openpower/transcendentals.mdwn

   1 # DRAFT Scalar Transcendentals
   2
   3 Summary:
   4
   5 *This proposal extends Power ISA scalar floating point operations to
   6 add IEEE754 transcendental functions (pow, log etc) and trigonometric
   7 functions (sin, cos etc). These functions are also 98% shared with the
   8 Khronos Group OpenCL Extended Instruction Set.*
   9
  10 Authors/Contributors:
  11
  12 * Luke Kenneth Casson Leighton
  13 * Jacob Lifshay
  14 * Dan Petroski
  15 * Mitch Alsup
  16 * Allen Baum
  17 * Andrew Waterman
  18 * Luis Vitorio Cargnini
  19
  20 [[!toc levels=2]]
  21
  22 See:
  23
  24 * <http://bugs.libre-soc.org/show_bug.cgi?id=127>
  25 * <https://bugs.libre-soc.org/show_bug.cgi?id=923> under review
  26 * <https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
  27 * [[power_trans_ops]] for opcode listing.
  28
  29 Extension subsets:
  30
  31 TODO: rename extension subsets -- we're not on RISC-V anymore.
  32
  33 * **Zftrans**: standard transcendentals (best suited to 3D)
  34 * **ZftransExt**: extra functions (useful, not generally needed for 3D,
  35   can be synthesised using Ztrans)
  36 * **Ztrigpi**: trig. xxx-pi sinpi cospi tanpi
  37 * **Ztrignpi**: trig non-xxx-pi sin cos tan
  38 * **Zarctrigpi**: arc-trig. a-xxx-pi: atan2pi asinpi acospi
  39 * **Zarctrignpi**: arc-trig. non-a-xxx-pi: atan2, asin, acos
  40 * **Zfhyp**: hyperbolic/inverse-hyperbolic.  sinh, cosh, tanh, asinh,
  41   acosh, atanh (can be synthesised - see below)
  42 * **ZftransAdv**: much more complex to implement in hardware
  43 * **Zfrsqrt**: Reciprocal square-root.
  44
  45 Minimum recommended requirements for 3D: Zftrans, Ztrignpi,
  46 Zarctrignpi, with Ztrigpi and Zarctrigpi as augmentations.
  47
  48 Minimum recommended requirements for Mobile-Embedded 3D:
  49 Ztrignpi, Zftrans, with Ztrigpi as an augmentation.
  50
  51 The Platform Requirements for 3D are driven by cost competitive
  52 factors and it is the Trademarked Vulkan Specification that provides
  53 clear direction for 3D GPU markets, but nothing else (IEEE754).
  54 Implementors must note that minimum
  55 Compliance with the Third Party Vulkan Specification (for power-area competitive
  56 reasons with other 3D GPU manufacturers) will not qualify for strict IEEE754 accuracy Compliance or vice-versa.
  57
  58 Implementors **must** make it clear which accuracy level is implemented and provide a switching mechanism and throw Illegal Instruction traps if fully compliant accuracy cannot be achieved.
  59 It is also the Implementor's responsibility to comply with all Third Party Certification Marks and Trademarks (Vulkan, OpenCL). Nothing in this specification in any way implies that any Third Party Certification Mark Compliance is granted, nullified, altered or overridden by this document.
  60
  61
  62 # TODO:
  63
  64 * Decision on accuracy, moved to [[zfpacc_proposal]]
  65 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002355.html>
  66 * Errors **MUST** be repeatable.
  67 * How about four Platform Specifications? 3DUNIX, UNIX, 3DEmbedded and Embedded?
  68 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002361.html>
  69   Accuracy requirements for dual (triple) purpose implementations must
  70   meet the higher standard.
  71 * Reciprocal Square-root is in its own separate extension (Zfrsqrt) as
  72   it is desirable on its own by other implementors.  This to be evaluated.
  73
  74 # Requirements <a name="requirements"></a>
  75
  76 This proposal is designed to meet a wide range of extremely diverse
  77 needs, allowing implementors from all of them to benefit from the tools
  78 and hardware cost reductions associated with common standards adoption
  79 in Power ISA (primarily IEEE754 and Vulkan).
  80
  81 **The use-cases are**:
  82
  83 * 3D GPUs
  84 * Numerical Computation
  85 * (Potentially) A.I. / Machine-learning (1)
  86
  87 (1) although approximations suffice in this field, making it more likely
  88 to use a custom extension.  High-end ML would inherently definitely
  89 be excluded.
