s_log1p.s

Glibc 2.3.2源代码(解压后有100多M)

.file "log1p.s" 

// Copyright (C) 2000, 2001, Intel Corporation
// All rights reserved.
// 
// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story,
// and Ping Tak Peter Tang of the Computational Software Lab, 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://developer.intel.com/opensource.
//
// History
//==============================================================
// 2/02/00  Initial version
// 4/04/00  Unwind support added
// 8/15/00  Bundle added after call to __libm_error_support to properly
//          set [the previously overwritten] GR_Parameter_RESULT.
//
// *********************************************************************
//
// Function:   log1p(x) = ln(x+1), for double precision x values
//
// *********************************************************************
//
// Accuracy:   Very accurate for double precision values
//
// *********************************************************************
//
// Resources Used:
//
//    Floating-Point Registers: f8 (Input and Return Value)
//                              f9,f33-f55,f99 
//
//    General Purpose Registers:
//      r32-r53
//      r54-r57 (Used to pass arguments to error handling routine)
//
//    Predicate Registers:      p6-p15
//
// *********************************************************************
//
// IEEE Special Conditions:
//
//    Denormal  fault raised on denormal inputs
//    Overflow exceptions cannot occur  
//    Underflow exceptions raised when appropriate for log1p 
//    (Error Handling Routine called for underflow)
//    Inexact raised when appropriate by algorithm
//
//    log1p(inf) = inf
//    log1p(-inf) = QNaN 
//    log1p(+/-0) = +/-0 
//    log1p(-1) =  -inf 
//    log1p(SNaN) = QNaN
//    log1p(QNaN) = QNaN
//    log1p(EM_special Values) = QNaN
//
// *********************************************************************
//
// Computation is based on the following kernel.
//
// ker_log_64( in_FR    :  X,
// 	    in_FR    :  E,
// 	    in_FR    :  Em1,
// 	    in_GR    :  Expo_Range,
// 	    out_FR   :  Y_hi,
// 	    out_FR   :  Y_lo,
// 	    out_FR   :  Scale,
// 	    out_PR   :  Safe  )
// 
// Overview
//
// The method consists of three cases.
//
// If	|X+Em1| < 2^(-80)	use case log1p_small;
// elseif	|X+Em1| < 2^(-7)	use case log_near1;
// else				use case log_regular;
//
// Case log1p_small:
//
// log( 1 + (X+Em1) ) can be approximated by (X+Em1).
//
// Case log_near1:
//
//   log( 1 + (X+Em1) ) can be approximated by a simple polynomial
//   in W = X+Em1. This polynomial resembles the truncated Taylor
//   series W - W^/2 + W^3/3 - ...
// 
// Case log_regular:
//
//   Here we use a table lookup method. The basic idea is that in
//   order to compute log(Arg) for an argument Arg in [1,2), we 
//   construct a value G such that G*Arg is close to 1 and that
//   log(1/G) is obtainable easily from a table of values calculated
//   beforehand. Thus
//
//	log(Arg) = log(1/G) + log(G*Arg)
//		 = log(1/G) + log(1 + (G*Arg - 1))
//
//   Because |G*Arg - 1| is small, the second term on the right hand
//   side can be approximated by a short polynomial. We elaborate
//   this method in four steps.
