openpower/transcendentals.mdwn

   1 # Transcendental operations
   2
   3 To be updated to OpenPOWER.
   4
   5 Summary:
   6
   7 *This proposal extends OpenPOWER scalar floating point operations to
   8 add IEEE754 transcendental functions (pow, log etc) and trigonometric
   9 functions (sin, cos etc). These functions are also 98% shared with the
  10 Khronos Group OpenCL Extended Instruction Set.*
  11
  12 With thanks to:
  13
  14 * Jacob Lifshay
  15 * Dan Petroski
  16 * Mitch Alsup
  17 * Allen Baum
  18 * Andrew Waterman
  19 * Luis Vitorio Cargnini
  20
  21 [[!toc levels=2]]
  22
  23 See:
  24
  25 * <http://bugs.libre-riscv.org/show_bug.cgi?id=127>
  26 * <https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
  27 * Discussion: <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002342.html>
  28 * [[rv_major_opcode_1010011]] for opcode listing.
  29 * [[zfpacc_proposal]] for accuracy settings proposal
  30
  31 Extension subsets:
  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: Ztrignpi, Zftrans, with Ztrigpi as an augmentation.
  49
  50 # TODO:
  51
  52 * Decision on accuracy, moved to [[zfpacc_proposal]]
  53 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002355.html>
  54 * Errors **MUST** be repeatable.
  55 * How about four Platform Specifications? 3DUNIX, UNIX, 3DEmbedded and Embedded?
  56 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002361.html>
  57   Accuracy requirements for dual (triple) purpose implementations must
  58   meet the higher standard.
  59 * Reciprocal Square-root is in its own separate extension (Zfrsqrt) as
  60   it is desirable on its own by other implementors.  This to be evaluated.
  61
  62 # Requirements <a name="requirements"></a>
  63
  64 This proposal is designed to meet a wide range of extremely diverse needs,
  65 allowing implementors from all of them to benefit from the tools and hardware
  66 cost reductions associated with common standards adoption in RISC-V (primarily IEEE754 and Vulkan).
  67
  68 **There are *four* different, disparate platform's needs (two new)**:
  69
  70 * 3D Embedded Platform (new)
  71 * Embedded Platform
  72 * 3D UNIX Platform (new)
  73 * UNIX Platform
  74
  75 **The use-cases are**:
  76
  77 * 3D GPUs
  78 * Numerical Computation
  79 * (Potentially) A.I. / Machine-learning (1)
  80
  81 (1) although approximations suffice in this field, making it more likely
  82 to use a custom extension.  High-end ML would inherently definitely
  83 be excluded.
  84
  85 **The power and die-area requirements vary from**:
  86
  87 * Ultra-low-power (smartwatches where GPU power budgets are in milliwatts)
  88 * Mobile-Embedded (good performance with high efficiency for battery life)
  89 * Desktop Computing
  90 * Server / HPC (2)
  91
  92 (2) Supercomputing is left out of the requirements as it is traditionally
  93 covered by Supercomputer Vectorisation Standards (such as RVV).
  94
  95 **The software requirements are**:
  96
  97 * Full public integration into GNU math libraries (libm)
  98 * Full public integration into well-known Numerical Computation systems (numpy)
  99 * Full public integration into upstream GNU and LLVM Compiler toolchains
 100 * Full public integration into Khronos OpenCL SPIR-V compatible Compilers
 101   seeking public Certification and Endorsement from the Khronos Group
 102   under their Trademarked Certification Programme.
 103
 104 **The "contra"-requirements are**:
 105
 106   Ultra Low Power Embedded platforms (smart watches) are sufficiently
 107   resource constrained that Vectorisation (of any kind) is likely to be
 108   unnecessary and inappropriate.
 109 * The requirements are **not** for the purposes of developing a full custom
 110   proprietary GPU with proprietary firmware driven by *hardware* centric
 111   optimised design decisions as a priority over collaboration.
 112 * A full custom proprietary GPU ASIC Manufacturer *may* benefit from
 113   this proposal however the fact that they typically develop proprietary
 114   software that is not shared with the rest of the community likely to
 115   use this proposal means that they have completely different needs.
