math: regularize build

This will be nicer to the automatic tools.
It requires a few more assembly stubs
but fewer Go files.

There are a few instances where it looks like
there are new blobs of code, but they are just
being copied out of deleted files.

There is no new code here.

Suppose you have a portable implementation for Sin
and a 386-specific assembly one.  The old way to
do this was to write three files

sin_decl.go
   func Sin(x float64) float64  // declaration only
sin_386.s
   assembly implementation

sin_port.go
   func Sin(x float64) float64 { ... }  // pure-Go impl

and then link in either sin_decl.go+sin_386.s or
just sin_port.go.  The Makefile actually did the magic
of linking in only the _port.go files for those without
assembly and only the _decl.go files for those with
assembly, or at least some of that magic.

The biggest problem with this, beyond being hard
to explain to the build system, is that once you do
explain it to the build system, godoc knows which
of sin_port.go or sin_decl.go are involved on a given
architecture, and it (correctly) ignores the other.
That means you have to put identical doc comments
in both files.

The new approach, which is more like what we did
in the later packages math/big and sync/atomic,
is to have

sin.go
   func Sin(x float64) float64  // decl only
   func sin(x float64) float64 {...}  // pure-Go impl

sin_386.s
   // assembly for Sin (ignores sin)
sin_amd64.s
   // assembly for Sin: jmp sin
sin_arm.s
   // assembly for Sin: jmp sin

Once we abandon Makefiles we can put all the assembly
stubs in one source file, so the number of files will
actually go down.

Chris asked whether the branches cost anything.
Given that they are branching to pure-Go implementations
that are not typically known for their speed, the single
direct branch is not going to be noticeable.  That is,
it's on the slow path.

An alternative would have been to preserve the old
"only write assembly files when there's an implementation"
and still have just one copy of the declaration of Sin
(and thus one doc comment) by doing:

sin.go
   func Sin(x float64) float64 { return sin(x) }

sin_decl.go
   func sin(x float64) float64 // declaration only
sin_386.s
   // assembly for sin

sin_port.go
   func sin(x float64) float64 { portable code }

In this version everyone would link in sin.go and
then either sin_decl.go+sin_386.s or sin_port.go.

This has an extra function call on all paths, including
the "fast path" to get to assembly, and it triples the
number of Go files involved compared to what I did
in this CL.  On the other hand you don't have to
write assembly stubs.  After starting down this path
I decided that the assembly stubs were the easier
approach.

As for generating the assembly stubs on the fly, much
of the goal here is to eliminate magic from the build
process, so that zero-configuration tools like goinstall
or the new go tool can handle this package.

R=golang-dev, r, cw, iant, r
CC=golang-dev
https://golang.org/cl/5488057

src/pkg/math/Makefile

@@ -6,44 +6,31 @@ include ../../Make.inc
 
 TARG=math
 
-OFILES_arm=\
-	sqrt_arm.$O\
-
-OFILES_amd64=\
-	abs_amd64.$O\
-	dim_amd64.$O\
-	exp_amd64.$O\
-	hypot_amd64.$O\
-	log_amd64.$O\
-	sqrt_amd64.$O\
-
-OFILES_386=\
-	abs_386.$O\
-	asin_386.$O\
-	atan_386.$O\
-	atan2_386.$O\
-	exp_386.$O\
-	exp2_386.$O\
-	expm1_386.$O\
-	floor_386.$O\
-	frexp_386.$O\
-	hypot_386.$O\
-	ldexp_386.$O\
-	log_386.$O\
-	log10_386.$O\
-	log1p_386.$O\
-	mod_386.$O\
-	modf_386.$O\
-	remainder_386.$O\
-	sin_386.$O\
-	sincos_386.$O\
-	sqrt_386.$O\
-	tan_386.$O\
-
 OFILES=\
-	$(OFILES_$(GOARCH))
+	abs_$(GOARCH).$O\
+	asin_$(GOARCH).$O\
+	atan_$(GOARCH).$O\
+	atan2_$(GOARCH).$O\
+	dim_$(GOARCH).$O\
+	exp_$(GOARCH).$O\
+	exp2_$(GOARCH).$O\
+	expm1_$(GOARCH).$O\
+	floor_$(GOARCH).$O\
+	frexp_$(GOARCH).$O\
+	hypot_$(GOARCH).$O\
+	ldexp_$(GOARCH).$O\
+	log_$(GOARCH).$O\
+	log10_$(GOARCH).$O\
+	log1p_$(GOARCH).$O\
+	mod_$(GOARCH).$O\
+	modf_$(GOARCH).$O\
+	remainder_$(GOARCH).$O\
+	sin_$(GOARCH).$O\
+	sincos_$(GOARCH).$O\
+	sqrt_$(GOARCH).$O\
+	tan_$(GOARCH).$O\
 
-ALLGOFILES=\
+GOFILES=\
 	abs.go\
 	acosh.go\
 	asin.go\
@@ -58,14 +45,11 @@ ALLGOFILES=\
 	dim.go\
 	erf.go\
 	exp.go\
-	exp_port.go\
-	exp2.go\
 	expm1.go\
 	floor.go\
 	frexp.go\
 	gamma.go\
 	hypot.go\
-	hypot_port.go\
 	j0.go\
 	j1.go\
 	jn.go\
@@ -86,16 +70,8 @@ ALLGOFILES=\
 	sincos.go\
 	sinh.go\
 	sqrt.go\
-	sqrt_port.go\
 	tan.go\
 	tanh.go\
 	unsafe.go\
 