//
//   Step 0: Initialization
//
//   We need to calculate log( E + X ). Obtain N, S_hi, S_lo such that
//
//	E + X = 2^N * ( S_hi + S_lo )	exactly
//
//   where S_hi in [1,2) and S_lo is a correction to S_hi in the sense
//   that |S_lo| <= ulp(S_hi).
//
//   Step 1: Argument Reduction
//
//   Based on S_hi, obtain G_1, G_2, G_3 from a table and calculate
//
//	G := G_1 * G_2 * G_3
//	r := (G * S_hi - 1)  + G * S_lo
//
//   These G_j's have the property that the product is exactly 
//   representable and that |r| < 2^(-12) as a result.
//
//   Step 2: Approximation
//
//
//   log(1 + r) is approximated by a short polynomial poly(r).
//
//   Step 3: Reconstruction
//
//
//   Finally, log( E + X ) is given by
//
//   log( E + X )   =   log( 2^N * (S_hi + S_lo) )
//                 ~=~  N*log(2) + log(1/G) + log(1 + r)
//                 ~=~  N*log(2) + log(1/G) + poly(r).
//
// **** Algorithm ****
//
// Case log1p_small:
//
// Although log(1 + (X+Em1)) is basically X+Em1, we would like to 
// preserve the inexactness nature as well as consistent behavior
// under different rounding modes. Note that this case can only be
// taken if E is set to be 1.0. In this case, Em1 is zero, and that
// X can be very tiny and thus the final result can possibly underflow.
// Thus, we compare X against a threshold that is dependent on the
// input Expo_Range. If |X| is smaller than this threshold, we set
// SAFE to be FALSE. 
//
// The result is returned as Y_hi, Y_lo, and in the case of SAFE 
// is FALSE, an additional value Scale is also returned. 
//
//	W    := X + Em1
//      Threshold := Threshold_Table( Expo_Range )
//      Tiny      := Tiny_Table( Expo_Range )
//
//      If ( |W| > Threshold ) then
//         Y_hi  := W
//         Y_lo  := -W*W
//      Else
//         Y_hi  := W
//         Y_lo  := -Tiny
//         Scale := 2^(-100)
//         Safe  := FALSE
//      EndIf
//
//
// One may think that Y_lo should be -W*W/2; however, it does not matter
// as Y_lo will be rounded off completely except for the correct effect in 
// directed rounding. Clearly -W*W is simplier to compute. Moreover,
// because of the difference in exponent value, Y_hi + Y_lo or 
// Y_hi + Scale*Y_lo is always inexact.
//
// Case log_near1:
//
// Here we compute a simple polynomial. To exploit parallelism, we split
// the polynomial into two portions.
// 
// 	W := X + Em1
// 	Wsq := W * W
// 	W4  := Wsq*Wsq
// 	W6  := W4*Wsq
// 	Y_hi := W + Wsq*(P_1 + W*(P_2 + W*(P_3 + W*P_4))
// 	Y_lo := W6*(P_5 + W*(P_6 + W*(P_7 + W*P_8)))
//      set lsb(Y_lo) to be 1
//
// Case log_regular:
//
// We present the algorithm in four steps.
//
//   Step 0. Initialization
//   ----------------------
//
//   Z := X + E
//   N := unbaised exponent of Z
//   S_hi := 2^(-N) * Z
//   S_lo := 2^(-N) * { (max(X,E)-Z) + min(X,E) }
//
//   Note that S_lo is always 0 for the case E = 0.
//
//   Step 1. Argument Reduction
//   --------------------------
//
//   Let
//
//	Z = 2^N * S_hi = 2^N * 1.d_1 d_2 d_3 ... d_63
//
//   We obtain G_1, G_2, G_3 by the following steps.
//
//
//	Define		X_0 := 1.d_1 d_2 ... d_14. This is extracted
//			from S_hi.
//
//	Define		A_1 := 1.d_1 d_2 d_3 d_4. This is X_0 truncated