 116 * This proposal is for *sharing* of effort in reducing development costs
 117
 118 # Proposed Opcodes vs Khronos OpenCL vs IEEE754-2019<a name="khronos_equiv"></a>
 119
 120 This list shows the (direct) equivalence between proposed opcodes,
 121 their Khronos OpenCL equivalents, and their IEEE754-2019 equivalents.
 122 98% of the opcodes in this proposal that are in the IEEE754-2019 standard
 123 are present in the Khronos Extended Instruction Set.
 124
 125 For RISCV opcode encodings see [[rv_major_opcode_1010011]]
 126 **TODO** replace with OpenPOWER
 127
 128 See
 129 <https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
 130 and <https://ieeexplore.ieee.org/document/8766229>
 131
 132 * Special FP16 opcodes are *not* being proposed, except by indirect / inherent
 133   use of the "fmt" field that is already present in the RISC-V Specification.
 134 * "Native" opcodes are *not* being proposed: implementors will be expected
 135   to use the (equivalent) proposed opcode covering the same function.
 136 * "Fast" opcodes are *not* being proposed, because the Khronos Specification
 137   fast\_length, fast\_normalise and fast\_distance OpenCL opcodes require
 138   vectors (or can be done as scalar operations using other RISC-V instructions).
 139
 140 The OpenCL FP32 opcodes are **direct** equivalents to the proposed opcodes.
 141 Deviation from conformance with the Khronos Specification - including the
 142 Khronos Specification accuracy requirements - is not an option, as it
 143 results in non-compliance, and the vendor may not use the Trademarked words
 144 "Vulkan" etc. in conjunction with their product.
 145
 146 IEEE754-2019 Table 9.1 lists "additional mathematical operations".
 147 Interestingly the only functions missing when compared to OpenCL are
 148 compound, exp2m1, exp10m1, log2p1, log10p1, pown (integer power) and powr.
 149
 150 [[!table data="""
 151 opcode   | OpenCL FP32 | OpenCL FP16 | OpenCL native | OpenCL fast | IEEE754  |
 152 FSIN     | sin         | half\_sin   | native\_sin   | NONE        | sin      |
 153 FCOS     | cos         | half\_cos   | native\_cos   | NONE        | cos      |
 154 FTAN     | tan         | half\_tan   | native\_tan   | NONE        | tan      |
 155 NONE (1) | sincos      | NONE        | NONE          | NONE        | NONE     |
 156 FASIN    | asin        | NONE        | NONE          | NONE        | asin     |
 157 FACOS    | acos        | NONE        | NONE          | NONE        | acos     |
 158 FATAN    | atan        | NONE        | NONE          | NONE        | atan     |
 159 FSINPI   | sinpi       | NONE        | NONE          | NONE        | sinPi    |
 160 FCOSPI   | cospi       | NONE        | NONE          | NONE        | cosPi    |
 161 FTANPI   | tanpi       | NONE        | NONE          | NONE        | tanPi    |
 162 FASINPI  | asinpi      | NONE        | NONE          | NONE        | asinPi   |
 163 FACOSPI  | acospi      | NONE        | NONE          | NONE        | acosPi   |
 164 FATANPI  | atanpi      | NONE        | NONE          | NONE        | atanPi   |
 165 FSINH    | sinh        | NONE        | NONE          | NONE        | sinh     |
 166 FCOSH    | cosh        | NONE        | NONE          | NONE        | cosh     |
 167 FTANH    | tanh        | NONE        | NONE          | NONE        | tanh     |
 168 FASINH   | asinh       | NONE        | NONE          | NONE        | asinh    |
