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Cider Isn't Darwin Emulation, Really
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Written by Eric Postpischil, July 2007.*/
.literal8
// Define the points where our algorithm changes.nPoint: .double -.62000000476837158pPoint: .double +.62000000476837158
// Define miscellaneous constants.nOne: .double -1pOne: .double +1HalfPi: .double 1.5707963267948966192313217Pi: .double 3.1415926535897932384626433
// Define a coefficient for center polynomial (used for x in [-.62, +.62]).C2: .double -0.7009458946846486978642201e-1
// Define coefficients for tail polynomial (used for x outside [-.62, +.62]).pT2: .double 0.1475254770791043897245664e-2nT2: .double -0.1475254770791043897245664e-2
// Define coefficients for reciprocal-square-root refinement.S0: .double 4.999999720603234088021257968S1: .double -3.333333463718495862077843807
.const .align 4
/* Define some coefficients for center polynomial (used for x in [-.62, +.62]). These are stored in pairs at aligned addresses for use in SIMD instructions.*/C1: .double +1.5714407739207253912688773, -1.5375796594506042231041545C0: .double +1.2521077245762256716713145, +1.8992998261460658570880250
/* Define some coefficients for positive tail polynomial (used for x in (+.62, 1)).*/T0: .double +14.7303701447222424478789710, +27.0927484663960477135094919T1: .double +2.7185038560132749878882294, -8.6073428141752546318770571
// 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 various symbols.
#define BaseP r0 // Base address for position-independent addressing.
#define p %xmm0 // Must be in %xmm0 for return on x86_64.#define x %xmm1#define x1 %xmm2#define w x1#define pa %xmm3#define e %xmm4#define rss %xmm5
#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 acosf(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 arcosine of 1, return +0 (C F.9.1.1).
For 1 < |x| (including infinity), return NaN (C F.9.1.1).
For a NaN, return the same NaN (C F.9 11 and 13). (If the NaN is a signalling NaN, we return the "same" NaN quieted.)
Otherwise:
If the rounding mode is round-to-nearest, return arccosine(x) faithfully rounded.
Return a value in [0, pi] (C 7.12.4.1 3). Note that this prohibits returning a correctly rounded value for acosf(-1), since pi rounded to a float lies outside that interval. Not implemented: In other rounding modes, return arccosine(x) possibly with slightly worse error, not necessarily honoring the rounding mode (Ali Sazegari narrowing C F.9 10).
Exceptions:
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 4.2.1) or 1 < |x| (including infinity) (C F.9.1.1) but not if the input is a quiet NaN (C F.9 11).
May not raise exceptions otherwise (C F.9 9).
Properties:
Monotonic, proven by exhaustive testing.
Exhaustive testing proved this routine returns faithfully rounded results. Since the rsqrtss instruction is specified to return a value in an interval, tests were performed using each possible result, showing that a valid result will be obtained regardless of which value rsqrtss provides.*/ .align 5#if !defined DevelopmentInstrumentation // This is the regular name used in the deployed implementation. .globl _acosf _acosf:#else // This is the name used for a special test version of the routine. .globl _acosfInstrumented _acosfInstrumented:#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.
#endif
/* We use different algorithms for different parts of the domain. There is a "negative tail" from -1 to nPoint, a center from nPoint to pPoint, and a "positive tail" from pPoint to +1. Here, we compare and branch to the appropriate code.
There are also special cases: NaNs, x < -1, and 1 < x. These are weeded out in PositiveTail or NegativeTail.*/
ucomisd Address(pPoint), x ja PositiveTail
ucomisd Address(nPoint), x jb NegativeTail
/* Here we have nPoint <= x <= pPoint. This is handled with a simple evaluation of a polynomial that approximates arccosine.
The polynomial has been arranged into the form:
pi/2 - x + x * c2 * (x**4 + c01 * x**2 + c00) * x**2 * (x**4 + c11 * x**2 * c10)).
The coefficients c00 and c10 are stored in a pair at C0, and c01 and c11 are stored at C1. c2 is at C2.
