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Cider Isn't Darwin Emulation, Really
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Written by Eric Postpischil, June 2007.*/
#define SignificandBits 24#define ExponentBits 8
.literal8
.align 3
NegativeOne: .double -1
.const
.align 4
/* Coefficients to calculate:
x * (x**4 + c02*x**2 + c00) * (x**4 + c12*x**2 + c10) * (x**4 + c22*x**2 + c20) * ( c32*x**2 + c30)
c02 and c12 are stored as a pair at the location below labeled c02, c00 and c10 are at c00, c22 and c23 at c22, and c20 and c30 at c20.
This polynomial was designed to cover two binades (supporting an exponent fold of one bit) for tanf with its period (pi) partitioned into two intervals. The expanded polynomial was factored to create the form above convenient for calculating with SIMD.
This was minimized for ULP error.*/c02: .double -0.34431297545117363561, -1.53503935362160698123c00: .double +0.75337415977492632831, +0.92783629599735333058c22: .double +0.91301179133375347365, +5.27860261872543679865c20: .double +0.58807104225499649417, +3.82127246397188494330
/* ExponentFold specifies how many bits of the exponent will not be used to look up entries in the table. This must match the ExponentFold used to generate the table.
The behavior for infinite inputs relies on the table contents -- if the table entries looked up for an infinite input are m0 and m1, then m0*infinity + m1*infinity should generate a NaN. This happens of either m0 or m1 is a zero or they have opposite signs. If this were not true, additional arrangements would have to be made for infinite inputs.*/#define ExponentFold 1
// Include the reduction table. See tanfMakeTable.c for more information.#include "tanfTable.s"
// Rename the general registers (just to make it easier to keep track of them).#if defined __i386__ #define r0 %eax #define r1 %ecx #define r2 %edx #define r3 %ebx #define r4 %esp #define r5 %ebp #define r6 %esi #define r7 %edi#elif defined __x86_64__ #define r0 %rax #define r1 %rcx #define r2 %rdx #define r3 %rbx #define r4 %rsp #define r5 %rbp #define r6 %rsi #define r7 %rdi#else #error "Unknown architecture."#endif
.text
// Define symbols for tanf.#define BaseP r0 // Base address for position-independent addressing.#define xInteger r1 // The input x in an integer register.#define ki r1 // x*m0 rounded to an integer.
#define p0 %xmm0 // Must be in %xmm0 for return on x86_64.#define p2 %xmm1#define x %xmm2#define k %xmm3#define m1 %xmm4#define x1 %xmm5#define Numerator %xmm6
#if defined __i386__
// Define location of argument x. #define Argx 4(%esp)
// Define how to address data. BaseP must contain the address of label 0. #define Address(label) label-0b(BaseP)
#elif defined __x86_64__
// Define location of argument x. #define Argx %xmm0
// Define how to address data. #define Address(label) label(%rip)
#endif
/* float tanf(float x).
Notes:
Citations in parentheses below indicate the source of a requirement.
"C" stands for ISO/IEC 9899:TC2.
The Open Group specification (IEEE Std 1003.1, 2004 edition) adds no requirements since it defers to C and requires errno behavior only if we choose to support it by arranging for "math_errhandling & MATH_ERRNO" to be non-zero, which we do not.
Return value:
For +/- zero, return zero with same sign (C F.9 12 and F.9.1.7).
For +/- infinity, return a NaN (C F.9.1.7).
For a NaN, return the same NaN (C F.9 11 and 13).
Otherwise:
If the rounding mode is round-to-nearest, return tangent(x) faithfully rounded. Not currently implemented: In other rounding modes, return tangent(x) possibly with slightly worse error, not necessarily honoring the rounding mode (Ali Sazegari narrowing C F.9 10).
Exceptions:
Raise underflow for a denormal result (C F.9 7 and Draft Standard for Floating-Point Arithmetic P754 Draft 1.2.5 9.5). If the input is the smallest normal, underflow may or may not be raised. This is stricter than the older 754 standard.
May or may not raise inexact, even if the result is exact (C F.9 8).
Raise invalid if the input is a signalling NaN (C 5.2.4.2.2 3, in spite of C F.2.1) or an infinity (C F.9.1.7) but not if the input is a quiet NaN (C F.9 11).
May not raise exceptions otherwise (C F.9 9).
Properties:
Desired to be monotonic. Not yet proven!
Exhaustive testing proved this routine returns faithfully rounded results.*/ .align 5 .globl _tanf_tanf:
// Put x into an integer register. #if defined __i386__ mov Argx, xInteger // Put x into xInteger. #elif defined __x86_64__ movd Argx, xInteger // Put x into xInteger. #endif
cvtss2sd Argx, x // Convert x to double precision.
#if defined __i386__
// Get address of 0 in BaseP. call 0f // Push program counter onto stack. 0: pop BaseP // Get program counter.
#elif defined __x86_64__
// Get address of reduction table in BaseP. lea ReductionTable(%rip), BaseP
#endif
movsd Address(NegativeOne), Numerator
unpcklpd x, x // Duplicate x.
/* Shift:
SignificandBits-1 bits right to remove significand.
ExponentFold bits right to remove unused bits of exponent.
