How to do unsigned saturating addition in C?

You probably want portable C code here, which your compiler will turn into proper ARM assembly. ARM has conditional moves, and these can be conditional on overflow. The algorithm then becomes: add and conditionally set the destination to unsigned(-1), if overflow was detected.

uint16_t add16(uint16_t a, uint16_t b)
{
  uint16_t c = a + b;
  if (c < a)  /* Can only happen due to overflow */
    c = -1;
  return c;
}

Note that this differs from the other algorithms in that it corrects overflow, instead of relying on another calculation to detect overflow.

x86-64 clang 3.7 -O3 output for adds32: significantly better than any other answer:

add     edi, esi
mov     eax, -1
cmovae  eax, edi
ret

ARMv7: gcc 4.8 -O3 -mcpu=cortex-a15 -fverbose-asm output for adds32:

adds    r0, r0, r1      @ c, a, b
it      cs
movcs   r0, #-1         @ conditional-move
bx      lr

16bit: still doesn't use ARM's unsigned-saturating add instruction (UADD16)

add     r1, r1, r0        @ tmp114, a
movw    r3, #65535      @ tmp116,
uxth    r1, r1  @ c, tmp114
cmp     r0, r1    @ a, c
ite     ls        @
movls   r0, r1        @,, c
movhi   r0, r3        @,, tmp116
bx      lr  @

In plain C:

uint16_t sadd16(uint16_t a, uint16_t b) {
  return (a > 0xFFFF - b) ? 0xFFFF : a + b;
}
     
uint32_t sadd32(uint32_t a, uint32_t b) {
  return (a > 0xFFFFFFFF - b) ? 0xFFFFFFFF : a + b;
}

which is almost macro-ized and directly conveys the meaning.

In IA32 without conditional jumps:

uint32_t sadd32(uint32_t a, uint32_t b)
{
#if defined IA32
  __asm
  {
    mov eax,a
    xor edx,edx
    add eax,b
    setnc dl
    dec edx
    or eax,edx
  }
#elif defined ARM
  // ARM code
#else
  // non-IA32/ARM way, copy from above
#endif
}

In ARM you may already have saturated arithmetic built-in. The ARMv5 DSP-extensions can saturate registers to any bit-length. Also on ARM saturation is usually cheap because you can excute most instructions conditional.

ARMv6 even has saturated addition, subtraction and all the other stuff for 32 bits and packed numbers.

On the x86 you get saturated arithmetic either via MMX or SSE.

All this needs assembler, so it's not what you've asked for.

There are C-tricks to do saturated arithmetic as well. This little code does saturated addition on four bytes of a dword. It's based on the idea to calculate 32 half-adders in parallel, e.g. adding numbers without carry overflow.

This is done first. Then the carries are calculated, added and replaced with a mask if the addition would overflow.

uint32_t SatAddUnsigned8(uint32_t x, uint32_t y) 
{
  uint32_t signmask = 0x80808080;
  uint32_t t0 = (y ^ x) & signmask;
  uint32_t t1 = (y & x) & signmask;
  x &= ~signmask;
  y &= ~signmask;
  x += y;
  t1 |= t0 & x;
  t1 = (t1 << 1) - (t1 >> 7);
  return (x ^ t0) | t1;
}

You can get the same for 16 bits (or any kind of bit-field) by changing the signmask constant and the shifts at the bottom like this:

uint32_t SatAddUnsigned16(uint32_t x, uint32_t y) 
{
  uint32_t signmask = 0x80008000;
  uint32_t t0 = (y ^ x) & signmask;
  uint32_t t1 = (y & x) & signmask;
  x &= ~signmask;
  y &= ~signmask;
  x += y;
  t1 |= t0 & x;
  t1 = (t1 << 1) - (t1 >> 15);
  return (x ^ t0) | t1;
}

uint32_t SatAddUnsigned32 (uint32_t x, uint32_t y)
{
  uint32_t signmask = 0x80000000;
  uint32_t t0 = (y ^ x) & signmask;
  uint32_t t1 = (y & x) & signmask;
  x &= ~signmask;
  y &= ~signmask;
  x += y;
  t1 |= t0 & x;
  t1 = (t1 << 1) - (t1 >> 31);
  return (x ^ t0) | t1;
}

Above code does the same for 16 and 32 bit values.

If you don't need the feature that the functions add and saturate multiple values in parallel just mask out the bits you need. On ARM you also want to change the signmask constant because ARM can't load all possible 32 bit constants in a single cycle.

Edit: The parallel versions are most likely slower than the straight forward methods, but they are faster if you have to saturate more than one value at a time.

If you care about performance, you really want to do this sort of stuff in SIMD, where x86 has native saturating arithmetic.

