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 // Copyright 2016 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 vector // This file contains a floating point math implementation of the vector // graphics rasterizer. import ( "math" ) func floatingMax(x, y float32) float32 { if x > y { return x } return y } func floatingMin(x, y float32) float32 { if x < y { return x } return y } func floatingFloor(x float32) int32 { return int32(math.Floor(float64(x))) } func floatingCeil(x float32) int32 { return int32(math.Ceil(float64(x))) } func (z *Rasterizer) floatingLineTo(bx, by float32) { ax, ay := z.penX, z.penY z.penX, z.penY = bx, by dir := float32(1) if ay > by { dir, ax, ay, bx, by = -1, bx, by, ax, ay } // Horizontal line segments yield no change in coverage. Almost horizontal // segments would yield some change, in ideal math, but the computation // further below, involving 1 / (by - ay), is unstable in floating point // math, so we treat the segment as if it was perfectly horizontal. if by-ay <= 0.000001 { return } dxdy := (bx - ax) / (by - ay) x := ax y := floatingFloor(ay) yMax := floatingCeil(by) if yMax > int32(z.size.Y) { yMax = int32(z.size.Y) } width := int32(z.size.X) for ; y < yMax; y++ { dy := floatingMin(float32(y+1), by) - floatingMax(float32(y), ay) // The "float32" in expressions like "float32(foo*bar)" here and below // look redundant, since foo and bar already have type float32, but are // explicit in order to disable the compiler's Fused Multiply Add (FMA) // instruction selection, which can improve performance but can result // in different rounding errors in floating point computations. // // This package aims to have bit-exact identical results across all // GOARCHes, and across pure Go code and assembly, so it disables FMA. // // See the discussion at // https://groups.google.com/d/topic/golang-dev/Sti0bl2xUXQ/discussion xNext := x + float32(dy*dxdy) if y < 0 { x = xNext continue } buf := z.bufF32[y*width:] d := float32(dy * dir) x0, x1 := x, xNext if x > xNext { x0, x1 = x1, x0 } x0i := floatingFloor(x0) x0Floor := float32(x0i) x1i := floatingCeil(x1) x1Ceil := float32(x1i) if x1i <= x0i+1 { xmf := float32(0.5*(x+xNext)) - x0Floor if i := clamp(x0i+0, width); i < uint(len(buf)) { buf[i] += d - float32(d*xmf) } if i := clamp(x0i+1, width); i < uint(len(buf)) { buf[i] += float32(d * xmf) } } else { s := 1 / (x1 - x0) x0f := x0 - x0Floor oneMinusX0f := 1 - x0f a0 := float32(0.5 * s * oneMinusX0f * oneMinusX0f) x1f := x1 - x1Ceil + 1 am := float32(0.5 * s * x1f * x1f) if i := clamp(x0i, width); i < uint(len(buf)) { buf[i] += float32(d * a0) } if x1i == x0i+2 { if i := clamp(x0i+1, width); i < uint(len(buf)) { buf[i] += float32(d * (1 - a0 - am)) } } else { a1 := float32(s * (1.5 - x0f)) if i := clamp(x0i+1, width); i < uint(len(buf)) { buf[i] += float32(d * (a1 - a0)) } dTimesS := float32(d * s) for xi := x0i + 2; xi < x1i-1; xi++ { if i := clamp(xi, width); i < uint(len(buf)) { buf[i] += dTimesS } } a2 := a1 + float32(s*float32(x1i-x0i-3)) if i := clamp(x1i-1, width); i < uint(len(buf)) { buf[i] += float32(d * (1 - a2 - am)) } } if i := clamp(x1i, width); i < uint(len(buf)) { buf[i] += float32(d * am) } } x = xNext } } const ( // almost256 scales a floating point value in the range [0, 1] to a uint8 // value in the range [0x00, 0xff]. // // 255 is too small. Floating point math accumulates rounding errors, so a // fully covered src value that would in ideal math be float32(1) might be // float32(1-ε), and uint8(255 * (1-ε)) would be 0xfe instead of 0xff. The // uint8 conversion rounds to zero, not to nearest. // // 256 is too big. If we multiplied by 256, below, then a fully covered src // value of float32(1) would translate to uint8(256 * 1), which can be 0x00 // instead of the maximal value 0xff. // // math.Float32bits(almost256) is 0x437fffff. almost256 = 255.99998 // almost65536 scales a floating point value in the range [0, 1] to a // uint16 value in the range [0x0000, 0xffff]. // // math.Float32bits(almost65536) is 0x477fffff. almost65536 = almost256 * 256 ) func floatingAccumulateOpOver(dst []uint8, src []float32) { // Sanity check that len(dst) >= len(src). if len(dst) < len(src) { return } acc := float32(0) for i, v := range src { acc += v a := acc if a < 0 { a = -a } if a > 1 { a = 1 } // This algorithm comes from the standard library's image/draw package. dstA := uint32(dst[i]) * 0x101 maskA := uint32(almost65536 * a) outA := dstA*(0xffff-maskA)/0xffff + maskA dst[i] = uint8(outA >> 8) } } func floatingAccumulateOpSrc(dst []uint8, src []float32) { // Sanity check that len(dst) >= len(src). if len(dst) < len(src) { return } acc := float32(0) for i, v := range src { acc += v a := acc if a < 0 { a = -a } if a > 1 { a = 1 } dst[i] = uint8(almost256 * a) } } func floatingAccumulateMask(dst []uint32, src []float32) { // Sanity check that len(dst) >= len(src). if len(dst) < len(src) { return } acc := float32(0) for i, v := range src { acc += v a := acc if a < 0 { a = -a } if a > 1 { a = 1 } dst[i] = uint32(almost65536 * a) } }