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# Zftrans - transcendental operations

With thanks to:

* Jacob Lifshay
* Dan Petroski
* Mitch Alsup
* Allen Baum
* Andrew Waterman
* Luis Vitorio Cargnini

[[!toc levels=2]]

See:

* <http://bugs.libre-riscv.org/show_bug.cgi?id=127>
* <https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
* Discussion: <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002342.html>
* [[rv_major_opcode_1010011]] for opcode listing.
* [[zfpacc_proposal]] for accuracy settings proposal

Extension subsets:

* **Zftrans**: standard transcendentals (best suited to 3D)
* **ZftransExt**: extra functions (useful, not generally needed for 3D,
  can be synthesised using Ztrans)
* **Ztrigpi**: trig. xxx-pi sinpi cospi tanpi
* **Ztrignpi**: trig non-xxx-pi sin cos tan
* **Zarctrigpi**: arc-trig. a-xxx-pi: atan2pi asinpi acospi
* **Zarctrignpi**: arc-trig. non-a-xxx-pi: atan2, asin, acos
* **Zfhyp**: hyperbolic/inverse-hyperbolic.  sinh, cosh, tanh, asinh,
  acosh, atanh (can be synthesised - see below)
* **ZftransAdv**: much more complex to implement in hardware
* **Zfrsqrt**: Reciprocal square-root.

Minimum recommended requirements for 3D: Zftrans, Ztrignpi,
Zarctrignpi, with Ztrigpi and Zarctrigpi as augmentations.

Minimum recommended requirements for Mobile-Embedded 3D: Ztrignpi, Zftrans, with Ztrigpi as an augmentation.

# TODO:

* Decision on accuracy, moved to [[zfpacc_proposal]]
<http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002355.html>
* Errors **MUST** be repeatable.
* How about four Platform Specifications? 3DUNIX, UNIX, 3DEmbedded and Embedded?
<http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002361.html>
  Accuracy requirements for dual (triple) purpose implementations must
  meet the higher standard.
* Reciprocal Square-root is in its own separate extension (Zfrsqrt) as
  it is desirable on its own by other implementors.  This to be evaluated.

# Requirements <a name="requirements"></a>

This proposal is designed to meet a wide range of extremely diverse needs,
allowing implementors from all of them to benefit from the tools and hardware
cost reductions associated with common standards adoption.

**There are *four* different, disparate platform's needs (two new)**:

* 3D Embedded Platform (new)
* Embedded Platform
* 3D UNIX Platform (new)
* UNIX Platform

**The use-cases are**:

* 3D GPUs
* Numerical Computation
* (Potentially) A.I. / Machine-learning (1)

(1) although approximations suffice in this field, making it more likely
to use a custom extension.  High-end ML would inherently definitely
be excluded.

**The power and die-area requirements vary from**:

* Ultra-low-power (smartwatches where GPU power budgets are in milliwatts)
* Mobile-Embedded (good performance with high efficiency for battery life)
* Desktop Computing
* Server / HPC (2)

(2) Supercomputing is left out of the requirements as it is traditionally
covered by Supercomputer Vectorisation Standards (such as RVV).

**The software requirements are**:

* Full public integration into GNU math libraries (libm)
* Full public integration into well-known Numerical Computation systems (numpy)
* Full public integration into upstream GNU and LLVM Compiler toolchains
* Full public integration into Khronos OpenCL SPIR-V compatible Compilers
  seeking public Certification and Endorsement from the Khronos Group
  under their Trademarked Certification Programme.

**The "contra"-requirements are**:

* NOT for use with RVV (RISC-V Vector Extension). These are *scalar* opcodes.
  Ultra Low Power Embedded platforms (smart watches) are sufficiently
  resource constrained that Vectorisation (of any kind) is likely to be
  unnecessary and inappropriate.
* The requirements are **not** for the purposes of developing a full custom
  proprietary GPU with proprietary firmware driven by *hardware* centric
  optimised design decisions as a priority over collaboration.
* A full custom proprietary GPU ASIC Manufacturer *may* benefit from
  this proposal however the fact that they typically develop proprietary
  software that is not shared with the rest of the community likely to
  use this proposal means that they have completely different needs.
* This proposal is for *sharing* of effort in reducing development costs

# Requirements Analysis <a name="requirements_analysis"></a>

**Platforms**:

3D Embedded will require significantly less accuracy and will need to make
power budget and die area compromises that other platforms (including Embedded)
will not need to make.