  90
  91 **The power and die-area requirements vary from**:
  92
  93 * Ultra-low-power (smartwatches where GPU power budgets are in milliwatts)
  94 * Mobile-Embedded (good performance with high efficiency for battery life)
  95 * Desktop Computing
  96 * Server / HPC / Supercomputing
  97
  98 **The software requirements are**:
  99
 100 * Full public integration into GNU math libraries (libm)
 101 * Full public integration into well-known Numerical Computation systems (numpy)
 102 * Full public integration into upstream GNU and LLVM Compiler toolchains
 103 * Full public integration into Khronos OpenCL SPIR-V compatible Compilers
 104   seeking public Certification and Endorsement from the Khronos Group
 105   under their Trademarked Certification Programme.
 106
 107 # Proposed Opcodes vs Khronos OpenCL vs IEEE754-2019<a name="khronos_equiv"></a>
 108
 109 This list shows the (direct) equivalence between proposed opcodes,
 110 their Khronos OpenCL equivalents, and their IEEE754-2019 equivalents.
 111 98% of the opcodes in this proposal that are in the IEEE754-2019 standard
 112 are present in the Khronos Extended Instruction Set.
 113
 114 See
 115 <https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
 116 and <https://ieeexplore.ieee.org/document/8766229>
 117
 118 * Special FP16 opcodes are *not* being proposed, except by indirect / inherent
 119   use of elwidth overrides that is already present in the SVP64 Specification.
 120 * "Native" opcodes are *not* being proposed: implementors will be expected
 121   to use the (equivalent) proposed opcode covering the same function.
 122 * "Fast" opcodes are *not* being proposed, because the Khronos Specification
 123   fast\_length, fast\_normalise and fast\_distance OpenCL opcodes require
 124   vectors (or can be done as scalar operations using other Power ISA
 125   instructions).
 126
 127 The OpenCL FP32 opcodes are **direct** equivalents to the proposed opcodes.
 128 Deviation from conformance with the Khronos Specification - including the
 129 Khronos Specification accuracy requirements - is not an option, as it
 130 results in non-compliance, and the vendor may not use the Trademarked words
 131 "Vulkan" etc. in conjunction with their product.
 132
 133 IEEE754-2019 Table 9.1 lists "additional mathematical operations".
 134 Interestingly the only functions missing when compared to OpenCL are
 135 compound, exp2m1, exp10m1, log2p1, log10p1, pown (integer power) and powr.
 136
 137 |opcode      |OpenCL FP32|OpenCL FP16|OpenCL native|IEEE754               |Power ISA               |
 138 |------------|-----------|-----------|-------------|--------------        |------------------------|
 139 |fsin        |sin        |half\_sin  |native\_sin  |sin                   |NONE                    |
 140 |fcos        |cos        |half\_cos  |native\_cos  |cos                   |NONE                    |
 141 |ftan        |tan        |half\_tan  |native\_tan  |tan                   |NONE                    |
 142 |NONE (1)    |sincos     |NONE       |NONE         |NONE                  |NONE                    |
 143 |fasin       |asin       |NONE       |NONE         |asin                  |NONE                    |
 144 |facos       |acos       |NONE       |NONE         |acos                  |NONE                    |
 145 |fatan       |atan       |NONE       |NONE         |atan                  |NONE                    |
 146 |fsinpi      |sinpi      |NONE       |NONE         |sinPi                 |NONE                    |
 147 |fcospi      |cospi      |NONE       |NONE         |cosPi                 |NONE                    |
 148 |ftanpi      |tanpi      |NONE       |NONE         |tanPi                 |NONE                    |
 149 |fasinpi     |asinpi     |NONE       |NONE         |asinPi                |NONE                    |
 150 |facospi     |acospi     |NONE       |NONE         |acosPi                |NONE                    |
 151 |fatanpi     |atanpi     |NONE       |NONE         |atanPi                |NONE                    |
 152 |fsinh       |sinh       |NONE       |NONE         |sinh                  |NONE                    |
 153 |fcosh       |cosh       |NONE       |NONE         |cosh                  |NONE                    |
 154 |ftanh       |tanh       |NONE       |NONE         |tanh                  |NONE                    |
 155 |fasinh      |asinh      |NONE       |NONE         |asinh                 |NONE                    |
