approx_exp.h

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#ifndef DataFormatsMathAPPROX_EXP_H
#define DataFormatsMathAPPROX_EXP_H
/*  Quick and not that dirty vectorizable exp implementations
    Author: Florent de Dinechin, Aric, ENS-Lyon 
    with advice from Vincenzo Innocente, CERN
    All right reserved

Warning + disclaimers:
 
Feel free to distribute or insert in other programs etc, as long as this notice is attached.
    Comments, requests etc: Florent.de.Dinechin@ens-lyon.fr

Polynomials were obtained using Sollya scripts (in comments): 
please also keep these comments attached to the code. 

If a derivative of this code ends up in the glibc I am too happy: the version with MANAGE_SUBNORMALS=1 and DEGREE=6 is faithful-accurate over the full 2^32 binary32 numbers and behaves well WRT exceptional cases. It is about 4 times faster than the stock expf on this PC, when compiled with gcc -O2.

This code is FMA-safe (i.e. accelerated and more accurate with an FMA) as long as my parentheses are respected. 

A remaining TODO is to try and manage the over/underflow using only integer tests as per Astasiev et al, RNC conf.
Not sure it makes that much sense in the vector context.

*/

// #define MANAGE_SUBNORMALS 1 // No measurable perf difference, so let's be clean.
// If set to zero we flush to zero the subnormal outputs, ie for x<-88 or so

// DEGREE
// 6 is perfect.
// 5 provides max 2-ulp error,
// 4 loses 44 ulps (6 bits) for an acceleration of 10% WRT 6
// (I don't subtract the loop and call overhead, so it would be more for constexprd code)

// see the comments in the code for the accuracy you get from a given degree

#include "DataFormats/Math/interface/approx_math.h"

template <int DEGREE>
constexpr float approx_expf_P(float p);

// degree =  2   => absolute accuracy is  8 bits
template <>
constexpr float approx_expf_P<2>(float y) {
  return float(0x2.p0) + y * (float(0x2.07b99p0) + y * float(0x1.025b84p0));
}
// degree =  3   => absolute accuracy is  12 bits
template <>
constexpr float approx_expf_P<3>(float 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
}
// degree =  4   => absolute accuracy is  17 bits
template <>
constexpr float approx_expf_P<4>(float y) {
  return float(0x2.p0) +
         y * (float(0x1.fffb1p0) + y * (float(0xf.ffe84p-4) + y * (float(0x5.5f9c1p-4) + y * float(0x1.57755p-4))));
}
// degree =  5   => absolute accuracy is  22 bits
template <>
constexpr float approx_expf_P<5>(float 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)))));
}
// degree =  6   => absolute accuracy is  27 bits
template <>
constexpr float approx_expf_P<6>(float 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;
}

// degree =  7   => absolute accuracy is  31 bits
template <>
constexpr float approx_expf_P<7>(float 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)))))));
}

/* The Sollya script that computes the polynomials above


f= 2*exp(y);
I=[-log(2)/2;log(2)/2];
filename="/tmp/polynomials";
print("") > filename;
for deg from 2 to 8 do begin
  p = fpminimax(f, deg,[|1,23...|],I, floating, absolute); 
  display=decimal;
  acc=floor(-log2(sup(supnorm(p, f, I, absolute, 2^(-40)))));
  print( "   // degree = ", deg, 
         "  => absolute accuracy is ",  acc, "bits" ) >> filename;
  print("#if ( DEGREE ==", deg, ")") >> filename;
  display=hexadecimal;
  print("   float p = ", horner(p) , ";") >> filename;
  print("#endif") >> filename;
end;

*/

// valid for -87.3365 < x < 88.7228
template <int DEGREE>
constexpr float unsafe_expf_impl(float x) {
  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;
}

#ifndef NO_APPROX_MATH

template <int DEGREE>
constexpr float unsafe_expf(float x) {
  return unsafe_expf_impl<DEGREE>(x);
}

template <int DEGREE>
constexpr float approx_expf(float 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;
}

#else
template <int DEGREE>
constexpr float unsafe_expf(float x) {
  return std::exp(x);
}
template <int DEGREE>
constexpr float approx_expf(float x) {
  return std::exp(x);
}
#endif  // NO_APPROX_MATH

#endif