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Implement lgamma (C99).
This uses an approximation based on the Stirling series for most positive values, but uses separate rational approximations for greater accuracy near 1 and 2. A reflection formula is used for negative values.
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Stephen Heumann
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afff478793
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97a295522c
357
math2.asm
357
math2.asm
@ -1906,6 +1906,363 @@ ret creturn 2:x return it
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rtl
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end
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****************************************************************
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*
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* double lgamma(double x);
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*
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* Computes the natural logarithm of the absolute value of the
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* gamma function of x.
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*
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* The rational approximations used near 1 and 2 are from
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* W. J. Cody and K. E. Hillstrom, "Chebyshev Approximations
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* for the Natural Logarithm of the Gamma Function".
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* For other positive x values, a computation based on
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* Stirling's formula is used.
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*
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****************************************************************
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*
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lgamma start
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lgammaf entry
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lgammal entry
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using MathCommon2
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csubroutine (10:x),0
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phb
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phk
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plb
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pha save env & set to default
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tsc
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inc a
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pea 0
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pha
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FPROCENTRY
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stz z z := 0
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stz z+2
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stz z+4
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stz z+6
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stz z+8
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stz reflect assume no reflection
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; For x < 0, lgamma(x) = ln(pi) - ln(abs(x*sin(pi*x))) - lgamma(-x)
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lda x+8 if x is negative then
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jpl positive
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lda x if x != -0 and x != -inf then
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ora x+2
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ora x+4
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ora x+6
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jeq abs_x
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inc reflect flag to do reflection
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tdc neg_adj := x REM 2
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clc
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adc #x
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pea 0
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pha
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ph4 #neg_adj
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FX2X
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ph4 #two
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ph4 #neg_adj
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FREMI
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ph4 #pi neg_adj := sin(neg_adj*pi)
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ph4 #neg_adj
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FMULX
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ph4 #neg_adj
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FSINX
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tdc neg_adj := neg_adj * x
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clc
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adc #x
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pea 0
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pha
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ph4 #neg_adj
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FMULX
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asl neg_adj+8 neg_adj := abs(neg_adj)
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lsr neg_adj+8
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ph4 #neg_adj neg_adj := ln(neg_adj)
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FLNX
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ph4 #ln_pi neg_adj := neg_adj - ln(pi)
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ph4 #neg_adj
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FSUBX
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lda neg_adj+8 if neg_adj is inf/nan then
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asl a
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cmp #32767*2
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jeq do_reflect skip to final reflection
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abs_x asl x+8 x := abs(x)
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lsr x+8
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positive tdc t1 := x
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clc
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adc #x
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pea 0
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pha
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ph4 #t1
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FX2X
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ph4 #rat2_hi if x is near 1 or 2 then
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ph4 #t1
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FCMPS
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jvc not_near_1_or_2
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; Rational approximation for x near 2
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ph4 #rat2_lo if x is near 2 then
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ph4 #t1
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FCMPS
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bvs small
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ph4 #z z := P2(t1)
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pea 7
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ph4 #P2
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jsl poly
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ph4 #y y := Q2(t1)
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pea 7
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ph4 #Q2
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jsl poly
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ph4 #two N := 2
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bra finish_rational_approx
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; Rational approximation for x near 1
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small ph4 #rat1_hi else if x is near 1 then
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ph4 #t1
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FCMPS
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jvc not_near_1_or_2
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ph4 #rat1_lo
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ph4 #t1
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FCMPS
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bvs not_near_1_or_2
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ph4 #z z := P1(t1)
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pea 7
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ph4 #P1
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jsl poly
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ph4 #y y := Q1(t1)
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pea 7
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ph4 #Q1
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jsl poly
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ph4 #one N := 1
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finish_rational_approx anop
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ph4 #t1 t1 := x - N
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FSUBI
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ph4 #y z := z/y
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ph4 #z
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FDIVX
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ph4 #t1 z := z * t1
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ph4 #z
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FMULX
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brl chk_reflect goto chk_reflect
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; Calculate based on Stirling's formula for other x values
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not_near_1_or_2 anop
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chk_big ph4 #cutoff while x < 7 do
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tdc
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clc
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adc #x
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pea 0
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pha
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FCMPI
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bvc stirling
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tdc t1 := ln(x)
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clc
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adc #x
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pea 0
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pha
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ph4 #t1
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FX2X
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ph4 #t1
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FLNX
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ph4 #t1 z := z + t1
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ph4 #z
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FADDX
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ph4 #one x := x + 1
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tdc
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clc
