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math: faster Lgamma
Converting from polynomial constants to counted array speeds up Lgamma from 51.3 to 37.7 ns/op. Variables renamed in Gamma to avoid overlap in Lgamma. R=rsc, golang-dev CC=golang-dev https://golang.org/cl/5359045
This commit is contained in:
Charles L. Dorian
Russ Cox
parent
3b860269ee
commit
300b443ade
@ -63,7 +63,7 @@ package math
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// Stephen L. Moshier
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// moshier@na-net.ornl.gov
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var _P = [...]float64{
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var _gamP = [...]float64{
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1.60119522476751861407e-04,
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1.19135147006586384913e-03,
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1.04213797561761569935e-02,
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@ -72,7 +72,7 @@ var _P = [...]float64{
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4.94214826801497100753e-01,
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9.99999999999999996796e-01,
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}
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var _Q = [...]float64{
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var _gamQ = [...]float64{
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-2.31581873324120129819e-05,
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5.39605580493303397842e-04,
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-4.45641913851797240494e-03,
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@ -82,7 +82,7 @@ var _Q = [...]float64{
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7.14304917030273074085e-02,
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1.00000000000000000320e+00,
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}
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var _S = [...]float64{
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var _gamS = [...]float64{
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7.87311395793093628397e-04,
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-2.29549961613378126380e-04,
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-2.68132617805781232825e-03,
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@ -98,7 +98,7 @@ func stirling(x float64) float64 {
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MaxStirling = 143.01608
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)
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w := 1 / x
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w = 1 + w*((((_S[0]*w+_S[1])*w+_S[2])*w+_S[3])*w+_S[4])
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w = 1 + w*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[4])
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y := Exp(x)
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if x > MaxStirling { // avoid Pow() overflow
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v := Pow(x, 0.5*x-0.25)
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@ -176,8 +176,8 @@ func Gamma(x float64) float64 {
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}
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x = x - 2
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p = (((((x*_P[0]+_P[1])*x+_P[2])*x+_P[3])*x+_P[4])*x+_P[5])*x + _P[6]
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q = ((((((x*_Q[0]+_Q[1])*x+_Q[2])*x+_Q[3])*x+_Q[4])*x+_Q[5])*x+_Q[6])*x + _Q[7]
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p = (((((x*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6]
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q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7]
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return z * p / q
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small:
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@ -88,6 +88,81 @@ package math
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//
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//
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var _lgamA = [...]float64{
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7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
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3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
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6.73523010531292681824e-02, // 0x3FB13E001A5562A7
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2.05808084325167332806e-02, // 0x3F951322AC92547B
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7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
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2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
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1.19270763183362067845e-03, // 0x3F538A94116F3F5D
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5.10069792153511336608e-04, // 0x3F40B6C689B99C00
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2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
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1.08011567247583939954e-04, // 0x3F1C5088987DFB07
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2.52144565451257326939e-05, // 0x3EFA7074428CFA52
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4.48640949618915160150e-05, // 0x3F07858E90A45837
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}
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var _lgamR = [...]float64{
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1.0, // placeholder
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1.39200533467621045958e+00, // 0x3FF645A762C4AB74
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7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
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1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
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1.86459191715652901344e-02, // 0x3F9317EA742ED475
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7.77942496381893596434e-04, // 0x3F497DDACA41A95B
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7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
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}
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var _lgamS = [...]float64{
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-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
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2.14982415960608852501e-01, // 0x3FCB848B36E20878
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3.25778796408930981787e-01, // 0x3FD4D98F4F139F59
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1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7
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2.66422703033638609560e-02, // 0x3F9B481C7E939961
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1.84028451407337715652e-03, // 0x3F5E26B67368F239
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3.19475326584100867617e-05, // 0x3F00BFECDD17E945
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}
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var _lgamT = [...]float64{
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4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2
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-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
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6.46249402391333854778e-02, // 0x3FB08B4294D5419B
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-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
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1.79706750811820387126e-02, // 0x3F9266E7970AF9EC
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-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
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6.10053870246291332635e-03, // 0x3F78FCE0E370E344
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-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
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2.25964780900612472250e-03, // 0x3F6282D32E15C915
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-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
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8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9
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-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
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3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7
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-3.12754168375120860518e-04, // 0xBF347F24ECC38C38
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3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4
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}
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var _lgamU = [...]float64{
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-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
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6.32827064025093366517e-01, // 0x3FE4401E8B005DFF
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1.45492250137234768737e+00, // 0x3FF7475CD119BD6F
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9.77717527963372745603e-01, // 0x3FEF497644EA8450
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2.28963728064692451092e-01, // 0x3FCD4EAEF6010924
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1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09
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}
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var _lgamV = [...]float64{
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1.0,
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2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
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2.12848976379893395361e+00, // 0x40010725A42B18F5
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7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
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1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
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3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
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}
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var _lgamW = [...]float64{
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4.18938533204672725052e-01, // 0x3FDACFE390C97D69
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8.33333333333329678849e-02, // 0x3FB555555555553B
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-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
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7.93650558643019558500e-04, // 0x3F4A019F98CF38B6
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-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
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8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1
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-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
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}
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// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
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//
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// Special cases are:
