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diff --git a/vendor/golang.org/x/image/math/fixed/fixed.go b/vendor/golang.org/x/image/math/fixed/fixed.go
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+++ b/vendor/golang.org/x/image/math/fixed/fixed.go
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+// Copyright 2015 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 fixed implements fixed-point integer types.
+package fixed // import "golang.org/x/image/math/fixed"
+
+import (
+ "fmt"
+)
+
+// TODO: implement fmt.Formatter for %f and %g.
+
+// I returns the integer value i as an Int26_6.
+//
+// For example, passing the integer value 2 yields Int26_6(128).
+func I(i int) Int26_6 {
+ return Int26_6(i << 6)
+}
+
+// Int26_6 is a signed 26.6 fixed-point number.
+//
+// The integer part ranges from -33554432 to 33554431, inclusive. The
+// fractional part has 6 bits of precision.
+//
+// For example, the number one-and-a-quarter is Int26_6(1<<6 + 1<<4).
+type Int26_6 int32
+
+// String returns a human-readable representation of a 26.6 fixed-point number.
+//
+// For example, the number one-and-a-quarter becomes "1:16".
+func (x Int26_6) String() string {
+ const shift, mask = 6, 1<<6 - 1
+ if x >= 0 {
+ return fmt.Sprintf("%d:%02d", int32(x>>shift), int32(x&mask))
+ }
+ x = -x
+ if x >= 0 {
+ return fmt.Sprintf("-%d:%02d", int32(x>>shift), int32(x&mask))
+ }
+ return "-33554432:00" // The minimum value is -(1<<25).
+}
+
+// Floor returns the greatest integer value less than or equal to x.
+//
+// Its return type is int, not Int26_6.
+func (x Int26_6) Floor() int { return int((x + 0x00) >> 6) }
+
+// Round returns the nearest integer value to x. Ties are rounded up.
+//
+// Its return type is int, not Int26_6.
+func (x Int26_6) Round() int { return int((x + 0x20) >> 6) }
+
+// Ceil returns the least integer value greater than or equal to x.
+//
+// Its return type is int, not Int26_6.
+func (x Int26_6) Ceil() int { return int((x + 0x3f) >> 6) }
+
+// Mul returns x*y in 26.6 fixed-point arithmetic.
+func (x Int26_6) Mul(y Int26_6) Int26_6 {
+ return Int26_6((int64(x)*int64(y) + 1<<5) >> 6)
+}
+
+// Int52_12 is a signed 52.12 fixed-point number.
+//
+// The integer part ranges from -2251799813685248 to 2251799813685247,
+// inclusive. The fractional part has 12 bits of precision.
+//
+// For example, the number one-and-a-quarter is Int52_12(1<<12 + 1<<10).
+type Int52_12 int64
+
+// String returns a human-readable representation of a 52.12 fixed-point
+// number.
+//
+// For example, the number one-and-a-quarter becomes "1:1024".
+func (x Int52_12) String() string {
+ const shift, mask = 12, 1<<12 - 1
+ if x >= 0 {
+ return fmt.Sprintf("%d:%04d", int64(x>>shift), int64(x&mask))
+ }
+ x = -x
+ if x >= 0 {
+ return fmt.Sprintf("-%d:%04d", int64(x>>shift), int64(x&mask))
+ }
+ return "-2251799813685248:0000" // The minimum value is -(1<<51).
+}
+
+// Floor returns the greatest integer value less than or equal to x.
+//
+// Its return type is int, not Int52_12.
+func (x Int52_12) Floor() int { return int((x + 0x000) >> 12) }
+
+// Round returns the nearest integer value to x. Ties are rounded up.
+//
+// Its return type is int, not Int52_12.
+func (x Int52_12) Round() int { return int((x + 0x800) >> 12) }
+
+// Ceil returns the least integer value greater than or equal to x.
+//
+// Its return type is int, not Int52_12.