3D UNIX Platform has to be performance-price-competitive: subtly-reduced
accuracy in FP32 is acceptable where, conversely, in the UNIX Platform,
IEEE754 compliance is a hard requirement that would compromise power
and efficiency on a 3D UNIX Platform.

Even in the Embedded platform, IEEE754 interoperability is beneficial,
where if it was a hard requirement the 3D Embedded platform would be severely
compromised in its ability to meet the demanding power budgets of that market.

Thus, learning from the lessons of
[SIMD considered harmful](https://www.sigarch.org/simd-instructions-considered-harmful/)
this proposal works in conjunction with the [[zfpacc_proposal]], so as
not to overburden the OP32 ISA space with extra "reduced-accuracy" opcodes.

**Use-cases**:

There really is little else in the way of suitable markets.  3D GPUs
have extremely competitive power-efficiency and power-budget requirements
that are completely at odds with the other market at the other end of
the spectrum: Numerical Computation.

Interoperability in Numerical Computation is absolutely critical: it
implies (correlates directly with) IEEE754 compliance.  However full
IEEE754 compliance automatically and inherently penalises a GPU on
performance and die area, where accuracy is simply just not necessary.

To meet the needs of both markets, the two new platforms have to be created,
and [[zfpacc_proposal]] is a critical dependency.  Runtime selection of
FP accuracy allows an implementation to be "Hybrid" - cover UNIX IEEE754
compliance *and* 3D performance in a single ASIC.

**Power and die-area requirements**:

This is where the conflicts really start to hit home.

A "Numerical High performance only" proposal (suitable for Server / HPC
only) would customise and target the Extension based on a quantitative
analysis of the value of certain opcodes *for HPC only*.  It would
conclude, reasonably and rationally, that it is worthwhile adding opcodes
to RVV as parallel Vector operations, and that further discussion of
the matter is pointless.

A "Proprietary GPU effort" (even one that was intended for publication
of its API through, for example, a public libre-licensed Vulkan SPIR-V
Compiler) would conclude, reasonably and rationally, that, likewise, the
opcodes were best suited to be added to RVV, and, further, that their
requirements conflict with the HPC world, due to the reduced accuracy.
This on the basis that the silicon die area required for IEEE754 is far
greater than that needed for reduced-accuracy, and thus their product
would be completely unacceptable in the market if it had to meet IEEE754,
unnecessarily.

An "Embedded 3D" GPU has radically different performance, power
and die-area requirements (and may even target SoftCores in FPGA).
Sharing of the silicon to cover multi-function uses (CORDIC for example)
is absolutely essential in order to keep cost and power down, and high
performance simply is not.  Multi-cycle FSMs instead of pipelines may
be considered acceptable, and so on.  Subsets of functionality are
also essential.

An "Embedded Numerical" platform has requirements that are separate and
distinct from all of the above!

Mobile Computing needs (tablets, smartphones) again pull in a different
direction: high performance, reasonable accuracy, but efficiency is
critical.  Screen sizes are not at the 4K range: they are within the
800x600 range at the low end (320x240 at the extreme budget end), and
only the high-performance smartphones and tablets provide 1080p (1920x1080).
With lower resolution, accuracy compromises are possible which the Desktop
market (4k and soon to be above) would find unacceptable.

Meeting these disparate markets may be achieved, again, through
[[zfpacc_proposal]], by subdividing into four platforms, yet, in addition
to that, subdividing the extension into subsets that best suit the different
market areas.

**Software requirements**:

A "custom" extension is developed in near-complete isolation from the
rest of the RISC-V Community.  Cost savings to the Corporation are
large, with no direct beneficial feedback to (or impact on) the rest
of the RISC-V ecosystem.

However given that 3D revolves around Standards - DirectX, Vulkan, OpenGL,
OpenCL - users have much more influence than first appears.  Compliance
with these standards is critical as the userbase (Games writers,
scientific applications) expects not to have to rewrite extremely large
and costly codebases to conform with *non-standards-compliant* hardware.

