.file "expf_m1.s"

// Copyright (c) 2000 - 2005, Intel Corporation

// All rights reserved.

//

// Contributed 2000 by the Intel Numerics Group, Intel Corporation

//

// Redistribution and use in source and binary forms, with or without

// modification, are permitted provided that the following conditions are

// met:

//

// * Redistributions of source code must retain the above copyright

// notice, this list of conditions and the following disclaimer.

//

// * Redistributions in binary form must reproduce the above copyright

// notice, this list of conditions and the following disclaimer in the

// documentation and/or other materials provided with the distribution.

//

// * The name of Intel Corporation may not be used to endorse or promote

// products derived from this software without specific prior written

// permission.

// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS

// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT

// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR

// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS

// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,

// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,

// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR

// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY

// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING

// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS

// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

//

// Intel Corporation is the author of this code, and requests that all

// problem reports or change requests be submitted to it directly at

// http://www.intel.com/software/products/opensource/libraries/num.htm.

// History

//*********************************************************************

// 02/02/00 Initial Version

// 04/04/00 Unwind support added

// 08/15/00 Bundle added after call to __libm_error_support to properly

//          set [the previously overwritten] GR_Parameter_RESULT.

// 07/07/01 Improved speed of all paths

// 05/20/02 Cleaned up namespace and sf0 syntax

// 11/20/02 Improved speed, algorithm based on expf

// 03/31/05 Reformatted delimiters between data tables

//

//

// API

//*********************************************************************

// float expm1f(float)

//

// Overview of operation

//*********************************************************************

// 1. Inputs of Nan, Inf, Zero, NatVal handled with special paths

//

// 2. |x| < 2^-40

//    Result = x, computed by x + x*x to handle appropriate flags and rounding

//

// 3. 2^-40 <= |x| < 2^-2

//    Result determined by 8th order Taylor series polynomial

//    expm1f(x) = x + A2*x^2 + ... + A8*x^8

//

// 4. x < -24.0

//    Here we know result is essentially -1 + eps, where eps only affects

//    rounded result.  Set I.

//

// 5. x >= 88.7228

//    Result overflows.  Set I, O, and call error support

//

// 6. 2^-2 <= x < 88.7228  or  -24.0 <= x < -2^-2

//    This is the main path.  The algorithm is described below:

// Take the input x. w is "how many log2/128 in x?"

//  w = x * 64/log2

//  NJ = int(w)

//  x = NJ*log2/64 + R

//  NJ = 64*n + j

//  x = n*log2 + (log2/64)*j + R

//

//  So, exp(x) = 2^n * 2^(j/64)* exp(R)

//

//  T =  2^n * 2^(j/64)

//       Construct 2^n

//       Get 2^(j/64) table

//           actually all the entries of 2^(j/64) table are stored in DP and

//           with exponent bits set to 0 -> multiplication on 2^n can be

//           performed by doing logical "or" operation with bits presenting 2^n

//  exp(R) = 1 + (exp(R) - 1)

//  P = exp(R) - 1 approximated by Taylor series of 3rd degree

//      P = A3*R^3 + A2*R^2 + R, A3 = 1/6, A2 = 1/2

//

//  The final result is reconstructed as follows

//  expm1f(x) = T*P + (T - 1.0)

// Special values

//*********************************************************************

// expm1f(+0)    = +0.0

// expm1f(-0)    = -0.0

// expm1f(+qnan) = +qnan

// expm1f(-qnan) = -qnan

// expm1f(+snan) = +qnan

// expm1f(-snan) = -qnan

// expm1f(-inf)  = -1.0

// expm1f(+inf)  = +inf

// Overflow and Underflow

//*********************************************************************

// expm1f(x) = largest single normal when

//     x = 88.7228 = 0x42b17217

//

// Underflow is handled as described in case 2 above.