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@ -103,68 +178,10 @@ func Lgamma(x float64) (lgamma float64, sign int) {
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Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
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Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
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Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
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A0 = 7.72156649015328655494e-02 // 0x3FB3C467E37DB0C8
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A1 = 3.22467033424113591611e-01 // 0x3FD4A34CC4A60FAD
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A2 = 6.73523010531292681824e-02 // 0x3FB13E001A5562A7
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A3 = 2.05808084325167332806e-02 // 0x3F951322AC92547B
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A4 = 7.38555086081402883957e-03 // 0x3F7E404FB68FEFE8
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A5 = 2.89051383673415629091e-03 // 0x3F67ADD8CCB7926B
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A6 = 1.19270763183362067845e-03 // 0x3F538A94116F3F5D
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A7 = 5.10069792153511336608e-04 // 0x3F40B6C689B99C00
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A8 = 2.20862790713908385557e-04 // 0x3F2CF2ECED10E54D
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A9 = 1.08011567247583939954e-04 // 0x3F1C5088987DFB07
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A10 = 2.52144565451257326939e-05 // 0x3EFA7074428CFA52
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A11 = 4.48640949618915160150e-05 // 0x3F07858E90A45837
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Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
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Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
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// Tt = -(tail of Tf)
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Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
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T0 = 4.83836122723810047042e-01 // 0x3FDEF72BC8EE38A2
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T1 = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509
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T2 = 6.46249402391333854778e-02 // 0x3FB08B4294D5419B
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T3 = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713
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T4 = 1.79706750811820387126e-02 // 0x3F9266E7970AF9EC
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T5 = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A
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T6 = 6.10053870246291332635e-03 // 0x3F78FCE0E370E344
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T7 = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7
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T8 = 2.25964780900612472250e-03 // 0x3F6282D32E15C915
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T9 = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1
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T10 = 8.81081882437654011382e-04 // 0x3F4CDF0CEF61A8E9
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T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC
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T12 = 3.15632070903625950361e-04 // 0x3F34AF6D6C0EBBF7
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T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38
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T14 = 3.35529192635519073543e-04 // 0x3F35FD3EE8C2D3F4
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U0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
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U1 = 6.32827064025093366517e-01 // 0x3FE4401E8B005DFF
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U2 = 1.45492250137234768737e+00 // 0x3FF7475CD119BD6F
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U3 = 9.77717527963372745603e-01 // 0x3FEF497644EA8450
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U4 = 2.28963728064692451092e-01 // 0x3FCD4EAEF6010924
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U5 = 1.33810918536787660377e-02 // 0x3F8B678BBF2BAB09
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V1 = 2.45597793713041134822e+00 // 0x4003A5D7C2BD619C
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V2 = 2.12848976379893395361e+00 // 0x40010725A42B18F5
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V3 = 7.69285150456672783825e-01 // 0x3FE89DFBE45050AF
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V4 = 1.04222645593369134254e-01 // 0x3FBAAE55D6537C88
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V5 = 3.21709242282423911810e-03 // 0x3F6A5ABB57D0CF61
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S0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
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S1 = 2.14982415960608852501e-01 // 0x3FCB848B36E20878
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S2 = 3.25778796408930981787e-01 // 0x3FD4D98F4F139F59
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S3 = 1.46350472652464452805e-01 // 0x3FC2BB9CBEE5F2F7
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S4 = 2.66422703033638609560e-02 // 0x3F9B481C7E939961
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S5 = 1.84028451407337715652e-03 // 0x3F5E26B67368F239
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S6 = 3.19475326584100867617e-05 // 0x3F00BFECDD17E945
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R1 = 1.39200533467621045958e+00 // 0x3FF645A762C4AB74
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R2 = 7.21935547567138069525e-01 // 0x3FE71A1893D3DCDC
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R3 = 1.71933865632803078993e-01 // 0x3FC601EDCCFBDF27
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R4 = 1.86459191715652901344e-02 // 0x3F9317EA742ED475
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R5 = 7.77942496381893596434e-04 // 0x3F497DDACA41A95B
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R6 = 7.32668430744625636189e-06 // 0x3EDEBAF7A5B38140
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W0 = 4.18938533204672725052e-01 // 0x3FDACFE390C97D69
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W1 = 8.33333333333329678849e-02 // 0x3FB555555555553B
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W2 = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C
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W3 = 7.93650558643019558500e-04 // 0x3F4A019F98CF38B6
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W4 = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741
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W5 = 8.36339918996282139126e-04 // 0x3F4B67BA4CDAD5D1
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W6 = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4
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Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
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)
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// TODO(rsc): Remove manual inlining of IsNaN, IsInf
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// when compiler does it for us
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@ -249,28 +266,28 @@ func Lgamma(x float64) (lgamma float64, sign int) {
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switch i {
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case 0:
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z := y * y
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p1 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10))))
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p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11)))))
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p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
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p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
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p := y*p1 + p2
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lgamma += (p - 0.5*y)
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case 1:
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z := y * y
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w := z * y
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p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp
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p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13)))
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p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14)))
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p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
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p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
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p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
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p := z*p1 - (Tt - w*(p2+y*p3))
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lgamma += (Tf + p)
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case 2:
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p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5)))))
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p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5))))
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p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
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p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
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lgamma += (-0.5*y + p1/p2)
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}
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case x < 8: // 2 <= x < 8
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i := int(x)
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y := x - float64(i)
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p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6))))))
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q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6)))))
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p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
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q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
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lgamma = 0.5*y + p/q
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z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
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switch i {
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@ -294,7 +311,7 @@ func Lgamma(x float64) (lgamma float64, sign int) {
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t := Log(x)
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z := 1 / x
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y := z * z
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w := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6)))))
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w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
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lgamma = (x-0.5)*(t-1) + w
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default: // 2**58 <= x <= Inf
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lgamma = x * (Log(x) - 1)
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