 156 |facosh      |acosh      |NONE       |NONE         |acosh                 |NONE                    |
 157 |fatanh      |atanh      |NONE       |NONE         |atanh                 |NONE                    |
 158 |fatan2      |atan2      |NONE       |NONE         |atan2                 |NONE                    |
 159 |fatan2pi    |atan2pi    |NONE       |NONE         |atan2pi               |NONE                    |
 160 |frsqrt      |rsqrt      |half\_rsqrt|native\_rsqrt|rSqrt                 |fsqrte, fsqrtes (4)     |
 161 |fcbrt       |cbrt       |NONE       |NONE         |NONE (2)              |NONE                    |
 162 |fexp2       |exp2       |half\_exp2 |native\_exp2 |exp2                  |NONE                    |
 163 |flog2       |log2       |half\_log2 |native\_log2 |log2                  |NONE                    |
 164 |fexpm1      |expm1      |NONE       |NONE         |expm1                 |NONE                    |
 165 |flog1p      |log1p      |NONE       |NONE         |logp1                 |NONE                    |
 166 |fexp        |exp        |half\_exp  |native\_exp  |exp                   |NONE                    |
 167 |flog        |log        |half\_log  |native\_log  |log                   |NONE                    |
 168 |fexp10      |exp10      |half\_exp10|native\_exp10|exp10                 |NONE                    |
 169 |flog10      |log10      |half\_log10|native\_log10|log10                 |NONE                    |
 170 |fpow        |pow        |NONE       |NONE         |pow                   |NONE                    |
 171 |fpown       |pown       |NONE       |NONE         |pown                  |NONE                    |
 172 |fpowr       |powr       |half\_powr |native\_powr |powr                  |NONE                    |
 173 |frootn      |rootn      |NONE       |NONE         |rootn                 |NONE                    |
 174 |fhypot      |hypot      |NONE       |NONE         |hypot                 |NONE                    |
 175 |frecip      |NONE       |half\_recip|native\_recip|NONE (3)              |fre, fres (4)           |
 176 |NONE        |NONE       |NONE       |NONE         |compound              |NONE                    |
 177 |fexp2m1     |NONE       |NONE       |NONE         |exp2m1                |NONE                    |
 178 |fexp10m1    |NONE       |NONE       |NONE         |exp10m1               |NONE                    |
 179 |flog2p1     |NONE       |NONE       |NONE         |log2p1                |NONE                    |
 180 |flog10p1    |NONE       |NONE       |NONE         |log10p1               |NONE                    |
 181 |fminnum08   |fmin       |fmin       |NONE         |minNum                |xsmindp (5)             |
 182 |fmaxnum08   |fmax       |fmax       |NONE         |maxNum                |xsmaxdp (5)             |
 183 |fmin19      |fmin       |fmin       |NONE         |minimum               |NONE                    |
 184 |fmax19      |fmax       |fmax       |NONE         |maximum               |NONE                    |
 185 |fminnum19   |fmin       |fmin       |NONE         |minimumNumber         |vminfp (6), xsminjdp (5)|
 186 |fmaxnum19   |fmax       |fmax       |NONE         |maximumNumber         |vmaxfp (6), xsmaxjdp (5)|
 187 |fminc       |fmin       |fmin       |NONE         |NONE                  |xsmincdp (5)            |
 188 |fmaxc       |fmax       |fmax       |NONE         |NONE                  |xsmaxcdp (5)            |
 189 |fminmagnum08|minmag     |minmag     |NONE         |minNumMag             |NONE                    |
 190 |fmaxmagnum08|maxmag     |maxmag     |NONE         |maxNumMag             |NONE                    |
 191 |fminmag19   |minmag     |minmag     |NONE         |minimumMagnitude      |NONE                    |
 192 |fmaxmag19   |maxmag     |maxmag     |NONE         |maximumMagnitude      |NONE                    |
 193 |fminmagnum19|minmag     |minmag     |NONE         |minimumMagnitudeNumber|NONE                    |
 194 |fmaxmagnum19|maxmag     |maxmag     |NONE         |maximumMagnitudeNumber|NONE                    |
 195 |fminmagc    |minmag     |minmag     |NONE         |NONE                  |NONE                    |
 196 |fmaxmagc    |maxmag     |maxmag     |NONE         |NONE                  |NONE                    |
 197 |fmod        |fmod       |fmod       |             |NONE                  |NONE                    |
 198 |fremainder  |remainder  |remainder  |             |remainder             |NONE                    |