-NOGOFILES=\
-	$(subst _$(GOARCH).$O,.go,$(OFILES_$(GOARCH)))
-
-GOFILES=\
-	$(filter-out $(NOGOFILES),$(ALLGOFILES))\
-	$(subst .go,_decl.go,$(NOGOFILES))\
-
 include ../../Make.pkg

src/pkg/math/abs.go

@@ -9,7 +9,9 @@ package math
 // Special cases are:
 //	Abs(±Inf) = +Inf
 //	Abs(NaN) = NaN
-func Abs(x float64) float64 {
+func Abs(x float64) float64
+
+func abs(x float64) float64 {
 	switch {
 	case x < 0:
 		return -x

src/pkg/math/abs_arm.s

@@ -1,7 +1,6 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
+// Copyright 2011 The Go Authors.  All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Abs(SB),7,$0
-
+	B ·abs(SB)
-func Log(x float64) float64

src/pkg/math/all_test.go

@@ -2062,6 +2062,34 @@ func TestHypot(t *testing.T) {
 	}
 }
 
+func TestHypotSqrtGo(t *testing.T) {
+	for i := 0; i < len(vf); i++ {
+		a := Abs(1e200 * tanh[i] * Sqrt(2))
+		if f := HypotSqrtGo(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+			t.Errorf("HypotSqrtGo(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+		}
+	}
+	for i := 0; i < len(vfhypotSC); i++ {
+		if f := HypotSqrtGo(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+			t.Errorf("HypotSqrtGo(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+		}
+	}
+}
+
+func TestHypotNoSqrtGo(t *testing.T) {
+	for i := 0; i < len(vf); i++ {
+		a := Abs(1e200 * tanh[i] * Sqrt(2))
+		if f := HypotNoSqrtGo(1e200*tanh[i], 1e200*tanh[i]); !veryclose(a, f) {
+			t.Errorf("HypotNoSqrtGo(%g, %g) = %g, want %g", 1e200*tanh[i], 1e200*tanh[i], f, a)
+		}
+	}
+	for i := 0; i < len(vfhypotSC); i++ {
+		if f := HypotNoSqrtGo(vfhypotSC[i][0], vfhypotSC[i][1]); !alike(hypotSC[i], f) {
+			t.Errorf("HypotNoSqrtGo(%g, %g) = %g, want %g", vfhypotSC[i][0], vfhypotSC[i][1], f, hypotSC[i])
+		}
+	}
+}
+
 func TestIlogb(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
 		a := frexp[i].i - 1 // adjust because fr in the interval [½, 1)
@@ -2713,9 +2741,15 @@ func BenchmarkHypot(b *testing.B) {
 	}
 }
 
-func BenchmarkHypotGo(b *testing.B) {
+func BenchmarkHypotNoSqrtGo(b *testing.B) {
 	for i := 0; i < b.N; i++ {
-		HypotGo(3, 4)
+		HypotNoSqrtGo(3, 4)
+	}
+}
+
+func BenchmarkHypotSqrtGo(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		HypotSqrtGo(3, 4)
 	}
 }
 

src/pkg/math/asin.go

@@ -16,7 +16,9 @@ package math
 // Special cases are:
 //	Asin(±0) = ±0
 //	Asin(x) = NaN if x < -1 or x > 1
-func Asin(x float64) float64 {
+func Asin(x float64) float64
+
+func asin(x float64) float64 {
 	if x == 0 {
 		return x // special case
 	}
@@ -46,4 +48,8 @@ func Asin(x float64) float64 {
 //
 // Special case is:
 //	Acos(x) = NaN if x < -1 or x > 1
-func Acos(x float64) float64 { return Pi/2 - Asin(x) }
+func Acos(x float64) float64
+
+func acos(x float64) float64 {
+	return Pi/2 - Asin(x)
+}

src/pkg/math/asin_amd64.s

@@ -0,0 +1,9 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Asin(SB),7,$0
+	JMP ·asin(SB)
+
+TEXT ·Acos(SB),7,$0
+	JMP ·acos(SB)

src/pkg/math/asin_arm.s

@@ -0,0 +1,9 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Asin(SB),7,$0
+	B ·asin(SB)
+
+TEXT ·Acos(SB),7,$0
+	B ·acos(SB)

src/pkg/math/asin_decl.go

@@ -1,8 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Acos(x float64) float64
-func Asin(x float64) float64

src/pkg/math/atan.go

@@ -51,7 +51,9 @@ func satan(arg float64) float64 {
 // Special cases are:
 //	Atan(±0) = ±0
 //	Atan(±Inf) = ±Pi/2
-func Atan(x float64) float64 {
+func Atan(x float64) float64
+
+func atan(x float64) float64 {
 	if x == 0 {
 		return x
 	}

src/pkg/math/atan2.go

@@ -26,7 +26,9 @@ package math
 //	Atan2(y<0, -Inf) = -Pi
 //	Atan2(+Inf, x) = +Pi/2
 //	Atan2(-Inf, x) = -Pi/2
-func Atan2(y, x float64) float64 {
+func Atan2(y, x float64) float64
+
+func atan2(y, x float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases

src/pkg/math/atan2_amd64.s

@@ -1,7 +1,6 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
+// Copyright 2011 The Go Authors.  All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Atan2(SB),7,$0
-
+	JMP ·atan2(SB)
-func Exp(x float64) float64

src/pkg/math/atan2_arm.s

@@ -1,7 +1,6 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
+// Copyright 2011 The Go Authors.  All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Atan2(SB),7,$0
-
+	B ·atan2(SB)
-func Tan(x float64) float64

src/pkg/math/atan_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Atan(SB),7,$0
+	JMP ·atan(SB)

src/pkg/math/atan_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Atan(SB),7,$0
+	B ·atan(SB)

src/pkg/math/dim.go

@@ -10,8 +10,10 @@ package math
 //	Dim(+Inf, +Inf) = NaN
 //	Dim(-Inf, -Inf) = NaN
 //	Dim(x, NaN) = Dim(NaN, x) = NaN
-func Dim(x, y float64) float64 {
+func Dim(x, y float64) float64
-	return Max(x-y, 0)
+
+func dim(x, y float64) float64 {
+	return max(x-y, 0)
 }
 