4 bits left to multiply by size of a table entry. */ shr $SignificandBits-1+ExponentFold-4, xInteger
// Clear bits other than those we are using from exponent. and $(1<<ExponentBits-ExponentFold)-1 << 4, xInteger je ExponentIsZero // Branch so we can get -0 right.
ucomisd x, x // Test for Nan. jpe HandleNaN
// Get table entry containing m0 and m1 and calculate x*m0 and x*m1. #if defined __i386__ mulpd ReductionTable-0b(BaseP, xInteger), x #elif defined __x86_64__ mulpd (BaseP, xInteger), x #endif /* Let the exponent of x be E, that is, E is the largest integer such that 2**E <= x. We use e to index the table, where e is E with the low ExponentFold bits set to zero. For example, if ExponentFold is 1, e = E & ~1.
The table entry indexed by e represents a number m that is (u/p % 1)/u*2**a, where u is the ULP of 2**e, p is the period (pi), and 2**a is the number of intervals we divide the period into (we use two, so a is one).
Observe that when 2**e <= x, x/u is an integer, and x/u < 2**(24+ExponentFold).
Then for some integer k, x*m = x*(u/p % 1)/u*2**a = x*(u/p - k)/u*2**a = x/p*2**a - x/u*k*2**a. x/u*k*2**a is an integer, so x*m is congruent to x/p*2**a modulo 1.
So tangent(x*m*p*2**-a) equals tangent(x).
However, we do not have m exactly. The table contains two numbers for 2**e, m0 and m1. m0 has the first 29 bits of m, and m1 has the next 53 bits.
Since x has 23 bits and m0 has 29 bits, their multiplication in double format is exact. From that product, we subtract the nearest integer, which changes nothing modulo 1 but reduces the magnitude. Then we add x*m1, yielding a precise but inexact approximation of x*m0 % 1 + x*m1.
The magnitude of x*m0 % 1 is at most 1/2, and x*m1 may make the sum slightly larger.
Recall that m is (u/p % 1)/u*2**a, so m's magnitude is at most .5/u*2**a. m0 contains 29 bits of this, so m1's magnitude is at most 2**-29*.5/u*2**a. With an exponent fold of 1 bit, x/u is less than 2**25, so |x*m1| <= |x*2**-29*.5/u*2**a| < 2**25**2**-29*.5*2**a = 2**(a-5).
Therefore, |x*m0 % 1 + x*m1| < .5 + 2**(a-5). The polynomial that approximates tangent(p*x*2**-a) has been engineered to be good in this interval. */
movapd Address(c02), p0 // Get c02 and c12. movapd Address(c22), p2 // Get c22 and c32.
cvtsd2si x, ki // Round to integer. // This requires round-to-nearest mode in the MXCSR. cvtsi2sd ki, k // Convert to floating-point. movhlps x, m1 // Get x*m1. subsd k, x // Subtract integer, leaving fraction. addsd m1, x // Get x*m0%1 + x*m1.
/* Below this point, references in the comments to "x" refer to the result of the reduction, not to the original input argument.*/ unpcklpd x, x // Duplicate x for SIMD work. test $1, ki // Is 2**0 bit on? movsd x, x1 // Save a copy of x for later. mulpd x, x // Get x**2.#define x2 x // Use the name x2 to emphasize we have x**2. addpd x2, p0 // Get x**2+c02 and x**2+c12. mulpd x2, p0 // Get x**4+c02*x**2 and x**4+c12*x**2. addpd Address(c00), p0 // Finish first pair of terms. addsd x2, p2 // Get x**2+c22 and c32. mulpd x2, p2 // Get x**4+c22*x**2 and c32*x**2. addpd Address(c20), p2 // Finish second pair of terms. mulpd p2, p0 // Multiply parallel pairs. mulsd x1, p0 // Multiply isolated x term. movhlps p0, p2 // Get high factor. mulsd p2, p0 // Combine parts from SIMD work. jne 2f // Branch if bit was on.
// ki was even, tangent is in p0.
cvtsd2ss p0, p0 // Convert result to single precision. #if defined __i386__ movss p0, Argx // Shuttle result through memory. // This uses the argument area for scratch space, which is allowed. flds Argx // Return result on floating-point stack. #else // On x86_64, the return value is now in p0, which is %xmm0. #endif
ret
// ki was odd, tangent is -1/p0.2: divsd p0, Numerator cvtsd2ss Numerator, p0 // Convert result to single precision.
ReturnSingle: #if defined __i386__ movss p0, Argx // Shuttle result through memory. // This uses the argument area for scratch space, which is allowed. flds Argx // Return result on floating-point stack. #else // On x86_64, the return value is now in p0, which is %xmm0. #endif
ret
// Handle a NaN input.HandleNaN:
#if defined __i386__ flds Argx // Return result on floating-point stack. #else cvtsd2ss x, %xmm0 // Return in %xmm0. /* We cannot just return the input, because we must quiet a signalling NaN. */ #endif
ret
/* The raw exponent field is zero, so the input is a zero or a denormal. We need to handle this separately to get -0 right, and it is faster.*/ExponentIsZero: .literal8Finagle: .double 1.00000001 .text mulsd Address(Finagle), x // Make non-zero results inexact. cvtsd2ss x, p0 // Return x. jmp ReturnSingle