+func (x Int52_12) Ceil() int { return int((x + 0xfff) >> 12) }
+
+// Mul returns x*y in 52.12 fixed-point arithmetic.
+func (x Int52_12) Mul(y Int52_12) Int52_12 {
+ const M, N = 52, 12
+ lo, hi := muli64(int64(x), int64(y))
+ ret := Int52_12(hi<<M | lo>>N)
+ ret += Int52_12((lo >> (N - 1)) & 1) // Round to nearest, instead of rounding down.
+ return ret
+}
+
+// muli64 multiplies two int64 values, returning the 128-bit signed integer
+// result as two uint64 values.
+//
+// This implementation is similar to $GOROOT/src/runtime/softfloat64.go's mullu
+// function, which is in turn adapted from Hacker's Delight.
+func muli64(u, v int64) (lo, hi uint64) {
+ const (
+ s = 32
+ mask = 1<<s - 1
+ )
+
+ u1 := uint64(u >> s)
+ u0 := uint64(u & mask)
+ v1 := uint64(v >> s)
+ v0 := uint64(v & mask)
+
+ w0 := u0 * v0
+ t := u1*v0 + w0>>s
+ w1 := t & mask
+ w2 := uint64(int64(t) >> s)
+ w1 += u0 * v1
+ return uint64(u) * uint64(v), u1*v1 + w2 + uint64(int64(w1)>>s)
+}
+
+// P returns the integer values x and y as a Point26_6.
+//
+// For example, passing the integer values (2, -3) yields Point26_6{128, -192}.
+func P(x, y int) Point26_6 {
+ return Point26_6{Int26_6(x << 6), Int26_6(y << 6)}
+}
+
+// Point26_6 is a 26.6 fixed-point coordinate pair.
+//
+// It is analogous to the image.Point type in the standard library.
+type Point26_6 struct {
+ X, Y Int26_6
+}
+
+// Add returns the vector p+q.
+func (p Point26_6) Add(q Point26_6) Point26_6 {
+ return Point26_6{p.X + q.X, p.Y + q.Y}
+}
+
+// Sub returns the vector p-q.
+func (p Point26_6) Sub(q Point26_6) Point26_6 {
+ return Point26_6{p.X - q.X, p.Y - q.Y}
+}
+
+// Mul returns the vector p*k.
+func (p Point26_6) Mul(k Int26_6) Point26_6 {
+ return Point26_6{p.X * k / 64, p.Y * k / 64}
+}
+
+// Div returns the vector p/k.
+func (p Point26_6) Div(k Int26_6) Point26_6 {
+ return Point26_6{p.X * 64 / k, p.Y * 64 / k}
+}
+
+// In returns whether p is in r.
+func (p Point26_6) In(r Rectangle26_6) bool {
+ return r.Min.X <= p.X && p.X < r.Max.X && r.Min.Y <= p.Y && p.Y < r.Max.Y
+}
+
+// Point52_12 is a 52.12 fixed-point coordinate pair.
+//
+// It is analogous to the image.Point type in the standard library.
+type Point52_12 struct {
+ X, Y Int52_12
+}
+
+// Add returns the vector p+q.
+func (p Point52_12) Add(q Point52_12) Point52_12 {
+ return Point52_12{p.X + q.X, p.Y + q.Y}
+}
+
+// Sub returns the vector p-q.
+func (p Point52_12) Sub(q Point52_12) Point52_12 {
+ return Point52_12{p.X - q.X, p.Y - q.Y}
+}
+
+// Mul returns the vector p*k.
+func (p Point52_12) Mul(k Int52_12) Point52_12 {
+ return Point52_12{p.X * k / 4096, p.Y * k / 4096}
+}
+
+// Div returns the vector p/k.
+func (p Point52_12) Div(k Int52_12) Point52_12 {
+ return Point52_12{p.X * 4096 / k, p.Y * 4096 / k}
+}
+
+// In returns whether p is in r.