 199
 200 Note (1) fsincos is macro-op fused (see below).
 201
 202 Note (2) synthesised in IEEE754-2019 as "rootn(x, 3)"
 203
 204 Note (3) synthesised in IEEE754-2019 using "1.0 / x"
 205
 206 Note (4) these are estimate opcodes that help accelerate
 207 software emulation
 208
 209 Note (5) f64-only (though can be used on f32 stored in f64 format), requires VSX.
 210
 211 Note (6) 4xf32-only, requires VMX.
 212
 213 ## List of 2-arg opcodes
 214
 215 | opcode       | Description                          | pseudocode                                     | Extension   |
 216 | ------       | ----------------                     | ----------------                               | ----------- |
 217 | fatan2       | atan2 arc tangent                    | FRT = atan2(FRB, FRA)                          | Zarctrignpi |
 218 | fatan2pi     | atan2 arc tangent / pi               | FRT = atan2(FRB, FRA) / pi                     | Zarctrigpi  |
 219 | fpow         | x power of y                         | FRT = pow(FRA, FRB)                            | ZftransAdv  |
 220 | fpown        | x power of n (n int)                 | FRT = pow(FRA, RB)                             | ZftransAdv  |
 221 | fpowr        | x power of y (x +ve)                 | FRT = exp(FRA log(FRB))                        | ZftransAdv  |
 222 | frootn       | x power 1/n (n integer)              | FRT = pow(FRA, 1/RB)                           | ZftransAdv  |
 223 | fhypot       | hypotenuse                           | FRT = sqrt(FRA^2 + FRB^2)                      | ZftransAdv  |
 224 | fminnum08    | IEEE 754-2008 minNum                 | FRT = minNum(FRA, FRB)  (1)                    | TBD         |
 225 | fmaxnum08    | IEEE 754-2008 maxNum                 | FRT = maxNum(FRA, FRB)  (1)                    | TBD         |
 226 | fmin19       | IEEE 754-2019 minimum                | FRT = minimum(FRA, FRB)                        | TBD         |
 227 | fmax19       | IEEE 754-2019 maximum                | FRT = maximum(FRA, FRB)                        | TBD         |
 228 | fminnum19    | IEEE 754-2019 minimumNumber          | FRT = minimumNumber(FRA, FRB)                  | TBD         |
 229 | fmaxnum19    | IEEE 754-2019 maximumNumber          | FRT = maximumNumber(FRA, FRB)                  | TBD         |
 230 | fminc        | C ternary-op minimum                 | FRT = FRA \< FRB ? FRA : FRB                   | TBD         |
 231 | fmaxc        | C ternary-op maximum                 | FRT = FRA > FRB ? FRA : FRB                    | TBD         |
 232 | fminmagnum08 | IEEE 754-2008 minNumMag              | FRT = minmaxmag(FRA, FRB, False, fminnum08) (2)| TBD         |
 233 | fmaxmagnum08 | IEEE 754-2008 maxNumMag              | FRT = minmaxmag(FRA, FRB, True, fmaxnum08) (2) | TBD         |
 234 | fminmag19    | IEEE 754-2019 minimumMagnitude       | FRT = minmaxmag(FRA, FRB, False, fmin19) (2)   | TBD         |
 235 | fmaxmag19    | IEEE 754-2019 maximumMagnitude       | FRT = minmaxmag(FRA, FRB, True, fmax19) (2)    | TBD         |
 236 | fminmagnum19 | IEEE 754-2019 minimumMagnitudeNumber | FRT = minmaxmag(FRA, FRB, False, fminnum19) (2)| TBD         |
 237 | fmaxmagnum19 | IEEE 754-2019 maximumMagnitudeNumber | FRT = minmaxmag(FRA, FRB, True, fmaxnum19) (2) | TBD         |
 238 | fminmagc     | C ternary-op minimum magnitude       | FRT = minmaxmag(FRA, FRB, False, fminc) (2)    | TBD         |
 239 | fmaxmagc     | C ternary-op maximum magnitude       | FRT = minmaxmag(FRA, FRB, True, fmaxc) (2)     | TBD         |
 240 | fmod         | modulus                              | FRT = fmod(FRA, FRB)                           | TBD         |
 241 | fremainder   | IEEE 754 remainder                   | FRT = remainder(FRA, FRB)                      | TBD         |
 242
 243 Note (1): for the purposes of minNum/maxNum, -0.0 is defined to be less than +0.0. This is not unspecified in IEEE 754-2008.
