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3+TITLE: Numerical precision

1 Control of the numerical precision

Controlling numerical precision enables optimizations. Here, the default parameters determining the target numerical precision and range are defined. Following the IEEE Standard for Floating-Point Arithmetic (IEEE 754), precision refers to the number of significand bits (including the sign bit) and range refers to the number of exponent bits.

QMCKL_DEFAULT_PRECISION 53
QMCKL_DEFAULT_RANGE 11 
typedef struct qmckl_numprec_struct {
  uint32_t  precision;
  uint32_t  range;
} qmckl_numprec_struct;

The following functions set and get the required precision and range. precision is an integer between 2 and 53, and range is an integer between 2 and 11.

The setter functions functions return a new context as a 64-bit integer. The getter functions return the value, as a 32-bit integer. The update functions return QMCKL_SUCCESS or QMCKL_FAILURE.

2 Precision

qmckl_context_set_numprec_precision modifies the parameter for the numerical precision in the context. 

qmckl_exit_code qmckl_set_numprec_precision(const qmckl_context context, const int precision) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
    return QMCKL_INVALID_CONTEXT;

  if (precision <  2) {
    return qmckl_failwith(context,
                          QMCKL_INVALID_ARG_2,
                          "qmckl_update_numprec_precision",
                          "precision < 2");
  }

  if (precision > 53) {
    return qmckl_failwith(context,
                          QMCKL_INVALID_ARG_2,
                          "qmckl_update_numprec_precision",
                          "precision > 53");
  }

  qmckl_context_struct* const ctx = (qmckl_context_struct*) context;

  /* This should be always true because the context is valid */
  assert (ctx != NULL);

  qmckl_lock(context);
  {
    ctx->numprec.precision = (uint32_t) precision;
  }
  qmckl_unlock(context);

  return QMCKL_SUCCESS;
}

interface
   integer (qmckl_exit_code) function qmckl_set_numprec_precision(context, precision) bind(C)
     use, intrinsic :: iso_c_binding
     import
     integer (qmckl_context), intent(in), value :: context
     integer (c_int32_t), intent(in), value :: precision
   end function qmckl_set_numprec_precision
end interface

qmckl_get_numprec_precision returns the value of the numerical precision in the context. 

int qmckl_get_numprec_precision(const qmckl_context context) {
  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
      return qmckl_failwith(context,
                      QMCKL_INVALID_CONTEXT,
                      "qmckl_get_numprec_precision",
                      "");
  }

  const qmckl_context_struct* const ctx = (qmckl_context_struct*) context;
  return ctx->numprec.precision;
}

interface
   integer (qmckl_exit_code) function qmckl_get_numprec_precision(context) bind(C)
     use, intrinsic :: iso_c_binding
     import
     integer (qmckl_context), intent(in), value :: context
   end function qmckl_get_numprec_precision
end interface

3 Range

qmckl_set_numprec_range modifies the parameter for the numerical range in a given context. 

qmckl_exit_code qmckl_set_numprec_range(const qmckl_context context, const int range) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
    return QMCKL_INVALID_CONTEXT;

  if (range <  2) {
    return qmckl_failwith(context,
                    QMCKL_INVALID_ARG_2,
                    "qmckl_set_numprec_range",
                    "range < 2");
  }

  if (range > 11) {
    return qmckl_failwith(context,
                    QMCKL_INVALID_ARG_2,
                    "qmckl_set_numprec_range",
                    "range > 11");
  }

  qmckl_context_struct* const ctx = (qmckl_context_struct*) context;

  /* This should be always true because the context is valid */
  assert (ctx != NULL);

  qmckl_lock(context);
  {
    ctx->numprec.range = (uint32_t) range;
  }
  qmckl_unlock(context);

  return QMCKL_SUCCESS;
}

interface
   integer (qmckl_exit_code) function qmckl_set_numprec_range(context, range) bind(C)
     use, intrinsic :: iso_c_binding
     import
     integer (qmckl_context), intent(in), value :: context
     integer (c_int32_t), intent(in), value :: range
   end function qmckl_set_numprec_range
end interface

qmckl_get_numprec_range returns the value of the numerical range in the context. 