Therefore, compliance with public APIs (Vulkan, OpenCL, OpenGL, DirectX)
is paramount, and compliance with Trademarked Standards is critical.
Any deviation from Trademarked Standards means that an implementation
may not be sold and also make a claim of being, for example, "Vulkan
compatible".

This in turn reinforces and makes a hard requirement a need for public
compliance with such standards, over-and-above what would otherwise be
set by a RISC-V Standards Development Process, including both the
software compliance and the knock-on implications that has for hardware.

**Collaboration**:

The case for collaboration on any Extension is already well-known.
In this particular case, the precedent for inclusion of Transcendentals
in other ISAs, both from Graphics and High-performance Computing, has
these primitives well-established in high-profile software libraries and
compilers in both GPU and HPC Computer Science divisions.  Collaboration
and shared public compliance with those standards brooks no argument.

The combined requirements of collaboration and multi accuracy requirements
mean that *overall this proposal is categorically and wholly unsuited
to relegation of "custom" status*.

# Quantitative Analysis <a name="analysis"></a>

This is extremely challenging.  Normally, an Extension would require full,
comprehensive and detailed analysis of every single instruction, for every
single possible use-case, in every single market.  The amount of silicon
area required would be balanced against the benefits of introducing extra
opcodes, as well as a full market analysis performed to see which divisions
of Computer Science benefit from the introduction of the instruction,
in each and every case.

With 34 instructions, four possible Platforms, and sub-categories of
implementations even within each Platform, over 136 separate and distinct
analyses is not a practical proposition.

A little more intelligence has to be applied to the problem space,
to reduce it down to manageable levels.

Fortunately, the subdivision by Platform, in combination with the
identification of only two primary markets (Numerical Computation and
3D), means that the logical reasoning applies *uniformly* and broadly
across *groups* of instructions rather than individually, making it a primarily
hardware-centric and accuracy-centric decision-making process.

In addition, hardware algorithms such as CORDIC can cover such a wide
range of operations (simply by changing the input parameters) that the
normal argument of compromising and excluding certain opcodes because they
would significantly increase the silicon area is knocked down.

However, CORDIC, whilst space-efficient, and thus well-suited to
Embedded, is an old iterative algorithm not well-suited to High-Performance
Computing or Mid to High-end GPUs, where commercially-competitive
FP32 pipeline lengths are only around 5 stages.

Not only that, but some operations such as LOG1P, which would normally
be excluded from one market (due to there being an alternative macro-op
fused sequence replacing it) are required for other markets due to
the higher accuracy obtainable at the lower range of input values when
compared to LOG(1+P).

(Thus we start to see why "proprietary" markets are excluded from this
proposal, because "proprietary" markets would make *hardware*-driven
optimisation decisions that would be completely inappropriate for a
common standard).

ATAN and ATAN2 is another example area in which one market's needs
conflict directly with another: the only viable solution, without compromising
one market to the detriment of the other, is to provide both opcodes
and let implementors make the call as to which (or both) to optimise,
at the *hardware* level.

Likewise it is well-known that loops involving "0 to 2 times pi", often
done in subdivisions of powers of two, are costly to do because they
involve floating-point multiplication by PI in each and every loop.
3D GPUs solved this by providing SINPI variants which range from 0 to 1
and perform the multiply *inside* the hardware itself.  In the case of
CORDIC, it turns out that the multiply by PI is not even needed (is a
loop invariant magic constant).

However, some markets may not wish to *use* CORDIC, for reasons mentioned
above, and, again, one market would be penalised if SINPI was prioritised
over SIN, or vice-versa.

In essence, then, even when only the two primary markets (3D and
Numerical Computation) have been identified, this still leaves two
(three) diametrically-opposed *accuracy* sub-markets as the prime
conflict drivers:

* Embedded Ultra Low Power
* IEEE754 compliance
* Khronos Vulkan compliance

Thus the best that can be done is to use Quantitative Analysis to work
out which "subsets" - sub-Extensions - to include, provide an additional
"accuracy" extension, be as "inclusive" as possible, and thus allow
implementors to decide what to add to their implementation, and how best
to optimise them.

This approach *only* works due to the uniformity of the function space,
and is **not** an appropriate methodology for use in other Extensions
with huge (non-uniform) market diversity even with similarly large
numbers of potential opcodes.  BitManip is the perfect counter-example.