 244
 245 Note (2): minmaxmag(x, y, cmp, fallback) is defined as:
 246
 247 ```python
 248 def minmaxmag(x, y, is_max, fallback):
 249     a = abs(x) < abs(y)
 250     b = abs(x) > abs(y)
 251     if is_max:
 252         a, b = b, a  # swap
 253     if a:
 254         return x
 255     if b:
 256         return y
 257     # equal magnitudes, or NaN input(s)
 258     return fallback(x, y)
 259 ```
 260
 261 ## List of 1-arg transcendental opcodes
 262
 263 | opcode   | Description            | pseudocode               | Extension  |
 264 | ------   | ----------------       | ----------------         | ---------- |
 265 | frsqrt   | Reciprocal Square-root | FRT = sqrt(FRA)          | Zfrsqrt    |
 266 | fcbrt    | Cube Root              | FRT = pow(FRA, 1.0 / 3)  | ZftransAdv |
 267 | frecip   | Reciprocal             | FRT = 1.0 / FRA          | Zftrans    |
 268 | fexp2m1  | power-2 minus 1        | FRT = pow(2, FRA) - 1.0  | ZftransExt |
 269 | flog2p1  | log2 plus 1            | FRT = log(2, 1 + FRA)    | ZftransExt |
 270 | fexp2    | power-of-2             | FRT = pow(2, FRA)        | Zftrans    |
 271 | flog2    | log2                   | FRT = log(2. FRA)        | Zftrans    |
 272 | fexpm1   | exponential minus 1    | FRT = pow(e, FRA) - 1.0  | ZftransExt |
 273 | flog1p   | log plus 1             | FRT = log(e, 1 + FRA)    | ZftransExt |
 274 | fexp     | exponential            | FRT = pow(e, FRA)        | ZftransExt |
 275 | flog     | natural log (base e)   | FRT = log(e, FRA)        | ZftransExt |
 276 | fexp10m1 | power-10 minus 1       | FRT = pow(10, FRA) - 1.0 | ZftransExt |
 277 | flog10p1 | log10 plus 1           | FRT = log(10, 1 + FRA)   | ZftransExt |
 278 | fexp10   | power-of-10            | FRT = pow(10, FRA)       | ZftransExt |
 279 | flog10   | log base 10            | FRT = log(10, FRA)       | ZftransExt |
 280
 281 ## List of 1-arg trigonometric opcodes
 282
 283 | opcode   | Description              | pseudocode               | Extension   |
 284 | -------- | ------------------------ | ------------------------ | ----------- |
 285 | fsin     | sin (radians)            | FRT = sin(FRA)           | Ztrignpi    |
 286 | fcos     | cos (radians)            | FRT = cos(FRA)           | Ztrignpi    |
 287 | ftan     | tan (radians)            | FRT = tan(FRA)           | Ztrignpi    |
 288 | fasin    | arcsin (radians)         | FRT = asin(FRA)          | Zarctrignpi |
 289 | facos    | arccos (radians)         | FRT = acos(FRA)          | Zarctrignpi |
 290 | fatan    | arctan (radians)         | FRT = atan(FRA)          | Zarctrignpi |
 291 | fsinpi   | sin times pi             | FRT = sin(pi * FRA)      | Ztrigpi     |
 292 | fcospi   | cos times pi             | FRT = cos(pi * FRA)      | Ztrigpi     |
 293 | ftanpi   | tan times pi             | FRT = tan(pi * FRA)      | Ztrigpi     |
 294 | fasinpi  | arcsin / pi              | FRT = asin(FRA) / pi     | Zarctrigpi  |
 295 | facospi  | arccos / pi              | FRT = acos(FRA) / pi     | Zarctrigpi  |
 296 | fatanpi  | arctan / pi              | FRT = atan(FRA) / pi     | Zarctrigpi  |
 297 | fsinh    | hyperbolic sin (radians) | FRT = sinh(FRA)          | Zfhyp       |
 298 | fcosh    | hyperbolic cos (radians) | FRT = cosh(FRA)          | Zfhyp       |
 299 | ftanh    | hyperbolic tan (radians) | FRT = tanh(FRA)          | Zfhyp       |
 300 | fasinh   | inverse hyperbolic sin   | FRT = asinh(FRA)         | Zfhyp       |
 301 | facosh   | inverse hyperbolic cos   | FRT = acosh(FRA)         | Zfhyp       |
 302 | fatanh   | inverse hyperbolic tan   | FRT = atanh(FRA)         | Zfhyp       |
 303
 304 [[!inline pages="openpower/power_trans_ops" raw=yes ]]
 305
 306 # Subsets
 307
 308 The full set is based on the Khronos OpenCL opcodes. If implemented
 309 entirely it would be too much for both Embedded and also 3D.