int qmckl_get_numprec_range(const qmckl_context context) {
  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
      return qmckl_failwith(context,
                      QMCKL_INVALID_CONTEXT,
                      "qmckl_get_numprec_range",
                      "");
  }

  const qmckl_context_struct* const ctx = (qmckl_context_struct*) context;
  return ctx->numprec.range;
}

4 Helper functions

4.1 Epsilon

qmckl_get_numprec_epsilon returns \(\epsilon = 2^{1-n}\) where n is the precision. We need to remove the sign bit from the precision. 

double qmckl_get_numprec_epsilon(const qmckl_context context) {
  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
    return QMCKL_INVALID_CONTEXT;
  const qmckl_context_struct* const ctx = (qmckl_context_struct*) context;
  const int precision = ctx->numprec.precision;
  return 1. /  (double) ( ((uint64_t) 1) << (precision-2));
}

4.2 Testing the number of unchanged bits

To test that a given approximation keeps a given number of bits unchanged, we need a function that returns the number of unchanged bits in the range, and in the precision.

For this, we first count by how many units in the last place (ulps) two numbers differ. 

int64_t countUlpDifference_64(double a, double b) {

  union int_or_float {
    int64_t i;
    double f;
  } x, y;

  x.f = a;
  y.f = b;

  // Handle sign bit discontinuity: if the signs are different and either value is not zero
  if ((x.i < 0) != (y.i < 0) && (x.f != 0.0) && (y.f != 0.0)) {
    // Use the absolute values and add the distance to zero for both numbers
    int64_t distanceToZeroForX = x.i < 0 ? INT64_MAX + x.i : INT64_MAX - x.i;
    int64_t distanceToZeroForY = y.i < 0 ? INT64_MAX + y.i : INT64_MAX - y.i;
    return distanceToZeroForX + distanceToZeroForY;
  }

  // Calculate the difference in their binary representations
  int64_t result = x.i - y.i;
  result = result > 0 ? result : -result;
  return result;
}

int32_t qmckl_test_precision_64(double a, double b) {

  int64_t diff = countUlpDifference_64(a,b);

  if (diff == 0) return 53;

  int32_t result = 53;

  for (int i=0 ; i<53 && diff != 0 ; ++i) {
    diff >>= 1;
    result--;
  }

  return result;
}

int32_t qmckl_test_precision_32(float a, float b) {
  return qmckl_test_precision_64( (double) a, (double) b );
}

5 Approximate functions

5.1 Exponential

Fast exponential function, adapted from Johan Rade's implementation (https://gist.github.com/jrade/293a73f89dfef51da6522428c857802d). It is based on Schraudolph's paper:

N. Schraudolph, "A Fast, Compact Approximation of the Exponential Function", Neural Computation 11, 853–862 (1999). (available at https://nic.schraudolph.org/pubs/Schraudolph99.pdf) 

float fastExpf(float x)
{
  const float a = 12102203.0;
  const float b = 1064986816.0;
  x = a * x + b;

  const float c = 8388608.0;
  const float d = 2139095040.0;
  if (x < c || x > d)
    x = (x < c) ? 0.0f : d;

  uint32_t n = (uint32_t) x;
  memcpy(&x, &n, 4);
  return x;
}


double fastExp(double x)
{
  const double a = 6497320848556798.0;
  const double b = 4606985713057410560.0;
  x = a * x + b;

  const double c = 4503599627370496.0;
  const double d = 9218868437227405312.0;
  if (x < c || x > d)
    x = (x < c) ? 0.0 : d;

  uint64_t n = (uint64_t) x;
  memcpy(&x, &n, 8);
  return x;
}

Author: TREX CoE

Created: 2024-03-29 Fri 08:29

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