// Registers used

//*********************************************************************

// Floating Point registers used:

// f8, input

// f6,f7, f9 -> f15,  f32 -> f45

// General registers used:

// r3, r20 -> r38

// Predicate registers used:

// p9 -> p15

// Assembly macros

//*********************************************************************

// integer registers used

// scratch

rNJ                   = r3

rExp_half             = r20

rSignexp_x            = r21

rExp_x                = r22

rExp_mask             = r23

rExp_bias             = r24

rTmp                  = r25

rM1_lim               = r25

rGt_ln                = r25

rJ                    = r26

rN                    = r27

rTblAddr              = r28

rLn2Div64             = r29

rRightShifter         = r30

r64DivLn2             = r31

// stacked

GR_SAVE_PFS           = r32

GR_SAVE_B0            = r33

GR_SAVE_GP            = r34

GR_Parameter_X        = r35

GR_Parameter_Y        = r36

GR_Parameter_RESULT   = r37

GR_Parameter_TAG      = r38

// floating point registers used

FR_X                  = f10

FR_Y                  = f1

FR_RESULT             = f8

// scratch

fRightShifter         = f6

f64DivLn2             = f7

fNormX                = f9

fNint                 = f10

fN                    = f11

fR                    = f12

fLn2Div64             = f13

fA2                   = f14

fA3                   = f15

// stacked

fP                    = f32

fX3                   = f33

fT                    = f34

fMIN_SGL_OFLOW_ARG    = f35

fMAX_SGL_NORM_ARG     = f36

fMAX_SGL_MINUS_1_ARG  = f37

fA4                   = f38

fA43                  = f38

fA432                 = f38

fRSqr                 = f39

fA5                   = f40

fTmp                  = f41

fGt_pln               = f41

fXsq                  = f41

fA7                   = f42

fA6                   = f43

fA65                  = f43

fTm1                  = f44

fA8                   = f45

fA87                  = f45

fA8765                = f45

fA8765432             = f45

fWre_urm_f8           = f45

RODATA

.align 16

LOCAL_OBJECT_START(_expf_table)

data8 0x3efa01a01a01a01a // A8 = 1/8!

data8 0x3f2a01a01a01a01a // A7 = 1/7!

data8 0x3f56c16c16c16c17 // A6 = 1/6!

data8 0x3f81111111111111 // A5 = 1/5!

data8 0x3fa5555555555555 // A4 = 1/4!

data8 0x3fc5555555555555 // A3 = 1/3!

//

data4 0x42b17218         // Smallest sgl arg to overflow sgl result

data4 0x42b17217         // Largest sgl arg to give sgl result

//

// 2^(j/64) table, j goes from 0 to 63

data8 0x0000000000000000 // 2^(0/64)

data8 0x00002C9A3E778061 // 2^(1/64)

data8 0x000059B0D3158574 // 2^(2/64)

data8 0x0000874518759BC8 // 2^(3/64)

data8 0x0000B5586CF9890F // 2^(4/64)

data8 0x0000E3EC32D3D1A2 // 2^(5/64)

data8 0x00011301D0125B51 // 2^(6/64)

data8 0x0001429AAEA92DE0 // 2^(7/64)

data8 0x000172B83C7D517B // 2^(8/64)

data8 0x0001A35BEB6FCB75 // 2^(9/64)

data8 0x0001D4873168B9AA // 2^(10/64)

data8 0x0002063B88628CD6 // 2^(11/64)

data8 0x0002387A6E756238 // 2^(12/64)

data8 0x00026B4565E27CDD // 2^(13/64)

data8 0x00029E9DF51FDEE1 // 2^(14/64)

data8 0x0002D285A6E4030B // 2^(15/64)

data8 0x000306FE0A31B715 // 2^(16/64)

data8 0x00033C08B26416FF // 2^(17/64)

data8 0x000371A7373AA9CB // 2^(18/64)

data8 0x0003A7DB34E59FF7 // 2^(19/64)

data8 0x0003DEA64C123422 // 2^(20/64)

data8 0x0004160A21F72E2A // 2^(21/64)