# Proposed Opcodes vs Khronos OpenCL vs IEEE754-2019<a name="khronos_equiv"></a>

This list shows the (direct) equivalence between proposed opcodes,
their Khronos OpenCL equivalents, and their IEEE754-2019 equivalents.
98% of the opcodes in this proposal that are in the IEEE754-2019 standard
are present in the Khronos Extended Instruction Set.

For RISCV opcode encodings see 
[[rv_major_opcode_1010011]]

See
<https://www.khronos.org/registry/spir-v/specs/unified1/OpenCL.ExtendedInstructionSet.100.html>
and <https://ieeexplore.ieee.org/document/8766229>

* Special FP16 opcodes are *not* being proposed, except by indirect / inherent
  use of the "fmt" field that is already present in the RISC-V Specification.
* "Native" opcodes are *not* being proposed: implementors will be expected
  to use the (equivalent) proposed opcode covering the same function.
* "Fast" opcodes are *not* being proposed, because the Khronos Specification
  fast\_length, fast\_normalise and fast\_distance OpenCL opcodes require
  vectors (or can be done as scalar operations using other RISC-V instructions).

The OpenCL FP32 opcodes are **direct** equivalents to the proposed opcodes.
Deviation from conformance with the Khronos Specification - including the
Khronos Specification accuracy requirements - is not an option, as it
results in non-compliance, and the vendor may not use the Trademarked words
"Vulkan" etc. in conjunction with their product.

IEEE754-2019 Table 9.1 lists "additional mathematical operations".
Interestingly the only functions missing when compared to OpenCL are
compound, exp2m1, exp10m1, log2p1, log10p1, pown (integer power) and powr.

[[!table data="""
opcode   | OpenCL FP32 | OpenCL FP16 | OpenCL native | OpenCL fast | IEEE754  |
FSIN     | sin         | half\_sin   | native\_sin   | NONE        | sin      |
FCOS     | cos         | half\_cos   | native\_cos   | NONE        | cos      |
FTAN     | tan         | half\_tan   | native\_tan   | NONE        | tan      |
NONE (1) | sincos      | NONE        | NONE          | NONE        | NONE     |
FASIN    | asin        | NONE        | NONE          | NONE        | asin     |
FACOS    | acos        | NONE        | NONE          | NONE        | acos     |
FATAN    | atan        | NONE        | NONE          | NONE        | atan     |
FSINPI   | sinpi       | NONE        | NONE          | NONE        | sinPi    |
FCOSPI   | cospi       | NONE        | NONE          | NONE        | cosPi    |
FTANPI   | tanpi       | NONE        | NONE          | NONE        | tanPi    |
FASINPI  | asinpi      | NONE        | NONE          | NONE        | asinPi   |
FACOSPI  | acospi      | NONE        | NONE          | NONE        | acosPi   |
FATANPI  | atanpi      | NONE        | NONE          | NONE        | atanPi   |
FSINH    | sinh        | NONE        | NONE          | NONE        | sinh     |
FCOSH    | cosh        | NONE        | NONE          | NONE        | cosh     |
FTANH    | tanh        | NONE        | NONE          | NONE        | tanh     |
FASINH   | asinh       | NONE        | NONE          | NONE        | asinh    |
FACOSH   | acosh       | NONE        | NONE          | NONE        | acosh    |
FATANH   | atanh       | NONE        | NONE          | NONE        | atanh    |
FATAN2   | atan2       | NONE        | NONE          | NONE        | atan2    |
FATAN2PI | atan2pi     | NONE        | NONE          | NONE        | atan2pi  |
FRSQRT   | rsqrt       | half\_rsqrt | native\_rsqrt | NONE        | rSqrt    |
FCBRT    | cbrt        | NONE        | NONE          | NONE        | NONE (2) |
FEXP2    | exp2        | half\_exp2  | native\_exp2  | NONE        | exp2     |
FLOG2    | log2        | half\_log2  | native\_log2  | NONE        | log2     |
FEXPM1   | expm1       | NONE        | NONE          | NONE        | expm1    |
FLOG1P   | log1p       | NONE        | NONE          | NONE        | logp1    |
FEXP     | exp         | half\_exp   | native\_exp   | NONE        | exp      |
FLOG     | log         | half\_log   | native\_log   | NONE        | log      |
FEXP10   | exp10       | half\_exp10 | native\_exp10 | NONE        | exp10    |
FLOG10   | log10       | half\_log10 | native\_log10 | NONE        | log10    |
FPOW     | pow         | NONE        | NONE          | NONE        | pow      |
FPOWN    | pown        | NONE        | NONE          | NONE        | pown     |
FPOWR    | powr        | NONE        | NONE          | NONE        | powr     |
FROOTN   | rootn       | NONE        | NONE          | NONE        | rootn    |
FHYPOT   | hypot       | NONE        | NONE          | NONE        | hypot    |
FRECIP   | NONE        | half\_recip | native\_recip | NONE        | NONE (3) |
NONE     | NONE        | NONE        | NONE          | NONE        | compound |
NONE     | NONE        | NONE        | NONE          | NONE        | exp2m1   |
NONE     | NONE        | NONE        | NONE          | NONE        | exp10m1  |
NONE     | NONE        | NONE        | NONE          | NONE        | log2p1   |
NONE     | NONE        | NONE        | NONE          | NONE        | log10p1  |
"""]]