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adc #x
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pea 0
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pha
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FADDI
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bra chk_big
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stirling tdc (push x arguments for below ops)
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clc
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adc #x
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ldy #0
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phy
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pha
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phy
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pha
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phy
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pha
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phy
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pha
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ph4 #t1 t1 := ln(x)
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FX2X
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ph4 #t1
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FLNX
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ph4 #y y := x - 1/2
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FX2X
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ph4 #one_half
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ph4 #y
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FSUBS
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ph4 #t1 y := t1 * y
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ph4 #y
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FMULX
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lda y+8 if y is not inf/nan then
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asl a
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cmp #32767*2
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beq stirling_poly
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tdc y := y - x
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clc
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adc #x
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pea 0
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pha
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ph4 #y
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FSUBX
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ph4 #half_ln_2pi y := y + 0.5*ln(2pi)
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ph4 #y
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FADDX
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ph4 #z y := y - z
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ph4 #y (this adjusts for any scaling up)
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FSUBX
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stirling_poly anop
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ph4 #t1 t1 := 1/x^2
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FX2X
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pea -2
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ph4 #t1
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FXPWRI
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ph4 #z z := P_stirling(t1)
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pea 13
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ph4 #P_stirling
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jsl poly
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ph4 #z z := z / x
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FDIVX
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ph4 #y z := y + z
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ph4 #z
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FADDX
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chk_reflect anop
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lda reflect if reflection is needed then
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beq done
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do_reflect anop
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ph4 #neg_adj z := z + neg_adj
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ph4 #z
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FADDX
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lda z+8 z := -z
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eor #$8000
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sta z+8
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done FPROCEXIT restore env & raise any new exceptions
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plb
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creturn 10:z return a pointer to the result
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; constants
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cutoff dc i2'7' above this, just use Stirling's formula
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;low/high limits for use of rational approximations near 1 and 2
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rat1_lo dc f'0.97265625'
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rat1_hi dc f'1.02880859375'
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rat2_lo dc f'1.77880859375'
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rat2_hi dc f'2.1875'
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one_half dc f'0.5'
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half_ln_2pi dc e'0.9189385332046727417803297'
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one dc i2'1'
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two dc i2'2'
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ln_pi dc e'1.14472988584940017414342735'
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pi dc e'3.1415926535897932384626433'
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; coefficients for rational approximation, 0.5 <= x <= 1.5
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P1 dc e'4.120843185847770031e00'
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dc e'8.568982062831317339e01'
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dc e'2.431752435244210223e02'
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dc e'-2.617218583856145190e02'
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dc e'-9.222613728801521582e02'
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dc e'-5.176383498023217924e02'
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dc e'-7.741064071332953034e01'
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dc e'-2.208843997216182306e00'
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Q1 dc e'1.000000000000000000e00'
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dc e'4.564677187585907957e01'
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dc e'3.778372484823942081e02'
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dc e'9.513235976797059772e02'
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dc e'8.460755362020782006e02'
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dc e'2.623083470269460180e02'
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dc e'2.443519662506311704e01'
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dc e'4.097792921092615065e-01'
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; coefficients for rational approximation, 1.5 <= x <= 4.0
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P2 dc e'5.1550576176408171704e00'
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dc e'3.7751067979721702241e02'
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dc e'5.2689832559149812458e03'
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dc e'1.9553605540630449846e04'
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dc e'1.2043173809871640151e04'
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dc e'-2.0648294205325283281e04'
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dc e'-1.5086302287667250272e04'
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dc e'-1.5138318341150667785e03'
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Q2 dc e'1.0000000000000000000e00'
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dc e'1.2890931890129576873e02'
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dc e'3.0399030414394398824e03'
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dc e'2.2029562144156636889e04'
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dc e'5.7120255396025029854e04'
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dc e'5.2622863838411992470e04'
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dc e'1.4402090371700852304e04'
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dc e'6.9832741405735102159e02'
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; coefficients for Stirling series approximation
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P_stirling anop
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dc e'-36108.77125372498935717327'
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dc e'2193.103333333333333333333'
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dc e'-156.8482846260020173063651'
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dc e'13.40286404416839199447895'
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dc e'-1.392432216905901116427432'
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dc e'0.1796443723688305731649385'
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dc e'-0.02955065359477124183006536'
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dc e'0.006410256410256410256410256'
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dc e'-0.001917526917526917526917527'
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dc e'0.0008417508417508417508417508'
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dc e'-0.0005952380952380952380952381'
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dc e'0.0007936507936507936507936508'
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dc e'-0.002777777777777777777777778'
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dc e'0.08333333333333333333333333'
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; temporaries/return values
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y ds 10
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z ds 10
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neg_adj ds 10 adjustment for negative x
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reflect ds 2 reflection flag
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end
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****************************************************************
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*
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* long long llrint(double x);
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@ -748,3 +748,8 @@
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.c
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~restm
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mend
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macro
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&l jvc &bp
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&l bvs *+5
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brl &bp
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mend
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