//			to lsb = 2^(-4).
//
//	Define		index_1 := [ d_1 d_2 d_3 d_4 ].
//
//	Fetch 		Z_1 := (1/A_1) rounded UP in fixed point with
//	fixed point	lsb = 2^(-15).
//			Z_1 looks like z_0.z_1 z_2 ... z_15
//		        Note that the fetching is done using index_1.
//			A_1 is actually not needed in the implementation
//			and is used here only to explain how is the value
//			Z_1 defined.
//
//	Fetch		G_1 := (1/A_1) truncated to 21 sig. bits.
//	floating pt.	Again, fetching is done using index_1. A_1
//			explains how G_1 is defined.
//
//	Calculate	X_1 := X_0 * Z_1 truncated to lsb = 2^(-14)
//			     = 1.0 0 0 0 d_5 ... d_14
//			This is accomplised by integer multiplication.
//			It is proved that X_1 indeed always begin
//			with 1.0000 in fixed point.
//
//
//	Define		A_2 := 1.0 0 0 0 d_5 d_6 d_7 d_8. This is X_1 
//			truncated to lsb = 2^(-8). Similar to A_1,
//			A_2 is not needed in actual implementation. It
//			helps explain how some of the values are defined.
//
//	Define		index_2 := [ d_5 d_6 d_7 d_8 ].
//
//	Fetch 		Z_2 := (1/A_2) rounded UP in fixed point with
//	fixed point	lsb = 2^(-15). Fetch done using index_2.
//			Z_2 looks like z_0.z_1 z_2 ... z_15
//
//	Fetch		G_2 := (1/A_2) truncated to 21 sig. bits.
//	floating pt.
//
//	Calculate	X_2 := X_1 * Z_2 truncated to lsb = 2^(-14)
//			     = 1.0 0 0 0 0 0 0 0 d_9 d_10 ... d_14
//			This is accomplised by integer multiplication.
//			It is proved that X_2 indeed always begin
//			with 1.00000000 in fixed point.
//
//
//	Define		A_3 := 1.0 0 0 0 0 0 0 0 d_9 d_10 d_11 d_12 d_13 1.
//			This is 2^(-14) + X_2 truncated to lsb = 2^(-13).
//
//	Define		index_3 := [ d_9 d_10 d_11 d_12 d_13 ].
//
//	Fetch		G_3 := (1/A_3) truncated to 21 sig. bits.
//	floating pt.	Fetch is done using index_3.
//
//	Compute		G := G_1 * G_2 * G_3. 
//
//	This is done exactly since each of G_j only has 21 sig. bits.
//
//	Compute   
//
//		r := (G*S_hi - 1) + G*S_lo   using 2 FMA operations.
//
//	thus, r approximates G*(S_hi+S_lo) - 1 to within a couple of 
//	rounding errors.
//
//
//  Step 2. Approximation
//  ---------------------
//
//   This step computes an approximation to log( 1 + r ) where r is the
//   reduced argument just obtained. It is proved that |r| <= 1.9*2^(-13);
//   thus log(1+r) can be approximated by a short polynomial:
//
//	log(1+r) ~=~ poly = r + Q1 r^2 + ... + Q4 r^5
//
//
//  Step 3. Reconstruction
//  ----------------------
//
//   This step computes the desired result of log(X+E):
//
//	log(X+E)  =   log( 2^N * (S_hi + S_lo) )
//		  =   N*log(2) + log( S_hi + S_lo )
//		  =   N*log(2) + log(1/G) +
//		      log(1 + C*(S_hi+S_lo) - 1 )
//
//   log(2), log(1/G_j) are stored as pairs of (single,double) numbers:
//   log2_hi, log2_lo, log1byGj_hi, log1byGj_lo. The high parts are
//   single-precision numbers and the low parts are double precision
//   numbers. These have the property that
//
//	N*log2_hi + SUM ( log1byGj_hi )
//
//   is computable exactly in double-extended precision (64 sig. bits).
//   Finally
//
//	Y_hi := N*log2_hi + SUM ( log1byGj_hi )
//	Y_lo := poly_hi + [ poly_lo + 
//	        ( SUM ( log1byGj_lo ) + N*log2_lo ) ]
//      set lsb(Y_lo) to be 1
//