data8 0x00044E086061892D // 2^(22/64)

data8 0x000486A2B5C13CD0 // 2^(23/64)

data8 0x0004BFDAD5362A27 // 2^(24/64)

data8 0x0004F9B2769D2CA7 // 2^(25/64)

data8 0x0005342B569D4F82 // 2^(26/64)

data8 0x00056F4736B527DA // 2^(27/64)

data8 0x0005AB07DD485429 // 2^(28/64)

data8 0x0005E76F15AD2148 // 2^(29/64)

data8 0x0006247EB03A5585 // 2^(30/64)

data8 0x0006623882552225 // 2^(31/64)

data8 0x0006A09E667F3BCD // 2^(32/64)

data8 0x0006DFB23C651A2F // 2^(33/64)

data8 0x00071F75E8EC5F74 // 2^(34/64)

data8 0x00075FEB564267C9 // 2^(35/64)

data8 0x0007A11473EB0187 // 2^(36/64)

data8 0x0007E2F336CF4E62 // 2^(37/64)

data8 0x00082589994CCE13 // 2^(38/64)

data8 0x000868D99B4492ED // 2^(39/64)

data8 0x0008ACE5422AA0DB // 2^(40/64)

data8 0x0008F1AE99157736 // 2^(41/64)

data8 0x00093737B0CDC5E5 // 2^(42/64)

data8 0x00097D829FDE4E50 // 2^(43/64)

data8 0x0009C49182A3F090 // 2^(44/64)

data8 0x000A0C667B5DE565 // 2^(45/64)

data8 0x000A5503B23E255D // 2^(46/64)

data8 0x000A9E6B5579FDBF // 2^(47/64)

data8 0x000AE89F995AD3AD // 2^(48/64)

data8 0x000B33A2B84F15FB // 2^(49/64)

data8 0x000B7F76F2FB5E47 // 2^(50/64)

data8 0x000BCC1E904BC1D2 // 2^(51/64)

data8 0x000C199BDD85529C // 2^(52/64)

data8 0x000C67F12E57D14B // 2^(53/64)

data8 0x000CB720DCEF9069 // 2^(54/64)

data8 0x000D072D4A07897C // 2^(55/64)

data8 0x000D5818DCFBA487 // 2^(56/64)

data8 0x000DA9E603DB3285 // 2^(57/64)

data8 0x000DFC97337B9B5F // 2^(58/64)

data8 0x000E502EE78B3FF6 // 2^(59/64)

data8 0x000EA4AFA2A490DA // 2^(60/64)

data8 0x000EFA1BEE615A27 // 2^(61/64)

data8 0x000F50765B6E4540 // 2^(62/64)

data8 0x000FA7C1819E90D8 // 2^(63/64)

LOCAL_OBJECT_END(_expf_table)

.section .text

GLOBAL_IEEE754_ENTRY(expm1f)

{ .mlx

      getf.exp        rSignexp_x = f8      // Must recompute if x unorm

      movl            r64DivLn2 = 0x40571547652B82FE // 64/ln(2)

}

{ .mlx

      addl            rTblAddr = @ltoff(_expf_table),gp

      movl            rRightShifter = 0x43E8000000000000 // DP Right Shifter

}

;;

{ .mfi

      // point to the beginning of the table

      ld8             rTblAddr = [rTblAddr]

      fclass.m        p14, p0 = f8 , 0x22  // test for -INF

      mov             rExp_mask = 0x1ffff   // Exponent mask

}

{ .mfi

      nop.m           0

      fnorm.s1        fNormX = f8 // normalized x

      nop.i           0

}

;;

{ .mfi

      setf.d          f64DivLn2 = r64DivLn2 // load 64/ln(2) to FP reg

      fclass.m        p9, p0 = f8 , 0x0b    // test for x unorm

      mov             rExp_bias = 0xffff    // Exponent bias

}

{ .mlx

      // load Right Shifter to FP reg

      setf.d          fRightShifter = rRightShifter

      movl            rLn2Div64 = 0x3F862E42FEFA39EF // DP ln(2)/64 in GR

}

;;

