above can cause overflow for floating-point types; the latter can also produce results that are not correctly rounded (by rounding in the subtraction and the addition). A third choice

``a/2+b/2``

prevents overflow but is not correctly rounded for subnormal inputs (due to rounding each separately). The library can easily provide overloads of `midpoint` for floating-point types that is correctly rounded for all inputs by switching between these strategies:

``````Float midpoint(Float a, Float b)
{return isnormal(a) && isnormal(b) ? a/2+b/2 : (a+b)/2;}``````

## Naming

The name `mean` is superior for the floating-point case, but for the other types (and the common application to binary search) the name `midpoint` used above is more appropriate. It would be possible to use both names (despite the possible confusion with the use of `mean` in [rand.dist]), but as it aims to replace the simple expression `a+(b-a)/2` regardless of type, a single name seems best.

# Linear interpolation

Both obvious approaches used in published implementations of floating-point linear interpolation have issues:

1. `a+t*(b-a)` does not in general reproduce b when `t==1`, and can overflow if a and b have the largest exponent and opposite signs.
2. `t*b+(1-t)*a` is not monotonic in general (unless `a*b<=0`).

Lacking an obvious, efficient means of obtaining a correctly rounded overall result, the goal is instead to guarantee the basic properties of

1. exactness: `linear(a,b,0)==a && linear(a,b,1)==b`
2. monotonicity: `(linear(a,b,t2)-linear(a,b,t1)) * (t2-t1) * (b-a) >= 0`
3. boundedness: `t<0 || t>1 || isfinite(linear(a,b,t))`
4. consistency: `linear(a,a,t)==a`

given that each argument `isfinite` (for monotonicity, t1 and t2 may also be infinite if `a!=b` and `t1!=t2`). These properties are useful in proofs of correctness of algorithms based on `linear`. When interpolating, consistency follows from the other properties, but it and monotonicity apply even when extrapolating.

To demonstrate the feasibility of satisfying these properties, a simple implementation that provides all of them is given here:

``````Float linear(Float a, Float b, Float t) {
// Exact, monotonic, bounded, and (for a=b=0) consistent:
if(a*b <= 0) return t*b + (1-t)*a;

if(t==1) return b;                        // exact
// Exact at t=0, monotonic except near t=1, bounded, and consistent:
const Float x = a + t*(b-a);
return t>1 == b>a ? max(b,x) : min(b,x);  // monotonic near t=1
}``````

Putting `b` first in the `min`/`max` prefers it to another equal value (i.e., `-0.` vs. `+0.`), which avoids returning `-0.` for `t==1` but `+0.` for other nearby values of t.

Whether it uses this implementation or not, the library can provide a function satisfying these mathematical properties.

## Naming

The common (if abstruse) name `lerp` is avoided because it might suggest a restriction to t on [0,1].

# Wording

## Midpoint

Add to the end of the synopsis in [numeric.ops.overview]:

``````  // 29.8.14, least common multiple
template<class M, class N>
constexpr common_type_t<M,N> lcm(M m, N n);``````

`  ``A midpoint(A a, A b); // A arithmetic`
`  ``template<class T>`
`    ``T* midpoint(T* a, T* b);`

`}`

Add a new subsection after [numeric.ops.lcm]:

### 29.8.15 Midpoint

``A midpoint(A a, A b);``
1. Returns: Half the sum of `a` and `b`. No overflow occurs. If `A` is an integer type and the sum is odd, the result is rounded towards `a`. If `A` is a floating-point type, at most one inexact operation occurs.
2. Remarks: An overload exists for each of `char` and all arithmetic types except `bool` being `A`.
``````template<class T>
T* midpoint(T* a, T* b);``````
1. Requires: `a` and `b` point to, respectively, elements `x[`i`]` and `x[`j`]` of the same array object `x` [ Footnote: An object that is not an array element is considered to belong to a single-element array for this purpose; see [expr.unary.op]. A pointer past the last element of an array x of n elements is considered to be equivalent to a pointer to a hypothetical element x[n] for this purpose; see [basic.compound]. — end footnote ].
2. Returns: A pointer to `x[`i+(j-i)/2`]`, where the result of the division is truncated towards zero.

## Linear interpolation

Add to the synopsis in [cmath.syn]:

``long double fmal(long double x, long double y, long double z);``

`// 29.9.4, linear interpolation`
`F linear(F a, F b, F t); // F floating-point`

``// 29.9.4, classification / comparison functions``

Add a new subsection after [c.math.hypot3]:

### 29.9.4 Linear interpolation

``F linear(F a, F b, F t);``
1. Returns: a+t(b-a). If each argument `isfinite`, the result satisfies these conditions:
1. `t<0 || t>1 || isfinite(linear(a,b,t))`
2. `linear(a,b,0)==a && linear(a,b,1)==b`
3. `linear(a,a,t)==a`
4. `(linear(a,b,t2)-linear(a,b,t1)) * (t2-t1) * (b-a) >= 0`
The last condition applies additionally if `!isnan(t1) && !isnan(t2) && t1!=t2 && a!=b`.
2. Remarks: An overload exists for each floating-point type being `F`.