 // Max returns the larger of x or y.
@@ -21,7 +23,9 @@ func Dim(x, y float64) float64 {
 //	Max(x, NaN) = Max(NaN, x) = NaN
 //	Max(+0, ±0) = Max(±0, +0) = +0
 //	Max(-0, -0) = -0
-func Max(x, y float64) float64 {
+func Max(x, y float64) float64
+
+func max(x, y float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases
@@ -48,7 +52,9 @@ func Max(x, y float64) float64 {
 //	Min(x, -Inf) = Min(-Inf, x) = -Inf
 //	Min(x, NaN) = Min(NaN, x) = NaN
 //	Min(-0, ±0) = Min(±0, -0) = -0
-func Min(x, y float64) float64 {
+func Min(x, y float64) float64
+
+func min(x, y float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases

src/pkg/math/dim_386.s

@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Dim(SB),7,$0
+	JMP ·dim(SB)
+
+TEXT ·Max(SB),7,$0
+	JMP ·max(SB)
+
+TEXT ·Min(SB),7,$0
+	JMP ·min(SB)

src/pkg/math/dim_arm.s

@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Dim(SB),7,$0
+	B ·dim(SB)
+
+TEXT ·Min(SB),7,$0
+	B ·min(SB)
+
+TEXT ·Max(SB),7,$0
+	B ·max(SB)

src/pkg/math/dim_decl.go

@@ -1,9 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Dim(x, y float64) float64
-func Max(x, y float64) float64
-func Min(x, y float64) float64

src/pkg/math/exp.go

@@ -11,4 +11,185 @@ package math
 //	Exp(NaN) = NaN
 // Very large values overflow to 0 or +Inf.
 // Very small values underflow to 1.
-func Exp(x float64) float64 { return expGo(x) }
+func Exp(x float64) float64
+
+// The original C code, the long comment, and the constants
+// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
+// and came with this notice.  The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+//
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+//
+// exp(x)
+// Returns the exponential of x.
+//
+// Method
+//   1. Argument reduction:
+//      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+//      Given x, find r and integer k such that
+//
+//               x = k*ln2 + r,  |r| <= 0.5*ln2.
+//
+//      Here r will be represented as r = hi-lo for better
+//      accuracy.
+//
+//   2. Approximation of exp(r) by a special rational function on
+//      the interval [0,0.34658]:
+//      Write
+//          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+//      We use a special Remes algorithm on [0,0.34658] to generate
+//      a polynomial of degree 5 to approximate R. The maximum error
+//      of this polynomial approximation is bounded by 2**-59. In
+//      other words,
+//          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+//      (where z=r*r, and the values of P1 to P5 are listed below)
+//      and
+//          |                  5          |     -59
+//          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
+//          |                             |
+//      The computation of exp(r) thus becomes
+//                             2*r
+//              exp(r) = 1 + -------
+//                            R - r
+//                                 r*R1(r)
+//                     = 1 + r + ----------- (for better accuracy)
+//                                2 - R1(r)
+//      where
+//                               2       4             10
+//              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+//
+//   3. Scale back to obtain exp(x):
+//      From step 1, we have
+//         exp(x) = 2**k * exp(r)
+//
+// Special cases:
+//      exp(INF) is INF, exp(NaN) is NaN;
+//      exp(-INF) is 0, and
+//      for finite argument, only exp(0)=1 is exact.
+//
+// Accuracy:
+//      according to an error analysis, the error is always less than
+//      1 ulp (unit in the last place).
+//
+// Misc. info.
+//      For IEEE double
+//          if x >  7.09782712893383973096e+02 then exp(x) overflow
+//          if x < -7.45133219101941108420e+02 then exp(x) underflow
+//
+// Constants:
+// The hexadecimal values are the intended ones for the following
+// constants. The decimal values may be used, provided that the
+// compiler will convert from decimal to binary accurately enough
+// to produce the hexadecimal values shown.
+
+func exp(x float64) float64 {
+	const (
+		Ln2Hi = 6.93147180369123816490e-01
+		Ln2Lo = 1.90821492927058770002e-10
+		Log2e = 1.44269504088896338700e+00
+
+		Overflow  = 7.09782712893383973096e+02
+		Underflow = -7.45133219101941108420e+02
+		NearZero  = 1.0 / (1 << 28) // 2**-28
+	)
+
+	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
+	// when compiler does it for us
+	// special cases
+	switch {
+	case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
+		return x
+	case x < -MaxFloat64: // IsInf(x, -1):
+		return 0
+	case x > Overflow:
+		return Inf(1)
+	case x < Underflow:
+		return 0
+	case -NearZero < x && x < NearZero:
+		return 1 + x
+	}
+
+	// reduce; computed as r = hi - lo for extra precision.
+	var k int
+	switch {
+	case x < 0:
+		k = int(Log2e*x - 0.5)
+	case x > 0:
+		k = int(Log2e*x + 0.5)
+	}
+	hi := x - float64(k)*Ln2Hi
+	lo := float64(k) * Ln2Lo
+
+	// compute
+	return expmulti(hi, lo, k)
+}
+
+// Exp2 returns 2**x, the base-2 exponential of x.
+//
+// Special cases are the same as Exp.
+func Exp2(x float64) float64
+
+func exp2(x float64) float64 {
+	const (
+		Ln2Hi = 6.93147180369123816490e-01
+		Ln2Lo = 1.90821492927058770002e-10
+
+		Overflow  = 1.0239999999999999e+03
+		Underflow = -1.0740e+03
+	)
+
+	// TODO: remove manual inlining of IsNaN and IsInf
+	// when compiler does it for us
+	// special cases
+	switch {
+	case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
+		return x
+	case x < -MaxFloat64: // IsInf(x, -1):
+		return 0
+	case x > Overflow:
+		return Inf(1)
+	case x < Underflow:
+		return 0
+	}
+
+	// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
+	// computed as r = hi - lo for extra precision.
+	var k int
+	switch {
+	case x > 0:
+		k = int(x + 0.5)
+	case x < 0:
+		k = int(x - 0.5)
+	}
+	t := x - float64(k)
+	hi := t * Ln2Hi
+	lo := -t * Ln2Lo
+
+	// compute
+	return expmulti(hi, lo, k)
+}
+
+// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
+func expmulti(hi, lo float64, k int) float64 {
+	const (
+		P1 = 1.66666666666666019037e-01  /* 0x3FC55555; 0x5555553E */
+		P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
+		P3 = 6.61375632143793436117e-05  /* 0x3F11566A; 0xAF25DE2C */
+		P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
+		P5 = 4.13813679705723846039e-08  /* 0x3E663769; 0x72BEA4D0 */
+	)
+
+	r := hi - lo
+	t := r * r
+	c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
+	y := 1 - ((lo - (r*c)/(2-c)) - hi)
+	// TODO(rsc): make sure Ldexp can handle boundary k
+	return Ldexp(y, k)
+}

src/pkg/math/exp2.go

@@ -1,10 +0,0 @@
-// Copyright 2010 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-// Exp2 returns 2**x, the base-2 exponential of x.
-//
-// Special cases are the same as Exp.
-func Exp2(x float64) float64 { return exp2Go(x) }