#include "libm_support.h"

#ifdef _LIBC
.rodata
#else
.data
#endif

// P_7, P_6, P_5, P_4, P_3, P_2, and P_1 

.align 64
Constants_P:
ASM_TYPE_DIRECTIVE(Constants_P,@object)
data4  0xEFD62B15,0xE3936754,0x00003FFB,0x00000000
data4  0xA5E56381,0x8003B271,0x0000BFFC,0x00000000
data4  0x73282DB0,0x9249248C,0x00003FFC,0x00000000
data4  0x47305052,0xAAAAAA9F,0x0000BFFC,0x00000000
data4  0xCCD17FC9,0xCCCCCCCC,0x00003FFC,0x00000000
data4  0x00067ED5,0x80000000,0x0000BFFD,0x00000000
data4  0xAAAAAAAA,0xAAAAAAAA,0x00003FFD,0x00000000
data4  0xFFFFFFFE,0xFFFFFFFF,0x0000BFFD,0x00000000
ASM_SIZE_DIRECTIVE(Constants_P)
 
// log2_hi, log2_lo, Q_4, Q_3, Q_2, and Q_1 

.align 64
Constants_Q:
ASM_TYPE_DIRECTIVE(Constants_Q,@object)
data4  0x00000000,0xB1721800,0x00003FFE,0x00000000 
data4  0x4361C4C6,0x82E30865,0x0000BFE2,0x00000000
data4  0x328833CB,0xCCCCCAF2,0x00003FFC,0x00000000
data4  0xA9D4BAFB,0x80000077,0x0000BFFD,0x00000000
data4  0xAAABE3D2,0xAAAAAAAA,0x00003FFD,0x00000000
data4  0xFFFFDAB7,0xFFFFFFFF,0x0000BFFD,0x00000000
ASM_SIZE_DIRECTIVE(Constants_Q)
 
// Z1 - 16 bit fixed, G1 and H1 - IEEE single 
 
.align 64
Constants_Z_G_H_h1:
ASM_TYPE_DIRECTIVE(Constants_Z_G_H_h1,@object)
data4  0x00008000,0x3F800000,0x00000000,0x00000000,0x00000000,0x00000000
data4  0x00007879,0x3F70F0F0,0x3D785196,0x00000000,0x617D741C,0x3DA163A6
data4  0x000071C8,0x3F638E38,0x3DF13843,0x00000000,0xCBD3D5BB,0x3E2C55E6
data4  0x00006BCB,0x3F579430,0x3E2FF9A0,0x00000000,0xD86EA5E7,0xBE3EB0BF
data4  0x00006667,0x3F4CCCC8,0x3E647FD6,0x00000000,0x86B12760,0x3E2E6A8C
data4  0x00006187,0x3F430C30,0x3E8B3AE7,0x00000000,0x5C0739BA,0x3E47574C
data4  0x00005D18,0x3F3A2E88,0x3EA30C68,0x00000000,0x13E8AF2F,0x3E20E30F
data4  0x0000590C,0x3F321640,0x3EB9CEC8,0x00000000,0xF2C630BD,0xBE42885B
data4  0x00005556,0x3F2AAAA8,0x3ECF9927,0x00000000,0x97E577C6,0x3E497F34
data4  0x000051EC,0x3F23D708,0x3EE47FC5,0x00000000,0xA6B0A5AB,0x3E3E6A6E
data4  0x00004EC5,0x3F1D89D8,0x3EF8947D,0x00000000,0xD328D9BE,0xBDF43E3C
data4  0x00004BDB,0x3F17B420,0x3F05F3A1,0x00000000,0x0ADB090A,0x3E4094C3
data4  0x00004925,0x3F124920,0x3F0F4303,0x00000000,0xFC1FE510,0xBE28FBB2
data4  0x0000469F,0x3F0D3DC8,0x3F183EBF,0x00000000,0x10FDE3FA,0x3E3A7895
data4  0x00004445,0x3F088888,0x3F20EC80,0x00000000,0x7CC8C98F,0x3E508CE5
data4  0x00004211,0x3F042108,0x3F29516A,0x00000000,0xA223106C,0xBE534874
ASM_SIZE_DIRECTIVE(Constants_Z_G_H_h1)
 