 310
 311 The subsets are organised by hardware complexity, need (3D, HPC), however
 312 due to synthesis producing inaccurate results at the range limits,
 313 the less common subsets are still required for IEEE754 HPC.
 314
 315 MALI Midgard, an embedded / mobile 3D GPU, for example only has the
 316 following opcodes:
 317
 318     E8 - fatan_pt2
 319     F0 - frcp (reciprocal)
 320     F2 - frsqrt (inverse square root, 1/sqrt(x))
 321     F3 - fsqrt (square root)
 322     F4 - fexp2 (2^x)
 323     F5 - flog2
 324     F6 - fsin1pi
 325     F7 - fcos1pi
 326     F9 - fatan_pt1
 327
 328 These in FP32 and FP16 only: no FP64 hardware, at all.
 329
 330 Vivante Embedded/Mobile 3D (etnaviv
 331 <https://github.com/laanwj/etna_viv/blob/master/rnndb/isa.xml>)
 332 only has the following:
 333
 334     sin, cos2pi
 335     cos, sin2pi
 336     log2, exp
 337     sqrt and rsqrt
 338     recip.
 339
 340 It also has fast variants of some of these, as a CSR Mode.
 341
 342 AMD's R600 GPU (R600\_Instruction\_Set\_Architecture.pdf) and the
 343 RDNA ISA (RDNA\_Shader\_ISA\_5August2019.pdf, Table 22, Section 6.3) have:
 344
 345     COS2PI (appx)
 346     EXP2
 347     LOG (IEEE754)
 348     RECIP
 349     RSQRT
 350     SQRT
 351     SIN2PI (appx)
 352
 353 AMD RDNA has F16 and F32 variants of all the above, and also has F64
 354 variants of SQRT, RSQRT and RECIP.  It is interesting that even the
 355 modern high-end AMD GPU does not have TAN or ATAN, where MALI Midgard
 356 does.
 357
 358 Also a general point, that customised optimised hardware targetting
 359 FP32 3D with less accuracy simply can neither be used for IEEE754 nor
 360 for FP64 (except as a starting point for hardware or software driven
 361 Newton Raphson or other iterative method).
 362
 363 Also in cost/area sensitive applications even the extra ROM lookup tables
 364 for certain algorithms may be too costly.
 365
 366 These wildly differing and incompatible driving factors lead to the
 367 subset subdivisions, below.
 368
 369 ## Transcendental Subsets
 370
 371 ### Zftrans
 372
 373 LOG2 EXP2 RECIP RSQRT
 374
 375 Zftrans contains the minimum standard transcendentals best suited to
 376 3D. They are also the minimum subset for synthesising log10, exp10,
 377 exp1m, log1p, the hyperbolic trigonometric functions sinh and so on.
 378
 379 They are therefore considered "base" (essential) transcendentals.
 380
 381 ### ZftransExt
 382
 383 LOG, EXP, EXP10, LOG10, LOGP1, EXP1M
 384
 385 These are extra transcendental functions that are useful, not generally
 386 needed for 3D, however for Numerical Computation they may be useful.
 387
 388 Although they can be synthesised using Ztrans (LOG2 multiplied
 389 by a constant), there is both a performance penalty as well as an
 390 accuracy penalty towards the limits, which for IEEE754 compliance is
 391 unacceptable. In particular, LOG(1+FRA) in hardware may give much better
 392 accuracy at the lower end (very small FRA) than LOG(FRA).