Note (1) FSINCOS is macro-op fused (see below).

Note (2) synthesised in IEEE754-2019 as "pown(x, 3)"

Note (3) synthesised in IEEE754-2019 using "1.0 / x"

## List of 2-arg opcodes

[[!table  data="""
opcode    | Description            | pseudocode                 | Extension   |
FATAN2    | atan2 arc tangent      | rd = atan2(rs2, rs1)       | Zarctrignpi |
FATAN2PI  | atan2 arc tangent / pi | rd = atan2(rs2, rs1) / pi  | Zarctrigpi  |
FPOW      | x power of y           | rd = pow(rs1, rs2)         | ZftransAdv  |
FPOWN     | x power of n (n int)   | rd = pow(rs1, rs2)         | ZftransAdv  |
FPOWR     | x power of y (x +ve)   | rd = exp(rs1 log(rs2))     | ZftransAdv  |
FROOTN    | x power 1/n (n integer)| rd = pow(rs1, 1/rs2)       | ZftransAdv  |
FHYPOT    | hypotenuse             | rd = sqrt(rs1^2 + rs2^2)   | ZftransAdv  |
"""]]

## List of 1-arg transcendental opcodes

[[!table  data="""
opcode   | Description              | pseudocode              | Extension  |
FRSQRT   | Reciprocal Square-root   | rd = sqrt(rs1)          | Zfrsqrt    |
FCBRT    | Cube Root                | rd = pow(rs1, 1.0 / 3)  | ZftransAdv |
FRECIP   | Reciprocal               | rd = 1.0 / rs1          | Zftrans    |
FEXP2    | power-of-2               | rd = pow(2, rs1)        | Zftrans    |
FLOG2    | log2                     | rd = log(2. rs1)        | Zftrans    |
FEXPM1   | exponential minus 1      | rd = pow(e, rs1) - 1.0  | ZftransExt |
FLOG1P   | log plus 1               | rd = log(e, 1 + rs1)    | ZftransExt |
FEXP     | exponential              | rd = pow(e, rs1)        | ZftransExt |
FLOG     | natural log (base e)     | rd = log(e, rs1)        | ZftransExt |
FEXP10   | power-of-10              | rd = pow(10, rs1)       | ZftransExt |
FLOG10   | log base 10              | rd = log(10, rs1)       | ZftransExt |
"""]]