// Z2 - 16 bit fixed, G2 and H2 - IEEE single 

.align 64 
Constants_Z_G_H_h2:
ASM_TYPE_DIRECTIVE(Constants_Z_G_H_h2,@object)
data4  0x00008000,0x3F800000,0x00000000,0x00000000,0x00000000,0x00000000
data4  0x00007F81,0x3F7F00F8,0x3B7F875D,0x00000000,0x22C42273,0x3DB5A116
data4  0x00007F02,0x3F7E03F8,0x3BFF015B,0x00000000,0x21F86ED3,0x3DE620CF
data4  0x00007E85,0x3F7D08E0,0x3C3EE393,0x00000000,0x484F34ED,0xBDAFA07E
data4  0x00007E08,0x3F7C0FC0,0x3C7E0586,0x00000000,0x3860BCF6,0xBDFE07F0
data4  0x00007D8D,0x3F7B1880,0x3C9E75D2,0x00000000,0xA78093D6,0x3DEA370F
data4  0x00007D12,0x3F7A2328,0x3CBDC97A,0x00000000,0x72A753D0,0x3DFF5791
data4  0x00007C98,0x3F792FB0,0x3CDCFE47,0x00000000,0xA7EF896B,0x3DFEBE6C
data4  0x00007C20,0x3F783E08,0x3CFC15D0,0x00000000,0x409ECB43,0x3E0CF156
data4  0x00007BA8,0x3F774E38,0x3D0D874D,0x00000000,0xFFEF71DF,0xBE0B6F97
data4  0x00007B31,0x3F766038,0x3D1CF49B,0x00000000,0x5D59EEE8,0xBE080483
data4  0x00007ABB,0x3F757400,0x3D2C531D,0x00000000,0xA9192A74,0x3E1F91E9
data4  0x00007A45,0x3F748988,0x3D3BA322,0x00000000,0xBF72A8CD,0xBE139A06
data4  0x000079D1,0x3F73A0D0,0x3D4AE46F,0x00000000,0xF8FBA6CF,0x3E1D9202
data4  0x0000795D,0x3F72B9D0,0x3D5A1756,0x00000000,0xBA796223,0xBE1DCCC4
data4  0x000078EB,0x3F71D488,0x3D693B9D,0x00000000,0xB6B7C239,0xBE049391
ASM_SIZE_DIRECTIVE(Constants_Z_G_H_h2)
 
// G3 and H3 - IEEE single and h3 -IEEE double 

.align 64 
Constants_Z_G_H_h3:
ASM_TYPE_DIRECTIVE(Constants_Z_G_H_h3,@object)
data4  0x3F7FFC00,0x38800100,0x562224CD,0x3D355595
data4  0x3F7FF400,0x39400480,0x06136FF6,0x3D8200A2
data4  0x3F7FEC00,0x39A00640,0xE8DE9AF0,0x3DA4D68D
data4  0x3F7FE400,0x39E00C41,0xB10238DC,0xBD8B4291
data4  0x3F7FDC00,0x3A100A21,0x3B1952CA,0xBD89CCB8
data4  0x3F7FD400,0x3A300F22,0x1DC46826,0xBDB10707
data4  0x3F7FCC08,0x3A4FF51C,0xF43307DB,0x3DB6FCB9
data4  0x3F7FC408,0x3A6FFC1D,0x62DC7872,0xBD9B7C47
data4  0x3F7FBC10,0x3A87F20B,0x3F89154A,0xBDC3725E
data4  0x3F7FB410,0x3A97F68B,0x62B9D392,0xBD93519D
data4  0x3F7FAC18,0x3AA7EB86,0x0F21BD9D,0x3DC18441
data4  0x3F7FA420,0x3AB7E101,0x2245E0A6,0xBDA64B95
data4  0x3F7F9C20,0x3AC7E701,0xAABB34B8,0x3DB4B0EC
data4  0x3F7F9428,0x3AD7DD7B,0x6DC40A7E,0x3D992337
data4  0x3F7F8C30,0x3AE7D474,0x4F2083D3,0x3DC6E17B
data4  0x3F7F8438,0x3AF7CBED,0x811D4394,0x3DAE314B
data4  0x3F7F7C40,0x3B03E1F3,0xB08F2DB1,0xBDD46F21
data4  0x3F7F7448,0x3B0BDE2F,0x6D34522B,0xBDDC30A4
data4  0x3F7F6C50,0x3B13DAAA,0xB1F473DB,0x3DCB0070
data4  0x3F7F6458,0x3B1BD766,0x6AD282FD,0xBDD65DDC
data4  0x3F7F5C68,0x3B23CC5C,0xF153761A,0xBDCDAB83
data4  0x3F7F5470,0x3B2BC997,0x341D0F8F,0xBDDADA40
data4  0x3F7F4C78,0x3B33C711,0xEBC394E8,0x3DCD1BD7
data4  0x3F7F4488,0x3B3BBCC6,0x52E3E695,0xBDC3532B
data4  0x3F7F3C90,0x3B43BAC0,0xE846B3DE,0xBDA3961E
data4  0x3F7F34A0,0x3B4BB0F4,0x785778D4,0xBDDADF06
data4  0x3F7F2CA8,0x3B53AF6D,0xE55CE212,0x3DCC3ED1
data4  0x3F7F24B8,0x3B5BA620,0x9E382C15,0xBDBA3103
data4  0x3F7F1CC8,0x3B639D12,0x5C5AF197,0x3D635A0B
data4  0x3F7F14D8,0x3B6B9444,0x71D34EFC,0xBDDCCB19
data4  0x3F7F0CE0,0x3B7393BC,0x52CD7ADA,0x3DC74502
data4  0x3F7F04F0,0x3B7B8B6D,0x7D7F2A42,0xBDB68F17
ASM_SIZE_DIRECTIVE(Constants_Z_G_H_h3)
 
