Floating-point trigonometric multiply-add coefficient
The FTMAD instruction calculates the series terms for either sin(x) or cos(x), where the argument x has been adjusted to be in the range -π/4 < x ≤ π/4.
To calculate the series terms of sin(x) and cos(x) the initial source operands of FTMAD should be zero in the first source vector and x2 in the second source vector. The FTMAD instruction is then executed eight times to calculate the sum of eight series terms, which gives a result of sufficient precision.
The FTMAD instruction multiplies each element of the first source vector by the absolute value of the corresponding element of the second source vector and performs a fused addition of each product with a value obtained from a table of hard-wired coefficients, and places the results destructively in the first source vector.
The coefficients are different for sin(x) and cos(x), and are selected by a combination of the sign bit in the second source element and an immediate index in the range 0 to 7.
Double-precision coefficient table for sin(x) (s2<63> == '0')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3ff0 0000 0000 0000 | 1.0 | = 1/1! |
| 1 | bfc5 5555 5555 5543 | -0.1666666666666661 | > -1/3! |
| 2 | 3f81 1111 1110 f30c | 0.8333333333320002e-02 | < 1/5! |
| 3 | bf2a 01a0 19b9 2fc6 | -0.1984126982840213e-03 | > -1/7! |
| 4 | 3ec7 1de3 51f3 d22b | 0.2755731329901505e-05 | < 1/9! |
| 5 | be5a e5e2 b60f 7b91 | -0.2505070584637887e-07 | > -1/11! |
| 6 | 3de5 d840 8868 552f | 0.1589413637195215e-09 | < 1/13! |
| 7 | 0000 0000 0000 0000 | 0.0 | > -1/15! |
Double-precision coefficient table for cos(x) (s2<63> == '1')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3ff0 0000 0000 0000 | 1.0 | = 1/0! |
| 1 | bfe0 0000 0000 0000 | -0.5000000000000000 | = -1/2! |
| 2 | 3fa5 5555 5555 5536 | 0.4166666666666645e-01 | < 1/4! |
| 3 | bf56 c16c 16c1 3a0b | -0.1388888888886111e-02 | > -1/6! |
| 4 | 3efa 01a0 19b1 e8d8 | 0.2480158728388683e-04 | < 1/8! |
| 5 | be92 7e4f 7282 f468 | -0.2755731309913950e-06 | > -1/10! |
| 6 | 3e21 ee96 d264 1b13 | 0.2087558253975872e-08 | < 1/12! |
| 7 | bda8 f763 80fb b401 | -0.1135338700720054e-10 | > -1/14! |
Single-precision coefficient table for sin(x) (s2<31> == '0')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3f80 0000 | 1.0 | = 1/1! |
| 1 | be2a aaab | -1.666666716337e-01 | > -1/3! |
| 2 | 3c08 8886 | 8.333330973983e-03 | < 1/5! |
| 3 | b950 08b9 | -1.983967522392e-04 | > -1/7! |
| 4 | 3636 9d6d | 2.721174723774e-06 | < 1/9! |
| 5 | 0000 0000 | 0.0 | > -1/11! |
| 6 | 0000 0000 | 0.0 | < 1/13! |
| 7 | 0000 0000 | 0.0 | > -1/15! |
Single-precision coefficient table for cos(x) (s2<31> == '1')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3f80 0000 | 1.0 | = 1/0! |
| 1 | bf00 0000 | -5.000000000000e-01 | = -1/2! |
| 2 | 3d2a aaa6 | 4.166664928198e-02 | < 1/4! |
| 3 | bab6 0705 | -1.388759003021e-03 | > -1/6! |
| 4 | 37cd 37cc | 2.446388680255e-05 | < 1/8! |
| 5 | 0000 0000 | 0.0 | > -1/10! |
| 6 | 0000 0000 | 0.0 | < 1/12! |
| 7 | 0000 0000 | 0.0 | > -1/14! |
Half-precision coefficient table for sin(x) (s2<15> == '0')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3c00 | 1.0 | = 1/1! |
| 1 | b155 | -1.666666716337e-01 | > -1/3! |
| 2 | 2030 | 8.333330973983e-03 | < 1/5! |
| 3 | 0000 | 0.0 | > -1/7! |
| 4 | 0000 | 0.0 | < 1/9! |
| 5 | 0000 | 0.0 | > -1/11! |
| 6 | 0000 | 0.0 | < 1/13! |
| 7 | 0000 | 0.0 | > -1/15! |
Half-precision coefficient table for cos(x) (s2<15> == '1')
| Index | Hexadecimal | Decimal | Exact Value |
|---|---|---|---|
| 0 | 3c00 | 1.0 | = 1/0! |
| 1 | b800 | -5.000000000000e-01 | = -1/2! |
| 2 | 293a | 4.166664928198e-02 | < 1/4! |
| 3 | 0000 | 0.0 | > -1/6! |
| 4 | 0000 | 0.0 | < 1/8! |
| 5 | 0000 | 0.0 | > -1/10! |
| 6 | 0000 | 0.0 | < 1/12! |
| 7 | 0000 | 0.0 | > -1/14! |
This instruction is illegal when executed in Streaming SVE mode, unless FEAT_SME_FA64 is implemented and enabled.
| 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | size | 0 | 1 | 0 | imm3 | 1 | 0 | 0 | 0 | 0 | 0 | Zm | Zdn | |||||||||||
if !HaveSVE() then UNDEFINED; if size == '00' then UNDEFINED; constant integer esize = 8 << UInt(size); integer dn = UInt(Zdn); integer m = UInt(Zm); integer imm = UInt(imm3);
| <Zdn> |
Is the name of the first source and destination scalable vector register, encoded in the "Zdn" field. |
| <T> |
Is the size specifier,
encoded in
|
| <Zm> |
Is the name of the second source scalable vector register, encoded in the "Zm" field. |
| <imm> |
Is the unsigned immediate operand, in the range 0 to 7, encoded in the "imm3" field. |
CheckNonStreamingSVEEnabled(); constant integer VL = CurrentVL; constant integer PL = VL DIV 8; constant integer elements = VL DIV esize; bits(VL) operand1 = Z[dn, VL]; bits(VL) operand2 = Z[m, VL]; bits(VL) result; for e = 0 to elements-1 bits(esize) element1 = Elem[operand1, e, esize]; bits(esize) element2 = Elem[operand2, e, esize]; Elem[result, e, esize] = FPTrigMAdd(imm, element1, element2, FPCR); Z[dn, VL] = result;
This instruction might be immediately preceded in program order by a MOVPRFX instruction. The MOVPRFX instruction must conform to all of the following requirements, otherwise the behavior of the MOVPRFX and this instruction is unpredictable:
Internal version only: aarchmrs v2023-12_rel, pseudocode v2023-12_rel, sve v2023-12_rel ; Build timestamp: 2023-12-15T16:46
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