{ .mfi

      ldfpd           fA8, fA7 = [rTblAddr], 16

      fcmp.eq.s1      p13, p0 = f0, f8      // test for x = 0.0

      mov             rExp_half = 0xfffe

}

{ .mfb

      setf.d          fLn2Div64 = rLn2Div64 // load ln(2)/64 to FP reg

      nop.f           0

(p9)  br.cond.spnt    EXPM1_UNORM // Branch if x unorm

}

;;

EXPM1_COMMON:

{ .mfb

      ldfpd           fA6, fA5 = [rTblAddr], 16

(p14) fms.s.s0        f8 = f0, f0, f1       // result if x = -inf

(p14) br.ret.spnt     b0                    // exit here if x = -inf

}

;;

{ .mfb

      ldfpd           fA4, fA3 = [rTblAddr], 16

      fclass.m        p15, p0 = f8 , 0x1e1  // test for NaT,NaN,+Inf

(p13) br.ret.spnt     b0                    // exit here if x =0.0, result is x

}

;;

{ .mfi

      // overflow thresholds

      ldfps           fMIN_SGL_OFLOW_ARG, fMAX_SGL_NORM_ARG = [rTblAddr], 8

      fma.s1          fXsq = fNormX, fNormX, f0      // x^2 for small path

      and             rExp_x = rExp_mask, rSignexp_x // Biased exponent of x

}

{ .mlx

      nop.m           0

      movl            rM1_lim = 0xc1c00000  // Minus -1 limit (-24.0), SP

}

;;

{ .mfi

      setf.exp        fA2 = rExp_half

      // x*(64/ln(2)) + Right Shifter

      fma.s1          fNint = fNormX, f64DivLn2, fRightShifter

      sub             rExp_x = rExp_x, rExp_bias     // True exponent of x

}

{ .mfb

      nop.m           0

(p15) fma.s.s0        f8 = f8, f1, f0       // result if x = NaT,NaN,+Inf

(p15) br.ret.spnt     b0                    // exit here if x = NaT,NaN,+Inf

}

;;

{ .mfi

      setf.s          fMAX_SGL_MINUS_1_ARG = rM1_lim // -1 threshold, -24.0

      nop.f           0

      cmp.gt          p7, p8 = -2, rExp_x      // Test |x| < 2^(-2)

}

;;

{ .mfi

(p7)  cmp.gt.unc      p6, p7 = -40, rExp_x     // Test |x| < 2^(-40)

      fma.s1          fA87 = fA8, fNormX, fA7  // Small path, A8*x+A7

      nop.i           0

}

{ .mfi

      nop.m           0

      fma.s1          fA65 = fA6, fNormX, fA5  // Small path, A6*x+A5

      nop.i           0

}

;;

{ .mfb

      nop.m           0

(p6)  fma.s.s0        f8 = f8, f8, f8          // If x < 2^-40, result=x+x*x

(p6)  br.ret.spnt     b0                       // Exit if x < 2^-40

}

;;

{ .mfi

      nop.m           0

      // check for overflow

      fcmp.gt.s1      p15, p14 = fNormX, fMIN_SGL_OFLOW_ARG

      nop.i           0

}

{ .mfi

      nop.m           0

      fms.s1          fN = fNint, f1, fRightShifter // n in FP register

      nop.i           0

}

;;

{ .mfi

      nop.m           0

(p7)  fma.s1          fA43 = fA4, fNormX, fA3   // Small path, A4*x+A3

      nop.i           0

}

;;

{ .mfi

      getf.sig        rNJ = fNint               // bits of n, j

(p7)  fma.s1          fA8765 = fA87, fXsq, fA65 // Small path, A87*xsq+A65

      nop.i           0

}

{ .mfb

      nop.m           0

(p7)  fma.s1          fX3 = fXsq, fNormX, f0    // Small path, x^3

      // branch out if overflow

(p15) br.cond.spnt    EXPM1_CERTAIN_OVERFLOW

}

;;