src/pkg/math/exp2_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Exp2(SB),7,$0
+	JMP ·exp2(SB)

src/pkg/math/exp2_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Exp2(SB),7,$0
+	B ·exp2(SB)

src/pkg/math/exp_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Exp(SB),7,$0
+	B ·exp(SB)

src/pkg/math/exp_port.go

@@ -1,191 +0,0 @@
-// Copyright 2009 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-// The original C code, the long comment, and the constants
-// below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
-// and came with this notice.  The go code is a simplified
-// version of the original C.
-//
-// ====================================================
-// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
-//
-// Permission to use, copy, modify, and distribute this
-// software is freely granted, provided that this notice
-// is preserved.
-// ====================================================
-//
-//
-// exp(x)
-// Returns the exponential of x.
-//
-// Method
-//   1. Argument reduction:
-//      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
-//      Given x, find r and integer k such that
-//
-//               x = k*ln2 + r,  |r| <= 0.5*ln2.
-//
-//      Here r will be represented as r = hi-lo for better
-//      accuracy.
-//
-//   2. Approximation of exp(r) by a special rational function on
-//      the interval [0,0.34658]:
-//      Write
-//          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
-//      We use a special Remes algorithm on [0,0.34658] to generate
-//      a polynomial of degree 5 to approximate R. The maximum error
-//      of this polynomial approximation is bounded by 2**-59. In
-//      other words,
-//          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
-//      (where z=r*r, and the values of P1 to P5 are listed below)
-//      and
-//          |                  5          |     -59
-//          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
-//          |                             |
-//      The computation of exp(r) thus becomes
-//                             2*r
-//              exp(r) = 1 + -------
-//                            R - r
-//                                 r*R1(r)
-//                     = 1 + r + ----------- (for better accuracy)
-//                                2 - R1(r)
-//      where
-//                               2       4             10
-//              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
-//
-//   3. Scale back to obtain exp(x):
-//      From step 1, we have
-//         exp(x) = 2**k * exp(r)
-//
-// Special cases:
-//      exp(INF) is INF, exp(NaN) is NaN;
-//      exp(-INF) is 0, and
-//      for finite argument, only exp(0)=1 is exact.
-//
-// Accuracy:
-//      according to an error analysis, the error is always less than
-//      1 ulp (unit in the last place).
-//
-// Misc. info.
-//      For IEEE double
-//          if x >  7.09782712893383973096e+02 then exp(x) overflow
-//          if x < -7.45133219101941108420e+02 then exp(x) underflow
-//
-// Constants:
-// The hexadecimal values are the intended ones for the following
-// constants. The decimal values may be used, provided that the
-// compiler will convert from decimal to binary accurately enough
-// to produce the hexadecimal values shown.
-
-// Exp returns e**x, the base-e exponential of x.
-//
-// Special cases are:
-//	Exp(+Inf) = +Inf
-//	Exp(NaN) = NaN
-// Very large values overflow to 0 or +Inf.
-// Very small values underflow to 1.
-func expGo(x float64) float64 {
-	const (
-		Ln2Hi = 6.93147180369123816490e-01
-		Ln2Lo = 1.90821492927058770002e-10
-		Log2e = 1.44269504088896338700e+00
-
-		Overflow  = 7.09782712893383973096e+02
-		Underflow = -7.45133219101941108420e+02
-		NearZero  = 1.0 / (1 << 28) // 2**-28
-	)
-
-	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
-	// when compiler does it for us
-	// special cases
-	switch {
-	case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
-		return x
-	case x < -MaxFloat64: // IsInf(x, -1):
-		return 0
-	case x > Overflow:
-		return Inf(1)
-	case x < Underflow:
-		return 0
-	case -NearZero < x && x < NearZero:
-		return 1 + x
-	}
-
-	// reduce; computed as r = hi - lo for extra precision.
-	var k int
-	switch {
-	case x < 0:
-		k = int(Log2e*x - 0.5)
-	case x > 0:
-		k = int(Log2e*x + 0.5)
-	}
-	hi := x - float64(k)*Ln2Hi
-	lo := float64(k) * Ln2Lo
-
-	// compute
-	return exp(hi, lo, k)
-}
-
-// Exp2 returns 2**x, the base-2 exponential of x.
-//
-// Special cases are the same as Exp.
-func exp2Go(x float64) float64 {
-	const (
-		Ln2Hi = 6.93147180369123816490e-01
-		Ln2Lo = 1.90821492927058770002e-10
-
-		Overflow  = 1.0239999999999999e+03
-		Underflow = -1.0740e+03
-	)
-
-	// TODO: remove manual inlining of IsNaN and IsInf
-	// when compiler does it for us
-	// special cases
-	switch {
-	case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
-		return x
-	case x < -MaxFloat64: // IsInf(x, -1):
-		return 0
-	case x > Overflow:
-		return Inf(1)
-	case x < Underflow:
-		return 0
-	}
-
-	// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
-	// computed as r = hi - lo for extra precision.
-	var k int
-	switch {
-	case x > 0:
-		k = int(x + 0.5)
-	case x < 0:
-		k = int(x - 0.5)
-	}
-	t := x - float64(k)
-	hi := t * Ln2Hi
-	lo := -t * Ln2Lo
-
-	// compute
-	return exp(hi, lo, k)
-}
-
-// exp returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
-func exp(hi, lo float64, k int) float64 {
-	const (
-		P1 = 1.66666666666666019037e-01  /* 0x3FC55555; 0x5555553E */
-		P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
-		P3 = 6.61375632143793436117e-05  /* 0x3F11566A; 0xAF25DE2C */
-		P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
-		P5 = 4.13813679705723846039e-08  /* 0x3E663769; 0x72BEA4D0 */
-	)
-
-	r := hi - lo
-	t := r * r
-	c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
-	y := 1 - ((lo - (r*c)/(2-c)) - hi)
-	// TODO(rsc): make sure Ldexp can handle boundary k
-	return Ldexp(y, k)
-}

src/pkg/math/exp_test.go

@@ -1,10 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-// Make expGo and exp2Go available for testing.
-
-func ExpGo(x float64) float64  { return expGo(x) }
-func Exp2Go(x float64) float64 { return exp2Go(x) }