The two quartic factors are evaluated in SIMD registers. For brevity, some comments below describe only one element of a register. The other is analagous.*/ movsd x, x1 // Save a copy of x for later. mulsd x, x // Form x**2. movapd Address(C1), p // Get first coefficient pair. unpcklpd x, x // Duplicate in SIMD register. addpd x, p // Form x**2 + c1. mulpd x, p // Form x**4 + c1 * x**2. movlpd Address(C2), x // Put c2 in low element, with x**2 in high. mulsd x1, x // Multiply by x. addpd Address(C0), p // Form x**4 + c1 * x**2 + c0. mulpd x, p // Multiply by c2*x in low and x**2 in high. movhlps p, pa // Get high element. subsd Address(HalfPi), x1 // Form x - pi/2. mulsd pa, p // Multiply two factors. subsd x1, p // Form pi/2 - x + x * c2 * ...
// Return the double-precision number currently in p.ReturnDoubleInp: cvtsd2ss p, p // Convert result to single precision.
// Return the single-precision number currently in p.ReturnSingleInp:
#if defined __i386__ movss p, Argx // Shuttle result through memory. // This uses the argument area for scratch space, which is allowed. flds Argx // Return input on floating-point stack. #else // On x86_64, the return value is now in p, which is %xmm0. #endif
ret
// Handle pPoint < x.PositiveTail:
movsd Address(pOne), w // Get +1 for math and comparison. ucomisd w, x // Compare x to +1. jae InputIsPositiveSpecial
/* Here we have pPoint < x < 1. The algorithm here is inspired by the identity arccosine(x) = 2 * arcsine(sqrt((1-x)/2)). Replacing arcsine with an approximating polynomial would give an odd polynomial in sqrt((1-x)/2), which is the same as sqrt(1-x) multiplied by some polynomial in x. So we have:
arccosine(x) ~= sqrt(1-x) * t(x).
Unfortunately, the square-root instruction (rsqrtss) takes too long, so we use the faster reciprocal-square-root-estimate instruction instead and refine its estimate. Given an estimate e from the rsqrtss instruction, the square root of 1-x is very nearly, e * (1-x) * s(e**2 * (1-x)). Let e2a be e**2 * (1-x), so sqrt(1-x) is nearly e * a * s(e2a). The leading coefficient of s, cl, has been removed (by dividing s by it) and multiplied into the polynomial p above. That leaves:
sqrt(1-x) / cl ~= e * a * s(e2a)/cl.
s(e2a)/cl is e2a**2 + s1 * e2a + s0, where s1 and s0 have been stored at labels S1 and S0, above.
t(x) has been arranged into the form:
t2 * (x**2 + t01 * x + t00) * (x**2 + t11 * x + t10).
The two quadratic factors are evaluated in SIMD registers. For brevity, some comments below describe only one element of a register. The other is analagous.
So, our job here is to evaluate:
a = 1-x.
e = estimate of 1/sqrt(a).
e2a = e*e*a.
acosf(x) ~= e * a * (e2a**2 + s1 * e2a + s0) * t2 * (x**2 + t01 * x + t00) * (x**2 + t11 * x + t10).*/
subsd x, w // Form 1-x. cvtsd2ss w, e // Convert to single precision for rsqrtss. movapd Address(T1), p // Start asinf polynomial. unpcklpd x, x // Duplicate x. addpd x, p // Form x + t1. #if !defined DevelopmentInstrumentation // This is the regular code used in the deployed implementation. rsqrtss e, e // Estimate 1/sqrt(1-x). #else /* This instruction uses an estimate of 1/sqrt(1-x) passed by the caller rather than the rsqrtss instruction. This allows us to test the implementation with all values that rsqrtss might return. */ movss 8(%esp), e // Use caller's estimate of 1/sqrt(1-x). #endif cvtss2sd e, e // Convert to double precision. mulpd x, p // Form x**2 + t1*x. mulsd e, w // Form e * (1-x). mulsd w, e // Form e**2 * (1-x). // "e" in comments refers to the initial estimate from rsqrtss. movhpd Address(pT2), w // Copy coefficient into high element. addpd Address(T0), p // Form x**2 + t1*x + t0. movsd Address(S1), rss // Form s1. addsd e, rss // Form e**2 * (1-x) + s1. mulpd w, p // Form e * (1-x) * p(x), split. mulsd e, rss // Form e2a**2 + s1 * e2a. movhlps p, pa // Separate high element. addsd Address(S0), rss // Form e2a**2 + s1 * e2a + s0. mulsd pa, p // Finish e * (1-x) * p(x). mulsd rss, p // Form e * (1-x) / sqrt(1-x) * p(x).
jmp ReturnDoubleInp
// Handle x < nPoint.NegativeTail:
movsd Address(pOne), w // Get +1 for math. ucomisd Address(nOne), x // Compare x to -1. jbe InputIsNegativeSpecial
/* Here we have -1 < x < nPoint. We use the same algorithm as in PositiveTail but adapted for -x.