 169 FACOSH   | acosh       | NONE        | NONE          | NONE        | acosh    |
 170 FATANH   | atanh       | NONE        | NONE          | NONE        | atanh    |
 171 FATAN2   | atan2       | NONE        | NONE          | NONE        | atan2    |
 172 FATAN2PI | atan2pi     | NONE        | NONE          | NONE        | atan2pi  |
 173 FRSQRT   | rsqrt       | half\_rsqrt | native\_rsqrt | NONE        | rSqrt    |
 174 FCBRT    | cbrt        | NONE        | NONE          | NONE        | NONE (2) |
 175 FEXP2    | exp2        | half\_exp2  | native\_exp2  | NONE        | exp2     |
 176 FLOG2    | log2        | half\_log2  | native\_log2  | NONE        | log2     |
 177 FEXPM1   | expm1       | NONE        | NONE          | NONE        | expm1    |
 178 FLOG1P   | log1p       | NONE        | NONE          | NONE        | logp1    |
 179 FEXP     | exp         | half\_exp   | native\_exp   | NONE        | exp      |
 180 FLOG     | log         | half\_log   | native\_log   | NONE        | log      |
 181 FEXP10   | exp10       | half\_exp10 | native\_exp10 | NONE        | exp10    |
 182 FLOG10   | log10       | half\_log10 | native\_log10 | NONE        | log10    |
 183 FPOW     | pow         | NONE        | NONE          | NONE        | pow      |
 184 FPOWN    | pown        | NONE        | NONE          | NONE        | pown     |
 185 FPOWR    | powr        | half\_powr  | native\_powr  | NONE        | powr     |
 186 FROOTN   | rootn       | NONE        | NONE          | NONE        | rootn    |
 187 FHYPOT   | hypot       | NONE        | NONE          | NONE        | hypot    |
 188 FRECIP   | NONE        | half\_recip | native\_recip | NONE        | NONE (3) |
 189 NONE     | NONE        | NONE        | NONE          | NONE        | compound |
 190 NONE     | NONE        | NONE        | NONE          | NONE        | exp2m1   |
 191 NONE     | NONE        | NONE        | NONE          | NONE        | exp10m1  |
 192 NONE     | NONE        | NONE        | NONE          | NONE        | log2p1   |
 193 NONE     | NONE        | NONE        | NONE          | NONE        | log10p1  |
 194 """]]
 195
 196 Note (1) FSINCOS is macro-op fused (see below).
 197
 198 Note (2) synthesised in IEEE754-2019 as "pown(x, 3)"
 199
 200 Note (3) synthesised in IEEE754-2019 using "1.0 / x"
 201
 202 ## List of 2-arg opcodes
 203
 204 [[!table  data="""
 205 opcode    | Description            | pseudocode                 | Extension   |
 206 FATAN2    | atan2 arc tangent      | rd = atan2(rs2, rs1)       | Zarctrignpi |
 207 FATAN2PI  | atan2 arc tangent / pi | rd = atan2(rs2, rs1) / pi  | Zarctrigpi  |
 208 FPOW      | x power of y           | rd = pow(rs1, rs2)         | ZftransAdv  |
 209 FPOWN     | x power of n (n int)   | rd = pow(rs1, rs2)         | ZftransAdv  |
 210 FPOWR     | x power of y (x +ve)   | rd = exp(rs1 log(rs2))     | ZftransAdv  |
 211 FROOTN    | x power 1/n (n integer)| rd = pow(rs1, 1/rs2)       | ZftransAdv  |
 212 FHYPOT    | hypotenuse             | rd = sqrt(rs1^2 + rs2^2)   | ZftransAdv  |
 213 """]]
 214
 215 ## List of 1-arg transcendental opcodes
 216
 217 [[!table  data="""
 218 opcode   | Description              | pseudocode              | Extension  |
 219 FRSQRT   | Reciprocal Square-root   | rd = sqrt(rs1)          | Zfrsqrt    |
 220 FCBRT    | Cube Root                | rd = pow(rs1, 1.0 / 3)  | ZftransAdv |
 221 FRECIP   | Reciprocal               | rd = 1.0 / rs1          | Zftrans    |