src/pkg/math/expm1.go

@@ -121,7 +121,9 @@ package math
 //	Expm1(-Inf) = -1
 //	Expm1(NaN) = NaN
 // Very large values overflow to -1 or +Inf.
-func Expm1(x float64) float64 {
+func Expm1(x float64) float64
+
+func expm1(x float64) float64 {
 	const (
 		Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
 		Ln2X56     = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1

src/pkg/math/expm1_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Expm1(SB),7,$0
+	JMP ·expm1(SB)

src/pkg/math/expm1_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Expm1(SB),7,$0
+	B ·expm1(SB)

src/pkg/math/expm1_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Expm1(x float64) float64

src/pkg/math/export_test.go

@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+// Export internal functions for testing.
+var ExpGo = exp
+var Exp2Go = exp2
+var HypotSqrtGo = hypotSqrt
+var HypotNoSqrtGo = hypotNoSqrt
+var SqrtGo = sqrt

src/pkg/math/floor.go

@@ -10,7 +10,9 @@ package math
 //	Floor(±0) = ±0
 //	Floor(±Inf) = ±Inf
 //	Floor(NaN) = NaN
-func Floor(x float64) float64 {
+func Floor(x float64) float64
+
+func floor(x float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	if x == 0 || x != x || x > MaxFloat64 || x < -MaxFloat64 { // x == 0 || IsNaN(x) || IsInf(x, 0)
@@ -33,7 +35,11 @@ func Floor(x float64) float64 {
 //	Ceil(±0) = ±0
 //	Ceil(±Inf) = ±Inf
 //	Ceil(NaN) = NaN
-func Ceil(x float64) float64 { return -Floor(-x) }
+func Ceil(x float64) float64
+
+func ceil(x float64) float64 {
+	return -Floor(-x)
+}
 
 // Trunc returns the integer value of x.
 //
@@ -41,7 +47,9 @@ func Ceil(x float64) float64 { return -Floor(-x) }
 //	Trunc(±0) = ±0
 //	Trunc(±Inf) = ±Inf
 //	Trunc(NaN) = NaN
-func Trunc(x float64) float64 {
+func Trunc(x float64) float64
+
+func trunc(x float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	if x == 0 || x != x || x > MaxFloat64 || x < -MaxFloat64 { // x == 0 || IsNaN(x) || IsInf(x, 0)

src/pkg/math/floor_amd64.s

@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Floor(SB),7,$0
+	JMP	·floor(SB)
+
+TEXT ·Ceil(SB),7,$0
+	JMP	·ceil(SB)
+
+TEXT ·Trunc(SB),7,$0
+	JMP	·trunc(SB)

src/pkg/math/floor_arm.s

@@ -0,0 +1,12 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Floor(SB),7,$0
+	B	·floor(SB)
+
+TEXT ·Ceil(SB),7,$0
+	B	·ceil(SB)
+
+TEXT ·Trunc(SB),7,$0
+	B	·trunc(SB)

src/pkg/math/floor_decl.go

@@ -1,9 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Ceil(x float64) float64
-func Floor(x float64) float64
-func Trunc(x float64) float64

src/pkg/math/frexp.go

@@ -13,7 +13,9 @@ package math
 //	Frexp(±0) = ±0, 0
 //	Frexp(±Inf) = ±Inf, 0
 //	Frexp(NaN) = NaN, 0
-func Frexp(f float64) (frac float64, exp int) {
+func Frexp(f float64) (frac float64, exp int)
+
+func frexp(f float64) (frac float64, exp int) {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases

src/pkg/math/frexp_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Frexp(SB),7,$0
+	JMP ·frexp(SB)

src/pkg/math/frexp_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Frexp(SB),7,$0
+	B ·frexp(SB)

src/pkg/math/frexp_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Frexp(x float64) (f float64, e int)

src/pkg/math/hypot.go

@@ -14,7 +14,9 @@ package math
 // Special cases are:
 //	Hypot(p, q) = +Inf if p or q is infinite
 //	Hypot(p, q) = NaN if p or q is NaN
-func Hypot(p, q float64) float64 {
+func Hypot(p, q float64) float64
+
+func hypotSqrt(p, q float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases
@@ -39,3 +41,46 @@ func Hypot(p, q float64) float64 {
 	q = q / p
 	return p * Sqrt(1+q*q)
 }
+
+func hypotNoSqrt(p, q float64) float64 {
+	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
+	// when compiler does it for us
+	// special cases
+	switch {
+	case p < -MaxFloat64 || p > MaxFloat64 || q < -MaxFloat64 || q > MaxFloat64: // IsInf(p, 0) || IsInf(q, 0):
+		return Inf(1)
+	case p != p || q != q: // IsNaN(p) || IsNaN(q):
+		return NaN()
+	}
+	if p < 0 {
+		p = -p
+	}
+	if q < 0 {
+		q = -q
+	}
+
+	if p < q {
+		p, q = q, p
+	}
+
+	if p == 0 {
+		return 0
+	}
+
+	pfac := p
+	q = q / p
+	r := q
+	p = 1
+	for {
+		r = r * r
+		s := r + 4
+		if s == 4 {
+			return p * pfac
+		}
+		r = r / s
+		p = p + 2*r*p
+		q = q * r
+		r = q / p
+	}
+	panic("unreachable")
+}

src/pkg/math/hypot_arm.s

@@ -1,7 +1,6 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
+// Copyright 2011 The Go Authors.  All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Hypot(SB),7,$0
-
+	B ·hypotNoSqrt(SB)
-func Atan(x float64) float64

src/pkg/math/hypot_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Hypot(x, y float64) float64

src/pkg/math/hypot_port.go

@@ -1,63 +0,0 @@
-// Copyright 2009-2010 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-/*
-	Hypot -- sqrt(p*p + q*q), but overflows only if the result does.
-	See:
-		Cleve Moler and Donald Morrison,
-		Replacing Square Roots by Pythagorean Sums
-		IBM Journal of Research and Development,
-		Vol. 27, Number 6, pp. 577-581, Nov. 1983
-*/
-
-// Hypot computes Sqrt(p*p + q*q), taking care to avoid
-// unnecessary overflow and underflow.
-//
-// Special cases are:
-//	Hypot(p, q) = +Inf if p or q is infinite
-//	Hypot(p, q) = NaN if p or q is NaN
-func hypotGo(p, q float64) float64 {
-	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
-	// when compiler does it for us
-	// special cases
-	switch {
-	case p < -MaxFloat64 || p > MaxFloat64 || q < -MaxFloat64 || q > MaxFloat64: // IsInf(p, 0) || IsInf(q, 0):
-		return Inf(1)
-	case p != p || q != q: // IsNaN(p) || IsNaN(q):
-		return NaN()
-	}
-	if p < 0 {
-		p = -p
-	}
-	if q < 0 {
-		q = -q
-	}
-
-	if p < q {
-		p, q = q, p
-	}
-
-	if p == 0 {
-		return 0
-	}
-
-	pfac := p
-	q = q / p
-	r := q
-	p = 1
-	for {
-		r = r * r
-		s := r + 4
-		if s == 4 {
-			return p * pfac
-		}
-		r = r / s
-		p = p + 2*r*p
-		q = q * r
-		r = q / p
-	}
-	panic("unreachable")
-}