+func (p Point52_12) In(r Rectangle52_12) bool {
+ return r.Min.X <= p.X && p.X < r.Max.X && r.Min.Y <= p.Y && p.Y < r.Max.Y
+}
+
+// R returns the integer values minX, minY, maxX, maxY as a Rectangle26_6.
+//
+// For example, passing the integer values (0, 1, 2, 3) yields
+// Rectangle26_6{Point26_6{0, 64}, Point26_6{128, 192}}.
+//
+// Like the image.Rect function in the standard library, the returned rectangle
+// has minimum and maximum coordinates swapped if necessary so that it is
+// well-formed.
+func R(minX, minY, maxX, maxY int) Rectangle26_6 {
+ if minX > maxX {
+ minX, maxX = maxX, minX
+ }
+ if minY > maxY {
+ minY, maxY = maxY, minY
+ }
+ return Rectangle26_6{
+ Point26_6{
+ Int26_6(minX << 6),
+ Int26_6(minY << 6),
+ },
+ Point26_6{
+ Int26_6(maxX << 6),
+ Int26_6(maxY << 6),
+ },
+ }
+}
+
+// Rectangle26_6 is a 26.6 fixed-point coordinate rectangle. The Min bound is
+// inclusive and the Max bound is exclusive. It is well-formed if Min.X <=
+// Max.X and likewise for Y.
+//
+// It is analogous to the image.Rectangle type in the standard library.
+type Rectangle26_6 struct {
+ Min, Max Point26_6
+}
+
+// Add returns the rectangle r translated by p.
+func (r Rectangle26_6) Add(p Point26_6) Rectangle26_6 {
+ return Rectangle26_6{
+ Point26_6{r.Min.X + p.X, r.Min.Y + p.Y},
+ Point26_6{r.Max.X + p.X, r.Max.Y + p.Y},
+ }
+}
+
+// Sub returns the rectangle r translated by -p.
+func (r Rectangle26_6) Sub(p Point26_6) Rectangle26_6 {
+ return Rectangle26_6{
+ Point26_6{r.Min.X - p.X, r.Min.Y - p.Y},
+ Point26_6{r.Max.X - p.X, r.Max.Y - p.Y},
+ }
+}
+
+// Intersect returns the largest rectangle contained by both r and s. If the
+// two rectangles do not overlap then the zero rectangle will be returned.
+func (r Rectangle26_6) Intersect(s Rectangle26_6) Rectangle26_6 {
+ if r.Min.X < s.Min.X {
+ r.Min.X = s.Min.X
+ }
+ if r.Min.Y < s.Min.Y {
+ r.Min.Y = s.Min.Y
+ }
+ if r.Max.X > s.Max.X {
+ r.Max.X = s.Max.X
+ }
+ if r.Max.Y > s.Max.Y {
+ r.Max.Y = s.Max.Y
+ }
+ // Letting r0 and s0 be the values of r and s at the time that the method
+ // is called, this next line is equivalent to:
+ //
+ // if max(r0.Min.X, s0.Min.X) >= min(r0.Max.X, s0.Max.X) || likewiseForY { etc }
+ if r.Empty() {
+ return Rectangle26_6{}
+ }
+ return r
+}
+
+// Union returns the smallest rectangle that contains both r and s.
+func (r Rectangle26_6) Union(s Rectangle26_6) Rectangle26_6 {
+ if r.Empty() {
+ return s
+ }
+ if s.Empty() {
+ return r
+ }
+ if r.Min.X > s.Min.X {
+ r.Min.X = s.Min.X
+ }
+ if r.Min.Y > s.Min.Y {
+ r.Min.Y = s.Min.Y
+ }
+ if r.Max.X < s.Max.X {
+ r.Max.X = s.Max.X
+ }
+ if r.Max.Y < s.Max.Y {
+ r.Max.Y = s.Max.Y
+ }
+ return r
+}
+
+// Empty returns whether the rectangle contains no points.
+func (r Rectangle26_6) Empty() bool {
+ return r.Min.X >= r.Max.X || r.Min.Y >= r.Max.Y
+}
+
+// In returns whether every point in r is in s.