Because of this lack of saturating arithmetic in scalar math, one can get cases in which operations done on 4-variable-wide SIMD is more than 4 times faster than the equivalent C (and correspondingly true with 8-variable-wide SIMD):

sub8x8_dct8_c: 1332 clocks
sub8x8_dct8_mmx: 182 clocks
sub8x8_dct8_sse2: 127 clocks

uint32_t saturate_add32(uint32_t a, uint32_t b)
{
    uint32_t sum = a + b;
    if ((sum < a) || (sum < b))
        return ~((uint32_t)0);
    else
        return sum;
} /* saturate_add32 */

uint16_t saturate_add16(uint16_t a, uint16_t b)
{
    uint16_t sum = a + b;
    if ((sum < a) || (sum < b))
        return ~((uint16_t)0);
    else
        return sum;
} /* saturate_add16 */

Edit: Now that you've posted your version, I'm not sure mine is any cleaner/better/more efficient/more studly.

The current implementation we are using is:

#define sadd16(a, b)  (uint16_t)( ((uint32_t)(a)+(uint32_t)(b)) > 0xffff ? 0xffff : ((a)+(b)))
#define sadd32(a, b)  (uint32_t)( ((uint64_t)(a)+(uint64_t)(b)) > 0xffffffff ? 0xffffffff : ((a)+(b)))

I'm not sure if this is faster than Skizz's solution (always profile), but here's an alternative no-branch assembly solution. Note that this requires the conditional move (CMOV) instruction, which I'm not sure is available on your target.

uint32_t sadd32(uint32_t a, uint32_t b)
{
    __asm
    {
        movl eax, a
        addl eax, b
        movl edx, 0xffffffff
        cmovc eax, edx
    }
}

I suppose, the best way for x86 is to use inline assembler to check overflow flag after addition. Something like:

add eax, ebx
jno @@1
or eax, 0FFFFFFFFh
@@1:
.......

It's not very portable, but IMHO the most efficient way.

The best performance will usually involve inline assembly (as some have already stated).

But for portable C, these functions only involve one comparison and no type-casting (and thus I believe optimal):

unsigned saturate_add_uint(unsigned x, unsigned y)
{
    if (y > UINT_MAX - x) return UINT_MAX;
    return x + y;
}

unsigned short saturate_add_ushort(unsigned short x, unsigned short y)
{
    if (y > USHRT_MAX - x) return USHRT_MAX;
    return x + y;
}

As macros, they become:

SATURATE_ADD_UINT(x, y) (((y)>UINT_MAX-(x)) ? UINT_MAX : ((x)+(y)))
SATURATE_ADD_USHORT(x, y) (((y)>SHRT_MAX-(x)) ? USHRT_MAX : ((x)+(y)))

I leave versions for 'unsigned long' and 'unsigned long long' as an exercise to the reader. ;-)

int saturating_add(int x, int y)
{
    int w = sizeof(int) << 3;
    int msb = 1 << (w-1);

    int s = x + y;
    int sign_x = msb & x;
    int sign_y = msb & y;
    int sign_s = msb & s;

    int nflow = sign_x && sign_y && !sign_s;
    int pflow = !sign_x && !sign_y && sign_s;

    int nmask = (~!nflow + 1);
    int pmask = (~!pflow + 1);

    return (nmask & ((pmask & s) | (~pmask & ~msb))) | (~nmask & msb);
}

This implementation doesn't use control flows, campare operators(==, !=) and the ?: operator. It just uses bitwise operators and logical operators.

Saturation arithmetic is not standard for C, but it's often implemented via compiler intrinsics, so the most efficient way will not be the cleanest. You must add #ifdef blocks to select the proper way. MSalters's answer is the fastest for x86 architecture. For ARM you need to use __qadd16 function (ARM compiler) of _arm_qadd16 (Microsoft Visual Studio) for 16 bit version and __qadd for 32-bit version. They'll be automatically translated to one ARM instruction.

Links:

• __qadd16
• _arm_qadd16
• __qadd

I'll add solutions that were not yet mentioned above.

There exists ADC instruction in Intel x86. It is represented as _addcarry_u32() intrinsic function. For ARM there should be similar intrinsic.

Which allows us to implement very fast uint32_t saturated addition for Intel x86:

Try it online!

#include <stdint.h>
#include <immintrin.h>

uint32_t add_sat_u32(uint32_t a, uint32_t b) {
    uint32_t r, carry = _addcarry_u32(0, a, b, &r);
    return r | (-carry);
}

Intel x86 MMX saturated addition instructions can be used to implement uint16_t variant:

Try it online!

#include <stdint.h>
#include <immintrin.h>

uint16_t add_sat_u16(uint16_t a, uint16_t b) {
    return _mm_cvtsi64_si32(_mm_adds_pu16(
        _mm_cvtsi32_si64(a),
        _mm_cvtsi32_si64(b)
    ));
}

I don't mention ARM solution, as it can be implemented by other generic solutions from other answers.