 222 FEXP2    | power-of-2               | rd = pow(2, rs1)        | Zftrans    |
 223 FLOG2    | log2                     | rd = log(2. rs1)        | Zftrans    |
 224 FEXPM1   | exponential minus 1      | rd = pow(e, rs1) - 1.0  | ZftransExt |
 225 FLOG1P   | log plus 1               | rd = log(e, 1 + rs1)    | ZftransExt |
 226 FEXP     | exponential              | rd = pow(e, rs1)        | ZftransExt |
 227 FLOG     | natural log (base e)     | rd = log(e, rs1)        | ZftransExt |
 228 FEXP10   | power-of-10              | rd = pow(10, rs1)       | ZftransExt |
 229 FLOG10   | log base 10              | rd = log(10, rs1)       | ZftransExt |
 230 """]]
 231
 232 ## List of 1-arg trigonometric opcodes
 233
 234 [[!table  data="""
 235 opcode      | Description              | pseudo-code             | Extension |
 236 FSIN        | sin (radians)            | rd = sin(rs1)           | Ztrignpi    |
 237 FCOS        | cos (radians)            | rd = cos(rs1)           | Ztrignpi    |
 238 FTAN        | tan (radians)            | rd = tan(rs1)           | Ztrignpi    |
 239 FASIN       | arcsin (radians)         | rd = asin(rs1)          | Zarctrignpi |
 240 FACOS       | arccos (radians)         | rd = acos(rs1)          | Zarctrignpi |
 241 FATAN       | arctan (radians)         | rd = atan(rs1)          | Zarctrignpi |
 242 FSINPI      | sin times pi             | rd = sin(pi * rs1)      | Ztrigpi |
 243 FCOSPI      | cos times pi             | rd = cos(pi * rs1)      | Ztrigpi |
 244 FTANPI      | tan times pi             | rd = tan(pi * rs1)      | Ztrigpi |
 245 FASINPI     | arcsin / pi              | rd = asin(rs1) / pi     | Zarctrigpi |
 246 FACOSPI     | arccos / pi              | rd = acos(rs1) / pi     | Zarctrigpi |
 247 FATANPI     | arctan / pi              | rd = atan(rs1) / pi     | Zarctrigpi |
 248 FSINH       | hyperbolic sin (radians) | rd = sinh(rs1)          | Zfhyp |
 249 FCOSH       | hyperbolic cos (radians) | rd = cosh(rs1)          | Zfhyp |
 250 FTANH       | hyperbolic tan (radians) | rd = tanh(rs1)          | Zfhyp |
 251 FASINH      | inverse hyperbolic sin   | rd = asinh(rs1)         | Zfhyp |
 252 FACOSH      | inverse hyperbolic cos   | rd = acosh(rs1)         | Zfhyp |
 253 FATANH      | inverse hyperbolic tan   | rd = atanh(rs1)         | Zfhyp |
 254 """]]
 255
 256 # Subsets
 257
 258 The full set is based on the Khronos OpenCL opcodes. If implemented
 259 entirely it would be too much for both Embedded and also 3D.
 260
 261 The subsets are organised by hardware complexity, need (3D, HPC), however
 262 due to synthesis producing inaccurate results at the range limits,
 263 the less common subsets are still required for IEEE754 HPC.
 264
 265 MALI Midgard, an embedded / mobile 3D GPU, for example only has the
 266 following opcodes:
 267
 268     E8 - fatan_pt2
 269     F0 - frcp (reciprocal)
 270     F2 - frsqrt (inverse square root, 1/sqrt(x))
 271     F3 - fsqrt (square root)
 272     F4 - fexp2 (2^x)
 273     F5 - flog2
 274     F6 - fsin1pi
 275     F7 - fcos1pi
 276     F9 - fatan_pt1
 277
 278 These in FP32 and FP16 only: no FP32 hardware, at all.
 279
 280 Vivante Embedded/Mobile 3D (etnaviv
 281 <https://github.com/laanwj/etna_viv/blob/master/rnndb/isa.xml>)
 282 only has the following:
 283
 284     sin, cos2pi
 285     cos, sin2pi
 286     log2, exp
 287     sqrt and rsqrt
 288     recip.