// 
//  Exponent Thresholds and Tiny Thresholds
//  for 8, 11, 15, and 17 bit exponents
// 
//  Expo_Range             Value
// 
//  0 (8  bits)            2^(-126)
//  1 (11 bits)            2^(-1022)
//  2 (15 bits)            2^(-16382)
//  3 (17 bits)            2^(-16382)
// 
//  Tiny_Table
//  ----------
//  Expo_Range             Value
// 
//  0 (8  bits)            2^(-16382)
//  1 (11 bits)            2^(-16382)
//  2 (15 bits)            2^(-16382)
//  3 (17 bits)            2^(-16382)
// 

.align 64 
Constants_Threshold:
ASM_TYPE_DIRECTIVE(Constants_Threshold,@object)
data4  0x00000000,0x80000000,0x00003F81,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
data4  0x00000000,0x80000000,0x00003C01,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
data4  0x00000000,0x80000000,0x00000001,0x00000000
ASM_SIZE_DIRECTIVE(Constants_Threshold)

.align 64
Constants_1_by_LN10:
ASM_TYPE_DIRECTIVE(Constants_1_by_LN10,@object)
data4  0x37287195,0xDE5BD8A9,0x00003FFD,0x00000000
data4  0xACCF70C8,0xD56EAABE,0x00003FBD,0x00000000
ASM_SIZE_DIRECTIVE(Constants_1_by_LN10)

FR_Input_X = f8 
FR_Neg_One = f9
FR_E       = f33
FR_Em1     = f34
FR_Y_hi    = f34  
// Shared with Em1
FR_Y_lo    = f35
FR_Scale   = f36
FR_X_Prime = f37 
FR_Z       = f38 
FR_S_hi    = f38  
// Shared with Z  
FR_W       = f39
FR_G       = f40
FR_wsq     = f40 
// Shared with G 
FR_H       = f41
FR_w4      = f41
// Shared with H  
FR_h       = f42
FR_w6      = f42  
// Shared with h     
FR_G_tmp   = f43
FR_poly_lo = f43
// Shared with G_tmp 
FR_P8      = f43  
// Shared with G_tmp 
FR_H_tmp   = f44
FR_poly_hi = f44
  // Shared with H_tmp
FR_P7      = f44  
// Shared with H_tmp
FR_h_tmp   = f45 
FR_rsq     = f45  
// Shared with h_tmp
FR_P6      = f45
// Shared with h_tmp
FR_abs_W   = f46
FR_r       = f46