src/pkg/math/hypot_test.go

@@ -1,9 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-// Make hypotGo available for testing.
-
-func HypotGo(x, y float64) float64 { return hypotGo(x, y) }

src/pkg/math/ldexp.go

@@ -11,7 +11,9 @@ package math
 //	Ldexp(±0, exp) = ±0
 //	Ldexp(±Inf, exp) = ±Inf
 //	Ldexp(NaN, exp) = NaN
-func Ldexp(frac float64, exp int) float64 {
+func Ldexp(frac float64, exp int) float64
+
+func ldexp(frac float64, exp int) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
 	// special cases

src/pkg/math/ldexp_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Ldexp(SB),7,$0
+	JMP ·ldexp(SB)

src/pkg/math/ldexp_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Ldexp(SB),7,$0
+	B ·ldexp(SB)

src/pkg/math/ldexp_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Ldexp(f float64, e int) float64

src/pkg/math/log.go

@@ -77,7 +77,9 @@ package math
 //	Log(0) = -Inf
 //	Log(x < 0) = NaN
 //	Log(NaN) = NaN
-func Log(x float64) float64 {
+func Log(x float64) float64
+
+func log(x float64) float64 {
 	const (
 		Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
 		Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */

src/pkg/math/log10.go

@@ -6,8 +6,16 @@ package math
 
 // Log10 returns the decimal logarithm of x.
 // The special cases are the same as for Log.
-func Log10(x float64) float64 { return Log(x) * (1 / Ln10) }
+func Log10(x float64) float64
+
+func log10(x float64) float64 {
+	return Log(x) * (1 / Ln10)
+}
 
 // Log2 returns the binary logarithm of x.
 // The special cases are the same as for Log.
-func Log2(x float64) float64 { return Log(x) * (1 / Ln2) }
+func Log2(x float64) float64
+
+func log2(x float64) float64 {
+	return Log(x) * (1 / Ln2)
+}

src/pkg/math/log10_amd64.s

@@ -0,0 +1,9 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Log10(SB),7,$0
+	JMP ·log10(SB)
+
+TEXT ·Log2(SB),7,$0
+	JMP ·log2(SB)

src/pkg/math/log10_arm.s

@@ -0,0 +1,9 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Log10(SB),7,$0
+	B ·log10(SB)
+
+TEXT ·Log2(SB),7,$0
+	B ·log2(SB)

src/pkg/math/log10_decl.go

@@ -1,8 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Log10(x float64) float64
-func Log2(x float64) float64

src/pkg/math/log1p.go

@@ -92,7 +92,9 @@ package math
 //	Log1p(-1) = -Inf
 //	Log1p(x < -1) = NaN
 //	Log1p(NaN) = NaN
-func Log1p(x float64) float64 {
+func Log1p(x float64) float64
+
+func log1p(x float64) float64 {
 	const (
 		Sqrt2M1     = 4.142135623730950488017e-01  // Sqrt(2)-1 = 0x3fda827999fcef34
 		Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866

src/pkg/math/log1p_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Log1p(SB),7,$0
+	JMP ·log1p(SB)

src/pkg/math/log1p_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Log1p(SB),7,$0
+	B ·log1p(SB)

src/pkg/math/log1p_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Log1p(x float64) float64

src/pkg/math/log_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Log(SB),7,$0
+	B ·log(SB)

src/pkg/math/mod.go

@@ -18,7 +18,9 @@ package math
 //	Mod(x, 0) = NaN
 //	Mod(x, ±Inf) = x
 //	Mod(x, NaN) = NaN
-func Mod(x, y float64) float64 {
+func Mod(x, y float64) float64
+
+func mod(x, y float64) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us.
 	if y == 0 || x > MaxFloat64 || x < -MaxFloat64 || x != x || y != y { // y == 0 || IsInf(x, 0) || IsNaN(x) || IsNan(y)

src/pkg/math/mod_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Mod(SB),7,$0
+	JMP ·mod(SB)

src/pkg/math/mod_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Mod(SB),7,$0
+	B ·mod(SB)

src/pkg/math/mod_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Mod(x, y float64) float64

src/pkg/math/modf.go

@@ -10,7 +10,9 @@ package math
 // Special cases are:
 //	Modf(±Inf) = ±Inf, NaN
 //	Modf(NaN) = NaN, NaN
-func Modf(f float64) (int float64, frac float64) {
+func Modf(f float64) (int float64, frac float64)
+
+func modf(f float64) (int float64, frac float64) {
 	if f < 1 {
 		if f < 0 {
 			int, frac = Modf(-f)

src/pkg/math/modf_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Modf(SB),7,$0
+	JMP ·modf(SB)

src/pkg/math/modf_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Modf(SB),7,$0
+	B ·modf(SB)

src/pkg/math/modf_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Modf(f float64) (int float64, frac float64)

src/pkg/math/remainder.go

@@ -34,7 +34,9 @@ package math
 //	Remainder(x, 0) = NaN
 //	Remainder(x, ±Inf) = x
 //	Remainder(x, NaN) = NaN
-func Remainder(x, y float64) float64 {
+func Remainder(x, y float64) float64
+
+func remainder(x, y float64) float64 {
 	const (
 		Tiny    = 4.45014771701440276618e-308 // 0x0020000000000000
 		HalfMax = MaxFloat64 / 2

src/pkg/math/remainder_amd64.s

@@ -1,7 +1,6 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
+// Copyright 2011 The Go Authors.  All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Remainder(SB),7,$0
-
+	JMP ·remainder(SB)
-func Atan2(y, x float64) float64