For brevity, some comments below describe only one element of a register. The other is analagous.
Our job here is to evaluate:
a = 1+x.
e = estimate of 1/sqrt(a).
e2a = e*e*a.
acosf(x) ~= pi - e * a * (e2a**2 + s1 * e2a + s0) * t2 * (x**2 - t01 * x + t00) * (x**2 - t11 * x + t10).
Note that the signs of terms involving x are negated from those in PositiveTail.
For convenience, the final two factors are negated:
* (-x**2 + t01 * x - t00) * (-x**2 + t11 * x - t10).*/
addsd x, w // Form 1+x. cvtsd2ss w, e // Convert to single precision for rsqrtss. movapd Address(T1), p // Start asinf polynomial. unpcklpd x, x // Duplicate x. subpd x, p // Form -x + t1. #if !defined DevelopmentInstrumentation // This is the regular code used in the deployed implementation. rsqrtss e, e // Estimate 1/sqrt(1+x). #else /* This instruction uses an estimate of 1/sqrt(1+x) passed by the caller rather than the rsqrtss instruction. This allows us to test the implementation with all values that rsqrtss might return. */ movss 8(%esp), e // Use caller's estimate of 1/sqrt(1+x). #endif cvtss2sd e, e // Convert to double precision. mulpd x, p // Form -x**2 + t1*x. mulsd e, w // Form e * (1+x). mulsd w, e // Form e**2 * (1+x). // "e" in comments refers to the initial estimate from rsqrtss. movhpd Address(nT2), w // Copy coefficient into high element. subpd Address(T0), p // Form -x**2 + t1*x - t0. movsd Address(S1), rss // Form s1. addsd e, rss // Form e**2 * (1+x) + s1. mulpd w, p // Form e * (1+x) * p(x), split. mulsd e, rss // Form e2a**2 + s1 * e2a. movhlps p, pa // Separate high element. addsd Address(S0), rss // Form e2a**2 + s1 * e2a + s0. mulsd pa, p // Finish e * (1+x) * p(x). mulsd rss, p // Form e * (1+x) / sqrt(1+x) * p(x). addsd Address(Pi), p // Form pi - 2*asin(sqrt((1+x)/2)).
jmp ReturnDoubleInp
/* Here we handle inputs greater than or equal to one, including infinity, but not including NaNs. The condition code must be set to indicate equal (zero flag is one) iff the input is one.*/InputIsPositiveSpecial: je InputIsPositiveOne jmp SignalInvalid
/* Here we handle inputs less than or equal to -1, including -infinity, and NaNs. The condition code must be set as if by using ucomisd to compare the input in the "source operand" to -1 in the "destination operand".*/InputIsNegativeSpecial: jp InputIsNaN je InputIsNegativeOne
// Continue into SignalInvalid.// jmp SignalInvalid
/* Here we handle inputs outside the domain of the function. We subtract infinity from itself to generate the invalid signal and return a NaN.*/
.literal4Infinity: .long 0x7f800000
.textSignalInvalid: movss Address(Infinity), p subss p, p // Generate invalid signal and NaN value. jmp ReturnSingleInp
/* Here we handle an input of +1 or -1. arccosine(-1) is pi, which increases when rounded to single precision. However, we are required to return results in [0, pi], so we return pi rounded down.*/
.literal8/* Define a value near pi that yields pi rounded down when converted to single precision. This allows us to generate inexact and return the desired value for acosf(-1).*/AlmostPi: .double 3.1415925
.textInputIsPositiveOne: xorps p, p // Return zero. jmp ReturnSingleInp
InputIsNegativeOne: cvtsd2ss Address(AlmostPi), p jmp ReturnSingleInp
InputIsNaN:
#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