## List of 1-arg trigonometric opcodes

[[!table  data="""
opcode      | Description              | pseudo-code             | Extension |
FSIN        | sin (radians)            | rd = sin(rs1)           | Ztrignpi    |
FCOS        | cos (radians)            | rd = cos(rs1)           | Ztrignpi    |
FTAN        | tan (radians)            | rd = tan(rs1)           | Ztrignpi    |
FASIN       | arcsin (radians)         | rd = asin(rs1)          | Zarctrignpi |
FACOS       | arccos (radians)         | rd = acos(rs1)          | Zarctrignpi |
FATAN       | arctan (radians)         | rd = atan(rs1)          | Zarctrignpi |
FSINPI      | sin times pi             | rd = sin(pi * rs1)      | Ztrigpi |
FCOSPI      | cos times pi             | rd = cos(pi * rs1)      | Ztrigpi |
FTANPI      | tan times pi             | rd = tan(pi * rs1)      | Ztrigpi |
FASINPI     | arcsin / pi              | rd = asin(rs1) / pi     | Zarctrigpi |
FACOSPI     | arccos / pi              | rd = acos(rs1) / pi     | Zarctrigpi |
FATANPI     | arctan / pi              | rd = atan(rs1) / pi     | Zarctrigpi |
FSINH       | hyperbolic sin (radians) | rd = sinh(rs1)          | Zfhyp |
FCOSH       | hyperbolic cos (radians) | rd = cosh(rs1)          | Zfhyp |
FTANH       | hyperbolic tan (radians) | rd = tanh(rs1)          | Zfhyp |
FASINH      | inverse hyperbolic sin   | rd = asinh(rs1)         | Zfhyp |
FACOSH      | inverse hyperbolic cos   | rd = acosh(rs1)         | Zfhyp |
FATANH      | inverse hyperbolic tan   | rd = atanh(rs1)         | Zfhyp |
"""]]

# Subsets

The full set is based on the Khronos OpenCL opcodes. If implemented
entirely it would be too much for both Embedded and also 3D.

The subsets are organised by hardware complexity, need (3D, HPC), however
due to synthesis producing inaccurate results at the range limits,
the less common subsets are still required for IEEE754 HPC.

MALI Midgard, an embedded / mobile 3D GPU, for example only has the
following opcodes:

    E8 - fatan_pt2
    F0 - frcp (reciprocal)
    F2 - frsqrt (inverse square root, 1/sqrt(x))
    F3 - fsqrt (square root)
    F4 - fexp2 (2^x)
    F5 - flog2
    F6 - fsin1pi
    F7 - fcos1pi
    F9 - fatan_pt1

These in FP32 and FP16 only: no FP32 hardware, at all.

Vivante Embedded/Mobile 3D (etnaviv <https://github.com/laanwj/etna_viv/blob/master/rnndb/isa.xml>) only has the following:

    sin, cos2pi
    cos, sin2pi
    log2, exp
    sqrt and rsqrt
    recip.

It also has fast variants of some of these, as a CSR Mode.

AMD's R600 GPU (R600\_Instruction\_Set\_Architecture.pdf) and the
RDNA ISA (RDNA\_Shader\_ISA\_5August2019.pdf, Table 22, Section 6.3) have:

    COS2PI (appx)
    EXP2
    LOG (IEEE754)
    RECIP
    RSQRT
    SQRT
    SIN2PI (appx)

AMD RDNA has F16 and F32 variants of all the above, and also has F64
variants of SQRT, RSQRT and RECIP.  It is interesting that even the
modern high-end AMD GPU does not have TAN or ATAN, where MALI Midgard
does.

Also a general point, that customised optimised hardware targetting
FP32 3D with less accuracy simply can neither be used for IEEE754 nor
for FP64 (except as a starting point for hardware or software driven
Newton Raphson or other iterative method).

Also in cost/area sensitive applications even the extra ROM lookup tables
for certain algorithms may be too costly.

These wildly differing and incompatible driving factors lead to the
subset subdivisions, below.

## Transcendental Subsets

### Zftrans

LOG2 EXP2 RECIP RSQRT

Zftrans contains the minimum standard transcendentals best suited to
3D. They are also the minimum subset for synthesising log10, exp10,
exp1m, log1p, the hyperbolic trigonometric functions sinh and so on.

They are therefore considered "base" (essential) transcendentals.

### ZftransExt

LOG, EXP, EXP10, LOG10, LOGP1, EXP1M

These are extra transcendental functions that are useful, not generally
needed for 3D, however for Numerical Computation they may be useful.

Although they can be synthesised using Ztrans (LOG2 multiplied
by a constant), there is both a performance penalty as well as an
accuracy penalty towards the limits, which for IEEE754 compliance is
unacceptable. In particular, LOG(1+rs1) in hardware may give much better
accuracy at the lower end (very small rs1) than LOG(rs1).

Their forced inclusion would be inappropriate as it would penalise
embedded systems with tight power and area budgets.  However if they
were completely excluded the HPC applications would be penalised on
performance and accuracy.