src/pkg/math/remainder_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Remainder(SB),7,$0
+	B ·remainder(SB)

src/pkg/math/remainder_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Remainder(x, y float64) float64

src/pkg/math/sin.go

@@ -113,7 +113,9 @@ var _cos = [...]float64{
 // Special cases are:
 //	Cos(±Inf) = NaN
 //	Cos(NaN) = NaN
-func Cos(x float64) float64 {
+func Cos(x float64) float64
+
+func cos(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 		PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
@@ -170,7 +172,9 @@ func Cos(x float64) float64 {
 //	Sin(±0) = ±0
 //	Sin(±Inf) = NaN
 //	Sin(NaN) = NaN
-func Sin(x float64) float64 {
+func Sin(x float64) float64
+
+func sin(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 		PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,

src/pkg/math/sin_amd64.s

@@ -2,6 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Sin(SB),7,$0
+	JMP	·sin(SB)
 
-func Exp2(x float64) float64
+TEXT ·Cos(SB),7,$0
+	JMP	·cos(SB)

src/pkg/math/sin_arm.s

@@ -2,6 +2,8 @@
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
-package math
+TEXT ·Sin(SB),7,$0
+	B	·sin(SB)
 
-func Abs(x float64) float64
+TEXT ·Cos(SB),7,$0
+	B	·cos(SB)

src/pkg/math/sin_decl.go

@@ -1,8 +0,0 @@
-// Copyright 2009 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Cos(x float64) float64
-func Sin(x float64) float64

src/pkg/math/sincos.go

@@ -12,7 +12,9 @@ package math
 //	Sincos(±0) = ±0, 1
 //	Sincos(±Inf) = NaN, NaN
 //	Sincos(NaN) = NaN, NaN
-func Sincos(x float64) (sin, cos float64) {
+func Sincos(x float64) (sin, cos float64)
+
+func sincos(x float64) (sin, cos float64) {
 	const (
 		PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 		PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,

src/pkg/math/sincos_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Sincos(SB),7,$0
+	B ·sincos(SB)

src/pkg/math/sincos_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Sincos(x float64) (sin, cos float64)

src/pkg/math/sqrt.go

@@ -11,4 +11,142 @@ package math
 //	Sqrt(±0) = ±0
 //	Sqrt(x < 0) = NaN
 //	Sqrt(NaN) = NaN
-func Sqrt(x float64) float64 { return sqrtGo(x) }
+func Sqrt(x float64) float64
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
+// came with this notice.  The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_sqrt(x)
+// Return correctly rounded sqrt.
+//           -----------------------------------------
+//           | Use the hardware sqrt if you have one |
+//           -----------------------------------------
+// Method:
+//   Bit by bit method using integer arithmetic. (Slow, but portable)
+//   1. Normalization
+//      Scale x to y in [1,4) with even powers of 2:
+//      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
+//              sqrt(x) = 2**k * sqrt(y)
+//   2. Bit by bit computation
+//      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
+//           i                                                   0
+//                                     i+1         2
+//          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
+//           i      i            i                 i
+//
+//      To compute q    from q , one checks whether
+//                  i+1       i
+//
+//                            -(i+1) 2
+//                      (q + 2      )  <= y.                     (2)
+//                        i
+//                                                            -(i+1)
+//      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
+//                             i+1   i             i+1   i
+//
+//      With some algebraic manipulation, it is not difficult to see
+//      that (2) is equivalent to
+//                             -(i+1)
+//                      s  +  2       <= y                       (3)
+//                       i                i
+//
+//      The advantage of (3) is that s  and y  can be computed by
+//                                    i      i
+//      the following recurrence formula:
+//          if (3) is false
+//
+//          s     =  s  ,       y    = y   ;                     (4)
+//           i+1      i          i+1    i
+//
+//      otherwise,
+//                         -i                      -(i+1)
+//          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
+//           i+1      i          i+1    i     i
+//
+//      One may easily use induction to prove (4) and (5).
+//      Note. Since the left hand side of (3) contain only i+2 bits,
+//            it does not necessary to do a full (53-bit) comparison
+//            in (3).
+//   3. Final rounding
+//      After generating the 53 bits result, we compute one more bit.
+//      Together with the remainder, we can decide whether the
+//      result is exact, bigger than 1/2ulp, or less than 1/2ulp
+//      (it will never equal to 1/2ulp).
+//      The rounding mode can be detected by checking whether
+//      huge + tiny is equal to huge, and whether huge - tiny is
+//      equal to huge for some floating point number "huge" and "tiny".
+//
+//
+// Notes:  Rounding mode detection omitted.  The constants "mask", "shift",
+// and "bias" are found in src/pkg/math/bits.go
+
+// Sqrt returns the square root of x.
+//
+// Special cases are:
+//	Sqrt(+Inf) = +Inf
+//	Sqrt(±0) = ±0
+//	Sqrt(x < 0) = NaN
+//	Sqrt(NaN) = NaN
+func sqrt(x float64) float64 {
+	// special cases
+	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
+	// when compiler does it for us
+	switch {
+	case x == 0 || x != x || x > MaxFloat64: // x == 0 || IsNaN(x) || IsInf(x, 1):
+		return x
+	case x < 0:
+		return NaN()
+	}
+	ix := Float64bits(x)
+	// normalize x
+	exp := int((ix >> shift) & mask)
+	if exp == 0 { // subnormal x
+		for ix&1<<shift == 0 {
+			ix <<= 1
+			exp--
+		}
+		exp++
+	}
+	exp -= bias // unbias exponent
+	ix &^= mask << shift
+	ix |= 1 << shift
+	if exp&1 == 1 { // odd exp, double x to make it even
+		ix <<= 1
+	}
+	exp >>= 1 // exp = exp/2, exponent of square root
+	// generate sqrt(x) bit by bit
+	ix <<= 1
+	var q, s uint64               // q = sqrt(x)
+	r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
+	for r != 0 {
+		t := s + r
+		if t <= ix {
+			s = t + r
+			ix -= t
+			q += r
+		}
+		ix <<= 1
+		r >>= 1
+	}
+	// final rounding
+	if ix != 0 { // remainder, result not exact
+		q += q & 1 // round according to extra bit
+	}
+	ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
+	return Float64frombits(ix)
+}
+
+func sqrtC(f float64, r *float64) {
+	*r = sqrt(f)
+}

