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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.nPoint0: .double -1pPoint0: .double +1nPoint1: .double -16777216pPoint1: .double +16777216
// Define miscellaneous constants.nHalfPi: .double -1.5707963267948966192313217One: .double 1pHalfPi: .double +1.5707963267948966192313217
// Define a coefficient for center polynomial (used for x in [-1, +1]).C2: .double 0.0029352921857004596570518
.const .align 4
/* Define some coefficients for center polynomial (used for x in [-1, +1]). These are stored in pairs at aligned addresses for use in SIMD instructions.*/C01: .double 2.2971562298314633761676433, 0.0207432003007420961489920C00: .double 2.4449692297316409651126041, 3.7888879287802702842997915C11: .double -2.9466967515109826289085300, -4.9721072376211623038916292C10: .double 5.4728447324456990092824269, 6.7197076223592378022736307
// Define coefficients for the tail polynomial (used for x outside [-1, +1]).T01: .double 0.9193672354545696501477531, -0.0052035944094405570566862T00: .double 0.3992772987220534996563340, 0.2521268658714555740707959T11: .double -0.5273186542866779494437308, -0.7201738803584184183894208T10: .double 0.1730466268612773143731748, 0.1408679409162453515360961
// 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 p0 %xmm0 // Must be in %xmm0 for return on x86_64.#define x %xmm1#define p1 %xmm2#define x1 %xmm3#define v0 %xmm4#define t0 %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 atanf(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 arctangent of +/- zero, return zero with same sign (C F.9 12 and F.9.1.3).
For arctangent of +/- infinity, return +/- pi/2 (C F.9.1.3).
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 arctangent(x) faithfully rounded.
Return a value in [-pi/2, +pi/2] (C 7.12.4.3 3). Note that this prohibits returning correctly rounded values for atanf of large positive or negative arguments, since pi/2 rounded to a float lies outside that interval. Not implemented: In other rounding modes, return arctangent(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 4.2.1) but not if the input is a quiet NaN (C F.9 11).
May not raise exceptions otherwise (C F.9 9).
Properties:
Monotonic.
Exhaustive testing proved this routine returns faithfully rounded results.*/ .align 5#if !defined DevelopmentInstrumentation // This is the regular name used in the deployed implementation. .globl _atanf _atanf:#else // This is the name used for a special test version of the routine. .globl _atanfInstrumented _atanfInstrumented:#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 -infinity to nPoint0, a center from nPoint0 to pPoint0, and a "positive tail" from pPoint0 to +infinity. Here, we compare and branch to the appropriate code.
NaNs are weeded out in PositiveTail or NegativeTail.*/
ucomisd Address(pPoint0), x ja PositiveTail
ucomisd Address(nPoint0), x jb NegativeTail
/* Here we have nPoint0 <= x <= pPoint0. This is handled with a simple evaluation of a polynomial that approximates arctangent.
The polynomial has been arranged into the form:
x * c2 * ((x**4 + c01 * x**2 + c00)) * ((x**4 + c11 * x**2 + c10)) * ((x**4 + c21 * x**2 + c20)) * ((x**4 + c31 * x**2 + c30))
The coefficients are stored in pairs, with c01 and c21 at C01, c00 and c20 at C00, c11 and c31 at C11, and c10 and c30 at C10. c2 is at C2.
The quartic factors are evaluated in SIMD registers. For brevity, some comments below describe only one element of a register. The other is analagous.*/ movlhps x, x1 // Save a copy of x for later. mulsd x, x // Form x**2.#define x2 x // Define name describing current register contents. movlhps x2, x2 // Duplicate x**2. movapd Address(C11), p1 addpd x2, p1 // Form x**2 + c11. mulpd x2, p1 // Form x**4 + c11 * x**2. addpd Address(C10), p1 // Form x**4 + c11 * x**2 + c10. movapd Address(C01), p0 // Get first coefficients. movlpd Address(C2), x1 // Put c2 in one element, with x in other. addpd x2, p0 // Form x**2 + c01. mulpd x2, p0 // Form x**4 + c01 * x**2. addpd Address(C00), p0 // Form x**4 + c01 * x**2 + c00. mulpd p1, p0 // Combine factors. mulpd x1, p0 // Multiply by x and c2. movhlps p0, p1 // Get high element. mulsd p1, p0 // Finish combining factors.#undef x2
// Return the double-precision number currently in p0.ReturnDoubleInp0: cvtsd2ss p0, p0 // Convert result to single precision.