Therefore they are their own subset extension.

### Zfhyp

SINH, COSH, TANH, ASINH, ACOSH, ATANH

These are the hyperbolic/inverse-hyperbolic functions. Their use in 3D is limited.

They can all be synthesised using LOG, SQRT and so on, so depend
on Zftrans.  However, once again, at the limits of the range, IEEE754
compliance becomes impossible, and thus a hardware implementation may
be required.

HPC and high-end GPUs are likely markets for these.

### ZftransAdv

CBRT, POW, POWN, POWR, ROOTN

These are simply much more complex to implement in hardware, and typically
will only be put into HPC applications.

* **Zfrsqrt**: Reciprocal square-root.

## Trigonometric subsets

### Ztrigpi vs Ztrignpi

* **Ztrigpi**: SINPI COSPI TANPI
* **Ztrignpi**: SIN COS TAN

Ztrignpi are the basic trigonometric functions through which all others
could be synthesised, and they are typically the base trigonometrics
provided by GPUs for 3D, warranting their own subset.

In the case of the Ztrigpi subset, these are commonly used in for loops
with a power of two number of subdivisions, and the cost of multiplying
by PI inside each loop (or cumulative addition, resulting in cumulative
errors) is not acceptable.

In for example CORDIC the multiplication by PI may be moved outside of
the hardware algorithm as a loop invariant, with no power or area penalty.

Again, therefore, if SINPI (etc.) were excluded, programmers would be penalised by being forced to divide by PI in some circumstances. Likewise if SIN were excluded, programmers would be penaslised by being forced to *multiply* by PI in some circumstances.

Thus again, a slightly different application of the same general argument applies to give Ztrignpi and
Ztrigpi as subsets.  3D GPUs will almost certainly provide both.

### Zarctrigpi and Zarctrignpi

* **Zarctrigpi**: ATAN2PI ASINPI ACOSPI
* **Zarctrignpi**: ATAN2 ACOS ASIN

These are extra trigonometric functions that are useful in some
applications, but even for 3D GPUs, particularly embedded and mobile class
GPUs, they are not so common and so are typically synthesised, there.

Although they can be synthesised using Ztrigpi and Ztrignpi, there is,
once again, both a performance penalty as well as an accuracy penalty
towards the limits, which for IEEE754 compliance is unacceptable, yet
is acceptable for 3D.

Therefore they are their own subset extensions.

# Synthesis, Pseudo-code ops and macro-ops

The pseudo-ops are best left up to the compiler rather than being actual
pseudo-ops, by allocating one scalar FP register for use as a constant
(loop invariant) set to "1.0" at the beginning of a function or other
suitable code block.

* FSINCOS - fused macro-op between FSIN and FCOS (issued in that order).
* FSINCOSPI - fused macro-op between FSINPI and FCOSPI (issued in that order).

FATANPI example pseudo-code:

    lui t0, 0x3F800 // upper bits of f32 1.0
    fmv.x.s ft0, t0
    fatan2pi.s rd, rs1, ft0

Hyperbolic function example (obviates need for Zfhyp except for
high-performance or correctly-rounding):

    ASINH( x ) = ln( x + SQRT(x**2+1))

# Evaluation and commentary

This section will move later to discussion.

## Reciprocal

Used to be an alias. Some implementors may wish to implement divide as
y times recip(x).

Others may have shared hardware for recip and divide, others may not.

To avoid penalising one implementor over another, recip stays.

## To evaluate: should LOG be replaced with LOG1P (and EXP with EXPM1)?

RISC principle says "exclude LOG because it's covered by LOGP1 plus an ADD".
Research needed to ensure that implementors are not compromised by such
a decision
<http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002358.html>

> > correctly-rounded LOG will return different results than LOGP1 and ADD.
> > Likewise for EXP and EXPM1

> ok, they stay in as real opcodes, then.