 289
 290 It also has fast variants of some of these, as a CSR Mode.
 291
 292 AMD's R600 GPU (R600\_Instruction\_Set\_Architecture.pdf) and the
 293 RDNA ISA (RDNA\_Shader\_ISA\_5August2019.pdf, Table 22, Section 6.3) have:
 294
 295     COS2PI (appx)
 296     EXP2
 297     LOG (IEEE754)
 298     RECIP
 299     RSQRT
 300     SQRT
 301     SIN2PI (appx)
 302
 303 AMD RDNA has F16 and F32 variants of all the above, and also has F64
 304 variants of SQRT, RSQRT and RECIP.  It is interesting that even the
 305 modern high-end AMD GPU does not have TAN or ATAN, where MALI Midgard
 306 does.
 307
 308 Also a general point, that customised optimised hardware targetting
 309 FP32 3D with less accuracy simply can neither be used for IEEE754 nor
 310 for FP64 (except as a starting point for hardware or software driven
 311 Newton Raphson or other iterative method).
 312
 313 Also in cost/area sensitive applications even the extra ROM lookup tables
 314 for certain algorithms may be too costly.
 315
 316 These wildly differing and incompatible driving factors lead to the
 317 subset subdivisions, below.
 318
 319 ## Transcendental Subsets
 320
 321 ### Zftrans
 322
 323 LOG2 EXP2 RECIP RSQRT
 324
 325 Zftrans contains the minimum standard transcendentals best suited to
 326 3D. They are also the minimum subset for synthesising log10, exp10,
 327 exp1m, log1p, the hyperbolic trigonometric functions sinh and so on.
 328
 329 They are therefore considered "base" (essential) transcendentals.
 330
 331 ### ZftransExt
 332
 333 LOG, EXP, EXP10, LOG10, LOGP1, EXP1M
 334
 335 These are extra transcendental functions that are useful, not generally
 336 needed for 3D, however for Numerical Computation they may be useful.
 337
 338 Although they can be synthesised using Ztrans (LOG2 multiplied
 339 by a constant), there is both a performance penalty as well as an
 340 accuracy penalty towards the limits, which for IEEE754 compliance is
 341 unacceptable. In particular, LOG(1+rs1) in hardware may give much better
 342 accuracy at the lower end (very small rs1) than LOG(rs1).
 343
 344 Their forced inclusion would be inappropriate as it would penalise
 345 embedded systems with tight power and area budgets.  However if they
 346 were completely excluded the HPC applications would be penalised on
 347 performance and accuracy.
 348
 349 Therefore they are their own subset extension.
 350
 351 ### Zfhyp
 352
 353 SINH, COSH, TANH, ASINH, ACOSH, ATANH
 354
 355 These are the hyperbolic/inverse-hyperbolic functions. Their use in 3D
 356 is limited.
 357
 358 They can all be synthesised using LOG, SQRT and so on, so depend
 359 on Zftrans.  However, once again, at the limits of the range, IEEE754
 360 compliance becomes impossible, and thus a hardware implementation may
 361 be required.
 362
 363 HPC and high-end GPUs are likely markets for these.
 364
 365 ### ZftransAdv
 366
 367 CBRT, POW, POWN, POWR, ROOTN
 368
 369 These are simply much more complex to implement in hardware, and typically
 370 will only be put into HPC applications.
 371
 372 * **Zfrsqrt**: Reciprocal square-root.
 373
 374 ## Trigonometric subsets
 375
 376 ### Ztrigpi vs Ztrignpi
 377
 378 * **Ztrigpi**: SINPI COSPI TANPI
 379 * **Ztrignpi**: SIN COS TAN
 380
 381 Ztrignpi are the basic trigonometric functions through which all others
 382 could be synthesised, and they are typically the base trigonometrics
 383 provided by GPUs for 3D, warranting their own subset.
 384
 385 In the case of the Ztrigpi subset, these are commonly used in for loops
 386 with a power of two number of subdivisions, and the cost of multiplying
 387 by PI inside each loop (or cumulative addition, resulting in cumulative
 388 errors) is not acceptable.
 389
 390 In for example CORDIC the multiplication by PI may be moved outside of
 391 the hardware algorithm as a loop invariant, with no power or area penalty.
 392
 393 Again, therefore, if SINPI (etc.) were excluded, programmers would be
 394 penalised by being forced to divide by PI in some circumstances. Likewise
 395 if SIN were excluded, programmers would be penaslised by being forced
 396 to *multiply* by PI in some circumstances.
