approx_exp.h File Reference

#include <cstdint>
#include <cmath>
#include <limits>
#include <algorithm>

Go to the source code of this file.

Classes
union	approx_math::binary32

Namespaces
	approx_math

Macros
#define	APPROX_MATH_N

Functions
template<int DEGREE>
float	approx_expf (float x)
template<int DEGREE>
float	approx_expf_P (float p)
template<>
float	approx_expf_P< 2 > (float y)
template<>
float	approx_expf_P< 3 > (float y)
template<>
float	approx_expf_P< 4 > (float y)
template<>
float	approx_expf_P< 5 > (float y)
template<>
float	approx_expf_P< 6 > (float y)
template<>
float	approx_expf_P< 7 > (float y)
float	approx_math::fpfloor (float x)
template<int DEGREE>
float	unsafe_expf (float x)
template<int DEGREE>
float	unsafe_expf_impl (float x)

Macro Definition Documentation

#define APPROX_MATH_N

Definition at line 43 of file approx_exp.h.

Function Documentation

template<int DEGREE>

float approx_expf ( float  x )

inline

Definition at line 203 of file approx_exp.h.

References constexpr, max(), bookConverter::min, alignCSCRings::r, and x.

                                  {

  constexpr float inf_threshold =float(0x5.8b90cp4);
  // log of the smallest normal
  constexpr float zero_threshold_ftz =-float(0x5.75628p4); // sollya: single(log(1b-126));
  // flush to zero on the output
  // manage infty output:
  // faster than directly on output!
  x = std::min(std::max(x,zero_threshold_ftz),inf_threshold);
  float r = unsafe_expf<DEGREE>(x); 

   return r;
}

template<int DEGREE>

float approx_expf_P ( float  p )

inline

template<>

float approx_expf_P< 2 > ( float  y )

inline

Definition at line 78 of file approx_exp.h.

References detailsBasic3DVector::y.

                                       {
  return   float(0x2.p0) + y * (float(0x2.07b99p0) + y * float(0x1.025b84p0)) ;
}

template<>

float approx_expf_P< 3 > ( float  y )

inline

Definition at line 83 of file approx_exp.h.

References detailsBasic3DVector::y.

                                       {
#ifdef HORNER  // HORNER 
  return   float(0x2.p0) + y * (float(0x1.fff798p0) + y * (float(0x1.02249p0) + y * float(0x5.62042p-4))) ;
#else // ESTRIN
  float p23 = (float(0x1.02249p0) + y * float(0x5.62042p-4)) ;
  float p01 = float(0x2.p0) + y * float(0x1.fff798p0);
  return p01 + y*y*p23;
#endif
}

template<>

float approx_expf_P< 4 > ( float  y )

inline

Definition at line 94 of file approx_exp.h.

References detailsBasic3DVector::y.

                                       {
  return   float(0x2.p0) + y * (float(0x1.fffb1p0) + y * (float(0xf.ffe84p-4) + y * (float(0x5.5f9c1p-4) + y * float(0x1.57755p-4)))) ;
}

template<>

float approx_expf_P< 5 > ( float  y )

inline

Definition at line 99 of file approx_exp.h.

References detailsBasic3DVector::y.

                                       {
  return   float(0x2.p0) + y * (float(0x2.p0) + y * (float(0xf.ffed8p-4) + y * (float(0x5.5551cp-4) + y * (float(0x1.5740d8p-4) + y * float(0x4.49368p-8))))) ;
}

template<>

float approx_expf_P< 6 > ( float  y )

inline

Definition at line 104 of file approx_exp.h.

References AlCaHLTBitMon_ParallelJobs::p, and detailsBasic3DVector::y.

                                       {
#ifdef HORNER  // HORNER 
  float p =  float(0x2.p0) + y * (float(0x2.p0) + y * (float(0x1.p0) + y * (float(0x5.55523p-4) + y * (float(0x1.5554dcp-4) + y * (float(0x4.48f41p-8) + y * float(0xb.6ad4p-12)))))) ;
#else // ESTRIN does seem to save a cycle or two
  float p56 = float(0x4.48f41p-8) + y * float(0xb.6ad4p-12);
  float p34 = float(0x5.55523p-4) + y * float(0x1.5554dcp-4);
  float y2 = y*y;
  float p12 = float(0x2.p0) + y; // By chance we save one operation here! Funny.
  float p36 = p34 + y2*p56;
  float p16 = p12 + y2*p36;
  float p =  float(0x2.p0) + y*p16;
#endif
  return p;
}

template<>

float approx_expf_P< 7 > ( float  y )

inline

Definition at line 121 of file approx_exp.h.

References detailsBasic3DVector::y.

                                       {
   return float(0x2.p0) + y * (float(0x2.p0) + y * (float(0x1.p0) + y * (float(0x5.55555p-4) + y * (float(0x1.5554e4p-4) + y * (float(0x4.444adp-8) + y * (float(0xb.6a8a6p-12) + y * float(0x1.9ec814p-12))))))) ;
}

template<int DEGREE>

float unsafe_expf ( float  x )

inline

Definition at line 198 of file approx_exp.h.

References x.

                                  {
 return  unsafe_expf_impl<DEGREE>(x); 
}

template<int DEGREE>

float unsafe_expf_impl ( float  x )

inline

Definition at line 149 of file approx_exp.h.

References constexpr, alignCSCRings::e, f, approx_math::fpfloor(), AlCaHLTBitMon_ParallelJobs::p, x, detailsBasic3DVector::y, and detailsBasic3DVector::z.

                                       {
  using namespace approx_math;
  /* Sollya for the following constants:
     display=hexadecimal;
     1b23+1b22;
     single(1/log(2));
     log2H=round(log(2), 16, RN);
     log2L = single(log(2)-log2H);
     log2H; log2L;
     
  */
  // constexpr float rnd_cst = float(0xc.p20);
  constexpr float inv_log2f = float(0x1.715476p0);
  constexpr float log2H = float(0xb.172p-4);
  constexpr float log2L = float(0x1.7f7d1cp-20);
  
   
  float y = x;
  // This is doing round(x*inv_log2f) to the nearest integer
  float z = fpfloor((x*inv_log2f) +0.5f);
  // Cody-and-Waite accurate range reduction. FMA-safe.
  y -= z*log2H;
  y -= z*log2L;
  // exponent 
  int32_t e = z;
  

  // we want RN above because it centers the interval around zero
  // but then we could have 2^e = below being infinity when it shouldn't 
  // (when e=128 but p<1)
  // so we avoid this case by reducing e and evaluating a polynomial for 2*exp
  e -=1; 

  // NaN inputs will propagate to the output as expected

  float p = approx_expf_P<DEGREE>(y);

  // cout << "x=" << x << "  e=" << e << "  y=" << y << "  p=" << p <<"\n";
  binary32 ef;
  uint32_t biased_exponent= e+127;
  ef.ui32=(biased_exponent<<23);
  
  return p * ef.f;
}