## ATAN / ATAN2 commentary

Discussion starts here:
<http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-August/002470.html>

from Mitch Alsup:

would like to point out that the general implementations of ATAN2 do a
bunch of special case checks and then simply call ATAN.

    double ATAN2( double y, double x )
    {   // IEEE 754-2008 quality ATAN2

        // deal with NANs
        if( ISNAN( x )             ) return x;
        if( ISNAN( y )             ) return y;

        // deal with infinities
        if( x == +∞    && |y|== +∞  ) return copysign(  π/4, y );
        if( x == +∞                 ) return copysign(  0.0, y );
        if( x == -∞    && |y|== +∞  ) return copysign( 3π/4, y );
        if( x == -∞                 ) return copysign(    π, y );
        if(               |y|== +∞  ) return copysign(  π/2, y );

        // deal with signed zeros
        if( x == 0.0  &&  y != 0.0 ) return copysign(  π/2, y );
        if( x >=+0.0  &&  y == 0.0 ) return copysign(  0.0, y );
        if( x <=-0.0  &&  y == 0.0 ) return copysign(    π, y );

        // calculate ATAN2 textbook style
        if( x  > 0.0               ) return     ATAN( |y / x| );
        if( x  < 0.0               ) return π - ATAN( |y / x| );
    }


Yet the proposed encoding makes ATAN2 the primitive and has ATAN invent
a constant and then call/use ATAN2.

When one considers an implementation of ATAN, one must consider several
ranges of evaluation::

     x  [  -∞, -1.0]:: ATAN( x ) = -π/2 + ATAN( 1/x );
     x  (-1.0, +1.0]:: ATAN( x ) =      + ATAN(   x );
     x  [ 1.0,   +∞]:: ATAN( x ) = +π/2 - ATAN( 1/x );

I should point out that the add/sub of π/2 can not lose significance
since the result of ATAN(1/x) is bounded 0..π/2

The bottom line is that I think you are choosing to make too many of
these into OpCodes, making the hardware function/calculation unit (and
sequencer) more complicated that necessary.

--------------------------------------------------------

We therefore I think have a case for bringing back ATAN and including ATAN2.

The reason is that whilst a microcode-like GPU-centric platform would do ATAN2 in terms of ATAN, a UNIX-centric platform would do it the other way round.

(that is the hypothesis, to be evaluated for correctness. feedback requested).

This because we cannot compromise or prioritise one platfrom's
speed/accuracy over another. That is not reasonable or desirable, to
penalise one implementor over another.

Thus, all implementors, to keep interoperability, must both have both
opcodes and may choose, at the architectural and routing level, which
one to implement in terms of the other.

Allowing implementors to choose to add either opcode and let traps sort it
out leaves an uncertainty in the software developer's mind: they cannot
trust the hardware, available from many vendors, to be performant right
across the board.

Standards are a pig.

---

I might suggest that if there were a way for a calculation to be performed
and the result of that calculation chained to a subsequent calculation
such that the precision of the result-becomes-operand is wider than
what will fit in a register, then you can dramatically reduce the count
of instructions in this category while retaining

acceptable accuracy:

     z = x / y

can be calculated as::

     z = x * (1/y)

Where 1/y has about 26-to-32 bits of fraction. No, it's not IEEE 754-2008
accurate, but GPUs want speed and

1/y is fully pipelined (F32) while x/y cannot be (at reasonable area). It
is also not "that inaccurate" displaying 0.625-to-0.52 ULP.

Given that one has the ability to carry (and process) more fraction bits,
one can then do high precision multiplies of  π or other transcendental
radixes.

And GPUs have been doing this almost since the dawn of 3D.

    // calculate ATAN2 high performance style
    // Note: at this point x != y
    //
    if( x  > 0.0             )
    {
        if( y < 0.0 && |y| < |x| ) return - π/2 - ATAN( x / y );
        if( y < 0.0 && |y| > |x| ) return       + ATAN( y / x );
        if( y > 0.0 && |y| < |x| ) return       + ATAN( y / x );
        if( y > 0.0 && |y| > |x| ) return + π/2 - ATAN( x / y );
    }
    if( x  < 0.0             )
    {
        if( y < 0.0 && |y| < |x| ) return + π/2 + ATAN( x / y );
        if( y < 0.0 && |y| > |x| ) return + π   - ATAN( y / x );
        if( y > 0.0 && |y| < |x| ) return + π   - ATAN( y / x );
        if( y > 0.0 && |y| > |x| ) return +3π/2 + ATAN( x / y );
    }

This way the adds and subtracts from the constant are not in a precision
precarious position.