 397
 398 Thus again, a slightly different application of the same general argument
 399 applies to give Ztrignpi and Ztrigpi as subsets.  3D GPUs will almost
 400 certainly provide both.
 401
 402 ### Zarctrigpi and Zarctrignpi
 403
 404 * **Zarctrigpi**: ATAN2PI ASINPI ACOSPI
 405 * **Zarctrignpi**: ATAN2 ACOS ASIN
 406
 407 These are extra trigonometric functions that are useful in some
 408 applications, but even for 3D GPUs, particularly embedded and mobile class
 409 GPUs, they are not so common and so are typically synthesised, there.
 410
 411 Although they can be synthesised using Ztrigpi and Ztrignpi, there is,
 412 once again, both a performance penalty as well as an accuracy penalty
 413 towards the limits, which for IEEE754 compliance is unacceptable, yet
 414 is acceptable for 3D.
 415
 416 Therefore they are their own subset extensions.
 417
 418 # Synthesis, Pseudo-code ops and macro-ops
 419
 420 The pseudo-ops are best left up to the compiler rather than being actual
 421 pseudo-ops, by allocating one scalar FP register for use as a constant
 422 (loop invariant) set to "1.0" at the beginning of a function or other
 423 suitable code block.
 424
 425 * FSINCOS - fused macro-op between FSIN and FCOS (issued in that order).
 426 * FSINCOSPI - fused macro-op between FSINPI and FCOSPI (issued in that order).
 427
 428 FATANPI example pseudo-code:
 429
 430     lui t0, 0x3F800 // upper bits of f32 1.0
 431     fmv.x.s ft0, t0
 432     fatan2pi.s rd, rs1, ft0
 433
 434 Hyperbolic function example (obviates need for Zfhyp except for
 435 high-performance or correctly-rounding):
 436
 437     ASINH( x ) = ln( x + SQRT(x**2+1))
 438
 439 # Evaluation and commentary
 440
 441 This section will move later to discussion.
 442
 443 ## Reciprocal
 444
 445 Used to be an alias. Some implementors may wish to implement divide as
 446 y times recip(x).
 447
 448 Others may have shared hardware for recip and divide, others may not.
 449
 450 To avoid penalising one implementor over another, recip stays.
 451
 452 ## To evaluate: should LOG be replaced with LOG1P (and EXP with EXPM1)?
 453
 454 RISC principle says "exclude LOG because it's covered by LOGP1 plus an ADD".
 455 Research needed to ensure that implementors are not compromised by such
 456 a decision
 457 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002358.html>
 458
 459 > > correctly-rounded LOG will return different results than LOGP1 and ADD.
 460 > > Likewise for EXP and EXPM1
 461
 462 > ok, they stay in as real opcodes, then.
 463
 464 ## ATAN / ATAN2 commentary
 465
 466 Discussion starts here:
 467 <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002470.html>
 468
 469 from Mitch Alsup:
 470
 471 would like to point out that the general implementations of ATAN2 do a
 472 bunch of special case checks and then simply call ATAN.
 473
 474     double ATAN2( double y, double x )
 475     {   // IEEE 754-2008 quality ATAN2
 476
 477         // deal with NANs
 478         if( ISNAN( x )             ) return x;
 479         if( ISNAN( y )             ) return y;
 480
 481         // deal with infinities
 482         if( x == +∞    && |y|== +∞  ) return copysign(  π/4, y );
 483         if( x == +∞                 ) return copysign(  0.0, y );
 484         if( x == -∞    && |y|== +∞  ) return copysign( 3π/4, y );
 485         if( x == -∞                 ) return copysign(    π, y );
 486         if(               |y|== +∞  ) return copysign(  π/2, y );
 487
 488         // deal with signed zeros
 489         if( x == 0.0  &&  y != 0.0 ) return copysign(  π/2, y );
 490         if( x >=+0.0  &&  y == 0.0 ) return copysign(  0.0, y );
 491         if( x <=-0.0  &&  y == 0.0 ) return copysign(    π, y );
 492
 493         // calculate ATAN2 textbook style
 494         if( x  > 0.0               ) return     ATAN( |y / x| );
 495         if( x  < 0.0               ) return π - ATAN( |y / x| );
 496     }
 497
 498
 499 Yet the proposed encoding makes ATAN2 the primitive and has ATAN invent
 500 a constant and then call/use ATAN2.