src/pkg/math/sqrt_decl.go

@@ -1,7 +0,0 @@
-// Copyright 2009 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-func Sqrt(x float64) float64

src/pkg/math/sqrt_port.go

@@ -1,147 +0,0 @@
-// Copyright 2009 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-/*
-	Floating-point square root.
-*/
-
-// The original C code and the long comment below are
-// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
-// came with this notice.  The go code is a simplified
-// version of the original C.
-//
-// ====================================================
-// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
-//
-// Developed at SunPro, a Sun Microsystems, Inc. business.
-// Permission to use, copy, modify, and distribute this
-// software is freely granted, provided that this notice
-// is preserved.
-// ====================================================
-//
-// __ieee754_sqrt(x)
-// Return correctly rounded sqrt.
-//           -----------------------------------------
-//           | Use the hardware sqrt if you have one |
-//           -----------------------------------------
-// Method:
-//   Bit by bit method using integer arithmetic. (Slow, but portable)
-//   1. Normalization
-//      Scale x to y in [1,4) with even powers of 2:
-//      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
-//              sqrt(x) = 2**k * sqrt(y)
-//   2. Bit by bit computation
-//      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
-//           i                                                   0
-//                                     i+1         2
-//          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
-//           i      i            i                 i
-//
-//      To compute q    from q , one checks whether
-//                  i+1       i
-//
-//                            -(i+1) 2
-//                      (q + 2      )  <= y.                     (2)
-//                        i
-//                                                            -(i+1)
-//      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
-//                             i+1   i             i+1   i
-//
-//      With some algebraic manipulation, it is not difficult to see
-//      that (2) is equivalent to
-//                             -(i+1)
-//                      s  +  2       <= y                       (3)
-//                       i                i
-//
-//      The advantage of (3) is that s  and y  can be computed by
-//                                    i      i
-//      the following recurrence formula:
-//          if (3) is false
-//
-//          s     =  s  ,       y    = y   ;                     (4)
-//           i+1      i          i+1    i
-//
-//      otherwise,
-//                         -i                      -(i+1)
-//          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
-//           i+1      i          i+1    i     i
-//
-//      One may easily use induction to prove (4) and (5).
-//      Note. Since the left hand side of (3) contain only i+2 bits,
-//            it does not necessary to do a full (53-bit) comparison
-//            in (3).
-//   3. Final rounding
-//      After generating the 53 bits result, we compute one more bit.
-//      Together with the remainder, we can decide whether the
-//      result is exact, bigger than 1/2ulp, or less than 1/2ulp
-//      (it will never equal to 1/2ulp).
-//      The rounding mode can be detected by checking whether
-//      huge + tiny is equal to huge, and whether huge - tiny is
-//      equal to huge for some floating point number "huge" and "tiny".
-//
-//
-// Notes:  Rounding mode detection omitted.  The constants "mask", "shift",
-// and "bias" are found in src/pkg/math/bits.go
-
-// Sqrt returns the square root of x.
-//
-// Special cases are:
-//	Sqrt(+Inf) = +Inf
-//	Sqrt(±0) = ±0
-//	Sqrt(x < 0) = NaN
-//	Sqrt(NaN) = NaN
-func sqrtGo(x float64) float64 {
-	// special cases
-	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
-	// when compiler does it for us
-	switch {
-	case x == 0 || x != x || x > MaxFloat64: // x == 0 || IsNaN(x) || IsInf(x, 1):
-		return x
-	case x < 0:
-		return NaN()
-	}
-	ix := Float64bits(x)
-	// normalize x
-	exp := int((ix >> shift) & mask)
-	if exp == 0 { // subnormal x
-		for ix&1<<shift == 0 {
-			ix <<= 1
-			exp--
-		}
-		exp++
-	}
-	exp -= bias // unbias exponent
-	ix &^= mask << shift
-	ix |= 1 << shift
-	if exp&1 == 1 { // odd exp, double x to make it even
-		ix <<= 1
-	}
-	exp >>= 1 // exp = exp/2, exponent of square root
-	// generate sqrt(x) bit by bit
-	ix <<= 1
-	var q, s uint64               // q = sqrt(x)
-	r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
-	for r != 0 {
-		t := s + r
-		if t <= ix {
-			s = t + r
-			ix -= t
-			q += r
-		}
-		ix <<= 1
-		r >>= 1
-	}
-	// final rounding
-	if ix != 0 { // remainder, result not exact
-		q += q & 1 // round according to extra bit
-	}
-	ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
-	return Float64frombits(ix)
-}
-
-func sqrtGoC(f float64, r *float64) {
-	*r = sqrtGo(f)
-}

src/pkg/math/sqrt_test.go

@@ -1,9 +0,0 @@
-// Copyright 2010 The Go Authors.  All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package math
-
-// Make sqrtGo available for testing.
-
-func SqrtGo(x float64) float64 { return sqrtGo(x) }

src/pkg/math/tan.go

@@ -79,7 +79,9 @@ var _tanQ = [...]float64{
 //	Tan(±0) = ±0
 //	Tan(±Inf) = NaN
 //	Tan(NaN) = NaN
-func Tan(x float64) float64 {
+func Tan(x float64) float64
+
+func tan(x float64) float64 {
 	const (
 		PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 		PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,

src/pkg/math/tan_amd64.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Tan(SB),7,$0
+	JMP ·tan(SB)

src/pkg/math/tan_arm.s

@@ -0,0 +1,6 @@
+// Copyright 2011 The Go Authors.  All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+TEXT ·Tan(SB),7,$0
+	B ·tan(SB)

src/pkg/runtime/arm/softfloat.c

@@ -15,7 +15,7 @@
 #define FLAGS_V (1 << 28)
 
 void	runtime·abort(void);
-void	math·sqrtGoC(uint64, uint64*);
+void	math·sqrtC(uint64, uint64*);
 
 static	uint32	trace = 0;
 
@@ -359,7 +359,7 @@ stage3:	// regd, regm are 4bit variables
 		break;
 
 	case 0xeeb10bc0:	// D[regd] = sqrt D[regm]
-		math·sqrtGoC(getd(regm), &uval);
+		math·sqrtC(getd(regm), &uval);
 		putd(regd, uval);
 
 		if(trace)