{ .mfi

      addl            rN = 0xffff-63, rNJ    // biased and shifted n

      fnma.s1         fR = fLn2Div64, fN, fNormX // R = x - N*ln(2)/64

      extr.u          rJ = rNJ , 0 , 6       // bits of j

}

;;

{ .mfi

      shladd          rJ = rJ, 3, rTblAddr   // address in the 2^(j/64) table

      // check for certain -1

      fcmp.le.s1      p13, p0 = fNormX, fMAX_SGL_MINUS_1_ARG

      shr             rN = rN, 6             // biased n

}

{ .mfi

      nop.m           0

(p7)  fma.s1          fA432 = fA43, fNormX, fA2 // Small path, A43*x+A2

      nop.i           0

}

;;

{ .mfi

      ld8             rJ = [rJ]

      nop.f           0

      shl             rN = rN , 52           // 2^n bits in DP format

}

;;

{ .mmi

      or              rN = rN, rJ        // bits of 2^n * 2^(j/64) in DP format

(p13) mov             rTmp = 1           // Make small value for -1 path

      nop.i           0

}

;;

{ .mfi

      setf.d          fT = rN            // 2^n

      // check for possible overflow (only happens if input higher precision)

(p14) fcmp.gt.s1      p14, p0 = fNormX, fMAX_SGL_NORM_ARG

      nop.i           0

}

{ .mfi

      nop.m           0

(p7)  fma.s1          fA8765432 = fA8765, fX3, fA432 // A8765*x^3+A432

      nop.i           0

}

;;

{ .mfi

(p13) setf.exp        fTmp = rTmp        // Make small value for -1 path

      fma.s1          fP = fA3, fR, fA2  // A3*R + A2

      nop.i           0

}

{ .mfb

      nop.m           0

      fma.s1          fRSqr = fR, fR, f0 // R^2

(p13) br.cond.spnt    EXPM1_CERTAIN_MINUS_ONE // Branch if x < -24.0

}

;;

{ .mfb

      nop.m           0

(p7)  fma.s.s0        f8 = fA8765432, fXsq, fNormX // Small path,

                                         // result=xsq*A8765432+x

(p7)  br.ret.spnt     b0                 // Exit if 2^-40 <= |x| < 2^-2

}

;;

{ .mfi

      nop.m           0

      fma.s1          fP = fP, fRSqr, fR // P = (A3*R + A2)*Rsqr + R

      nop.i           0

}

;;

{ .mfb

      nop.m           0

      fms.s1          fTm1 = fT, f1, f1  // T - 1.0

(p14) br.cond.spnt    EXPM1_POSSIBLE_OVERFLOW

}

;;

{ .mfb

      nop.m           0

      fma.s.s0        f8 = fP, fT, fTm1

      br.ret.sptk     b0                 // Result for main path

                                         // minus_one_limit < x < -2^-2

                                         // and +2^-2 <= x < overflow_limit

}

;;

// Here if x unorm

EXPM1_UNORM:

{ .mfb

      getf.exp        rSignexp_x = fNormX // Must recompute if x unorm

      fcmp.eq.s0      p6, p0 = f8, f0     // Set D flag

      br.cond.sptk    EXPM1_COMMON

}

;;

// here if result will be -1 and inexact, x <= -24.0

EXPM1_CERTAIN_MINUS_ONE:

{ .mfb

      nop.m           0

      fms.s.s0        f8 = fTmp, fTmp, f1  // Result -1, and Inexact set

      br.ret.sptk     b0

}

;;

EXPM1_POSSIBLE_OVERFLOW:

// Here if fMAX_SGL_NORM_ARG < x < fMIN_SGL_OFLOW_ARG

// This cannot happen if input is a single, only if input higher precision.

// Overflow is a possibility, not a certainty.