 501
 502 When one considers an implementation of ATAN, one must consider several
 503 ranges of evaluation::
 504
 505      x  [  -∞, -1.0]:: ATAN( x ) = -π/2 + ATAN( 1/x );
 506      x  (-1.0, +1.0]:: ATAN( x ) =      + ATAN(   x );
 507      x  [ 1.0,   +∞]:: ATAN( x ) = +π/2 - ATAN( 1/x );
 508
 509 I should point out that the add/sub of π/2 can not lose significance
 510 since the result of ATAN(1/x) is bounded 0..π/2
 511
 512 The bottom line is that I think you are choosing to make too many of
 513 these into OpCodes, making the hardware function/calculation unit (and
 514 sequencer) more complicated that necessary.
 515
 516 --------------------------------------------------------
 517
 518 We therefore I think have a case for bringing back ATAN and including ATAN2.
 519
 520 The reason is that whilst a microcode-like GPU-centric platform would
 521 do ATAN2 in terms of ATAN, a UNIX-centric platform would do it the other
 522 way round.
 523
 524 (that is the hypothesis, to be evaluated for correctness. feedback requested).
 525
 526 This because we cannot compromise or prioritise one platfrom's
 527 speed/accuracy over another. That is not reasonable or desirable, to
 528 penalise one implementor over another.
 529
 530 Thus, all implementors, to keep interoperability, must both have both
 531 opcodes and may choose, at the architectural and routing level, which
 532 one to implement in terms of the other.
 533
 534 Allowing implementors to choose to add either opcode and let traps sort it
 535 out leaves an uncertainty in the software developer's mind: they cannot
 536 trust the hardware, available from many vendors, to be performant right
 537 across the board.
 538
 539 Standards are a pig.
 540
 541 ---
 542
 543 I might suggest that if there were a way for a calculation to be performed
 544 and the result of that calculation chained to a subsequent calculation
 545 such that the precision of the result-becomes-operand is wider than
 546 what will fit in a register, then you can dramatically reduce the count
 547 of instructions in this category while retaining
 548
 549 acceptable accuracy:
 550
 551      z = x / y
 552
 553 can be calculated as::
 554
 555      z = x * (1/y)
 556
 557 Where 1/y has about 26-to-32 bits of fraction. No, it's not IEEE 754-2008
 558 accurate, but GPUs want speed and
 559
 560 1/y is fully pipelined (F32) while x/y cannot be (at reasonable area). It
 561 is also not "that inaccurate" displaying 0.625-to-0.52 ULP.
 562
 563 Given that one has the ability to carry (and process) more fraction bits,
 564 one can then do high precision multiplies of  π or other transcendental
 565 radixes.
 566
 567 And GPUs have been doing this almost since the dawn of 3D.
 568
 569     // calculate ATAN2 high performance style
 570     // Note: at this point x != y
 571     //
 572     if( x  > 0.0             )
 573     {
 574         if( y < 0.0 && |y| < |x| ) return - π/2 - ATAN( x / y );
 575         if( y < 0.0 && |y| > |x| ) return       + ATAN( y / x );
 576         if( y > 0.0 && |y| < |x| ) return       + ATAN( y / x );
 577         if( y > 0.0 && |y| > |x| ) return + π/2 - ATAN( x / y );
 578     }
 579     if( x  < 0.0             )
 580     {
 581         if( y < 0.0 && |y| < |x| ) return + π/2 + ATAN( x / y );
 582         if( y < 0.0 && |y| > |x| ) return + π   - ATAN( y / x );
 583         if( y > 0.0 && |y| < |x| ) return + π   - ATAN( y / x );
 584         if( y > 0.0 && |y| > |x| ) return +3π/2 + ATAN( x / y );
 585     }
 586
 587 This way the adds and subtracts from the constant are not in a precision
 588 precarious position.