+func (r Rectangle26_6) In(s Rectangle26_6) bool {
+ if r.Empty() {
+ return true
+ }
+ // Note that r.Max is an exclusive bound for r, so that r.In(s)
+ // does not require that r.Max.In(s).
+ return s.Min.X <= r.Min.X && r.Max.X <= s.Max.X &&
+ s.Min.Y <= r.Min.Y && r.Max.Y <= s.Max.Y
+}
+
+// Rectangle52_12 is a 52.12 fixed-point coordinate rectangle. The Min bound is
+// inclusive and the Max bound is exclusive. It is well-formed if Min.X <=
+// Max.X and likewise for Y.
+//
+// It is analogous to the image.Rectangle type in the standard library.
+type Rectangle52_12 struct {
+ Min, Max Point52_12
+}
+
+// Add returns the rectangle r translated by p.
+func (r Rectangle52_12) Add(p Point52_12) Rectangle52_12 {
+ return Rectangle52_12{
+ Point52_12{r.Min.X + p.X, r.Min.Y + p.Y},
+ Point52_12{r.Max.X + p.X, r.Max.Y + p.Y},
+ }
+}
+
+// Sub returns the rectangle r translated by -p.
+func (r Rectangle52_12) Sub(p Point52_12) Rectangle52_12 {
+ return Rectangle52_12{
+ Point52_12{r.Min.X - p.X, r.Min.Y - p.Y},
+ Point52_12{r.Max.X - p.X, r.Max.Y - p.Y},
+ }
+}
+
+// Intersect returns the largest rectangle contained by both r and s. If the
+// two rectangles do not overlap then the zero rectangle will be returned.
+func (r Rectangle52_12) Intersect(s Rectangle52_12) Rectangle52_12 {
+ if r.Min.X < s.Min.X {
+ r.Min.X = s.Min.X
+ }
+ if r.Min.Y < s.Min.Y {
+ r.Min.Y = s.Min.Y
+ }
+ if r.Max.X > s.Max.X {
+ r.Max.X = s.Max.X
+ }
+ if r.Max.Y > s.Max.Y {
+ r.Max.Y = s.Max.Y
+ }
+ // Letting r0 and s0 be the values of r and s at the time that the method
+ // is called, this next line is equivalent to:
+ //
+ // if max(r0.Min.X, s0.Min.X) >= min(r0.Max.X, s0.Max.X) || likewiseForY { etc }
+ if r.Empty() {
+ return Rectangle52_12{}
+ }
+ return r
+}
+
+// Union returns the smallest rectangle that contains both r and s.
+func (r Rectangle52_12) Union(s Rectangle52_12) Rectangle52_12 {
+ if r.Empty() {
+ return s
+ }
+ if s.Empty() {
+ return r
+ }
+ if r.Min.X > s.Min.X {
+ r.Min.X = s.Min.X
+ }
+ if r.Min.Y > s.Min.Y {
+ r.Min.Y = s.Min.Y
+ }
+ if r.Max.X < s.Max.X {
+ r.Max.X = s.Max.X
+ }
+ if r.Max.Y < s.Max.Y {
+ r.Max.Y = s.Max.Y
+ }
+ return r
+}
+
+// Empty returns whether the rectangle contains no points.
+func (r Rectangle52_12) Empty() bool {
+ return r.Min.X >= r.Max.X || r.Min.Y >= r.Max.Y
+}
+
+// In returns whether every point in r is in s.
+func (r Rectangle52_12) In(s Rectangle52_12) bool {
+ if r.Empty() {
+ return true
+ }
+ // Note that r.Max is an exclusive bound for r, so that r.In(s)
+ // does not require that r.Max.In(s).
+ return s.Min.X <= r.Min.X && r.Max.X <= s.Max.X &&
+ s.Min.Y <= r.Min.Y && r.Max.Y <= s.Max.Y
+}

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