 393
 394 Their forced inclusion would be inappropriate as it would penalise
 395 embedded systems with tight power and area budgets.  However if they
 396 were completely excluded the HPC applications would be penalised on
 397 performance and accuracy.
 398
 399 Therefore they are their own subset extension.
 400
 401 ### Zfhyp
 402
 403 SINH, COSH, TANH, ASINH, ACOSH, ATANH
 404
 405 These are the hyperbolic/inverse-hyperbolic functions. Their use in 3D
 406 is limited.
 407
 408 They can all be synthesised using LOG, SQRT and so on, so depend
 409 on Zftrans.  However, once again, at the limits of the range, IEEE754
 410 compliance becomes impossible, and thus a hardware implementation may
 411 be required.
 412
 413 HPC and high-end GPUs are likely markets for these.
 414
 415 ### ZftransAdv
 416
 417 CBRT, POW, POWN, POWR, ROOTN
 418
 419 These are simply much more complex to implement in hardware, and typically
 420 will only be put into HPC applications.
 421
 422 * **Zfrsqrt**: Reciprocal square-root.
 423
 424 ## Trigonometric subsets
 425
 426 ### Ztrigpi vs Ztrignpi
 427
 428 * **Ztrigpi**: SINPI COSPI TANPI
 429 * **Ztrignpi**: SIN COS TAN
 430
 431 Ztrignpi are the basic trigonometric functions through which all others
 432 could be synthesised, and they are typically the base trigonometrics
 433 provided by GPUs for 3D, warranting their own subset.
 434
 435 (programmerjake: actually, all other GPU ISAs mentioned in this document have sinpi/cospi or equivalent, and often not sin/cos, because sinpi/cospi are actually *waay* easier to implement because range reduction is simply a bitwise mask, whereas for sin/cos range reduction is a full division by pi)
 436
 437 In the case of the Ztrigpi subset, these are commonly used in for loops
 438 with a power of two number of subdivisions, and the cost of multiplying
 439 by PI inside each loop (or cumulative addition, resulting in cumulative
 440 errors) is not acceptable.
 441
 442 In for example CORDIC the multiplication by PI may be moved outside of
 443 the hardware algorithm as a loop invariant, with no power or area penalty.
 444
 445 Again, therefore, if SINPI (etc.) were excluded, programmers would be
 446 penalised by being forced to divide by PI in some circumstances. Likewise
 447 if SIN were excluded, programmers would be penaslised by being forced
 448 to *multiply* by PI in some circumstances.
 449
 450 Thus again, a slightly different application of the same general argument
 451 applies to give Ztrignpi and Ztrigpi as subsets.  3D GPUs will almost
 452 certainly provide both.
 453
 454 ### Zarctrigpi and Zarctrignpi
 455
 456 * **Zarctrigpi**: ATAN2PI ASINPI ACOSPI
 457 * **Zarctrignpi**: ATAN2 ACOS ASIN
 458
 459 These are extra trigonometric functions that are useful in some
 460 applications, but even for 3D GPUs, particularly embedded and mobile class
 461 GPUs, they are not so common and so are typically synthesised, there.
 462
 463 Although they can be synthesised using Ztrigpi and Ztrignpi, there is,
 464 once again, both a performance penalty as well as an accuracy penalty
 465 towards the limits, which for IEEE754 compliance is unacceptable, yet
 466 is acceptable for 3D.
 467
 468 Therefore they are their own subset extensions.
 469
 470 # Synthesis, Pseudo-code ops and macro-ops
 471
 472 The pseudo-ops are best left up to the compiler rather than being actual
 473 pseudo-ops, by allocating one scalar FP register for use as a constant
 474 (loop invariant) set to "1.0" at the beginning of a function or other
 475 suitable code block.
 476
 477 * fsincos - fused macro-op between fsin and fcos (issued in that order).
 478 * fsincospi - fused macro-op between fsinpi and fcospi (issued in that order).
 479
 480 fatanpi example pseudo-code:
 481
 482     fmvis ft0, 0x3F800 // upper bits of f32 1.0 (BF16)
 483     fatan2pis FRT, FRA, ft0
 484
 485 Hyperbolic function example (obviates need for Zfhyp except for
 486 high-performance or correctly-rounding):
 487
 488     ASINH( x ) = ln( x + SQRT(x**2+1))
 489
 490 # Evaluation and commentary
 491
 492 Moved to [[discussion]]
 493