// Return the single-precision number currently in p0.ReturnSingleInp0:
#if defined __i386__ movss p0, 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 p0, which is %xmm0. #endif
ret
// Handle pPoint0 < x.PositiveTail:
ucomisd Address(pPoint1), x // Compare x to algorithm change point. jae InputIsPositiveSpecial
movsd Address(pHalfPi), p0 // Prepare +pi/2.
CommonTail:
/* Here we have pPoint0 < x < pPoint1. The algorithm here is inspired by the identity (for positive x) arctangent(x) = +pi/2 - arctangent(1/x). This is approximated by evaluating:
+pi/2 - * ((x**4 + t01 * x**2 + t00)) * ((x**4 + t11 * x**2 + t10)) * ((x**4 + t21 * x**2 + t20)) * ((x**4 + t31 * x**2 + t30)) * (1/x) * (1/x)**16.
The quartic factors are evaluated in SIMD registers. For brevity, some comments below describe only one element of a register. The other is analagous.*/
movsd Address(One), v0 // Get constant. divsd x, v0 // Form 1/x, which we will call r. /* I tried rcpss, but using it requires a precision conversion, two adds, and three multiplies. divsd performance is not bad on Core2, and I was not able to get the rcpss version to run as fast. */ mulsd x, x // Form x**2.#define x2 x // Define name describing current register contents. movlhps v0, v0 // Copy finished reciprocal to high element. movlhps x2, x2 // Duplicate x**2. mulsd v0, v0 // Form r**2. mulsd v0, v0 // Form r**4. mulsd v0, v0 // Form r**8. movapd Address(T11), p1 // Get first coefficients. mulsd v0, v0 // Form r**16.
movapd Address(T01), t0 // Get first coefficients. addpd x2, p1 // Form x**2 + t11. mulpd x2, p1 // Form x**4 + t11 * x**2. addpd Address(T10), p1 // Form x**4 + t11 * x**2 + t10. addpd x2, t0 // Form x**2 + t01. mulpd x2, t0 // Form x**4 + t01 * x**2. addpd Address(T00), t0 // Form x**4 + t01 * x**2 + t00. mulpd p1, t0 // Combine factors. mulpd v0, t0 // Multiply by r**16 and r. movhlps t0, p1 // Get high element. mulsd p1, t0 // Finish combining factors. subsd t0, p0 // Subtract from pi/2.#undef x2
jmp ReturnDoubleInp0
// Handle x < nPoint0.NegativeTail:
ucomisd Address(nPoint1), x // Compare x to algorithm change point. jbe InputIsNegativeSpecial
movsd Address(nHalfPi), p0 // Prepare -pi/2.
jmp CommonTail
.literal8/* Define values near +/- pi/2 that yield +/- pi/2 rounded toward zero when converted to single precision. This allows us to generate inexact and return the desired values for atanf of big positive and negative numbers.*/AlmostpHalfPi: .double +1.5707962AlmostnHalfPi: .double -1.5707962
.text
/* Here we handle inputs greater or equal to pPoint1, including infinity, but not including NaNs.*/InputIsPositiveSpecial:
/* The input is a big positive number. Return +pi/2 rounded toward zero and generate an inexact exception. */ cvtsd2ss Address(AlmostpHalfPi), p0 jmp ReturnSingleInp0
/* 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 nPoint1 in the "destination operand".*/InputIsNegativeSpecial:
jp InputIsNaN
/* The input is a big negative number. Return -pi/2 rounded toward zero and generate an inexact exception. */ cvtsd2ss Address(AlmostnHalfPi), p0 jmp ReturnSingleInp0
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