// Recompute result using status field 2 with user's rounding mode,

// and wre set.  If result is larger than largest single, then we have

// overflow

{ .mfi

      mov             rGt_ln  = 0x1007f // Exponent for largest sgl + 1 ulp

      fsetc.s2        0x7F,0x42         // Get user's round mode, set wre

      nop.i           0

}

;;

{ .mfi

      setf.exp        fGt_pln = rGt_ln  // Create largest single + 1 ulp

      fma.s.s2        fWre_urm_f8 = fP, fT, fTm1  // Result with wre set

      nop.i           0

}

;;

{ .mfi

      nop.m           0

      fsetc.s2        0x7F,0x40                   // Turn off wre in sf2

      nop.i           0

}

;;

{ .mfi

      nop.m           0

      fcmp.ge.s1      p6, p0 =  fWre_urm_f8, fGt_pln // Test for overflow

      nop.i           0

}

;;

{ .mfb

      nop.m           0

      nop.f           0

(p6)  br.cond.spnt    EXPM1_CERTAIN_OVERFLOW // Branch if overflow

}

;;

{ .mfb

      nop.m           0

      fma.s.s0        f8 = fP, fT, fTm1

      br.ret.sptk     b0                     // Exit if really no overflow

}

;;

// here if overflow

EXPM1_CERTAIN_OVERFLOW:

{ .mmi

      addl            rTmp = 0x1FFFE, r0;;

      setf.exp        fTmp = rTmp

      nop.i 999

}

;;

{ .mfi

      alloc           r32 = ar.pfs, 0, 3, 4, 0 // get some registers

      fmerge.s        FR_X = fNormX,fNormX

      nop.i           0

}

{ .mfb

      mov             GR_Parameter_TAG = 43

      fma.s.s0        FR_RESULT = fTmp, fTmp, f0 // Set I,O and +INF result

      br.cond.sptk    __libm_error_region

}

;;

GLOBAL_IEEE754_END(expm1f)

libm_alias_float_other (__expm1, expm1)

LOCAL_LIBM_ENTRY(__libm_error_region)

.prologue

{ .mfi

      add   GR_Parameter_Y=-32,sp             // Parameter 2 value

      nop.f 999

.save   ar.pfs,GR_SAVE_PFS

      mov  GR_SAVE_PFS=ar.pfs                 // Save ar.pfs

}

{ .mfi

.fframe 64

      add sp=-64,sp                           // Create new stack

      nop.f 0

      mov GR_SAVE_GP=gp                       // Save gp

};;

{ .mmi

      stfs [GR_Parameter_Y] = FR_Y,16         // Store Parameter 2 on stack

      add GR_Parameter_X = 16,sp              // Parameter 1 address

.save   b0, GR_SAVE_B0

      mov GR_SAVE_B0=b0                       // Save b0

};;

.body

{ .mfi

      stfs [GR_Parameter_X] = FR_X            // Store Parameter 1 on stack

      nop.f 0

      add   GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address

}

{ .mib

      stfs [GR_Parameter_Y] = FR_RESULT       // Store Parameter 3 on stack

      add   GR_Parameter_Y = -16,GR_Parameter_Y

      br.call.sptk b0=__libm_error_support#   // Call error handling function

};;

{ .mmi

      add   GR_Parameter_RESULT = 48,sp

      nop.m 0

      nop.i 0

};;

{ .mmi

      ldfs  f8 = [GR_Parameter_RESULT]       // Get return result off stack

.restore sp

      add   sp = 64,sp                       // Restore stack pointer

      mov   b0 = GR_SAVE_B0                  // Restore return address

};;

{ .mib

      mov   gp = GR_SAVE_GP                  // Restore gp

      mov   ar.pfs = GR_SAVE_PFS             // Restore ar.pfs

      br.ret.sptk     b0                     // Return

};;

LOCAL_LIBM_END(__libm_error_region)

.type   __libm_error_support#,@function

.global __libm_error_support#