如何快速将浮点数打包为 4 个字节？

Question

我一直在寻找一种在 WebGL 纹理上存储浮点数的方法。我在互联网上找到了some solutions，但那些只处理 [0..1) 范围内的浮点数。我希望能够存储任意浮点数，为此，需要扩展这样的函数以存储指数（比如在第一个字节上）。不过，我不太明白这些是如何工作的，所以如何做到这一点并不明显。简而言之：

将浮点数打包成 4 个字节的有效算法是什么？

Answer 1

这并不快，但可行。 （请注意，GLSL 1.00 浮点文字在编译器中存在转换错误）。

struct Bitset8Bits {
    mediump vec4 bit0;
    mediump vec4 bit1;
    mediump vec4 bit2;
    mediump vec4 bit3;
    mediump vec4 bit4;
    mediump vec4 bit5;
    mediump vec4 bit6;
    mediump vec4 bit7;
};


vec4 when_gt (vec4 l, vec4 r) {
  return max(sign(l - r), 0.0);
}


Bitset8Bits unpack_4_bytes (lowp vec4 byte) {
    Bitset8Bits result;

    result.bit7 = when_gt(byte, vec4(127.5));
    vec4 bits0to6 = byte - 128.0 * result.bit7;

    result.bit6 = when_gt(bits0to6, vec4(63.5));
    vec4 bits0to5 = bits0to6 - 64.0 * result.bit6;

    result.bit5 = when_gt(bits0to5, vec4(31.5));
    vec4 bits0to4 = bits0to5 - 32.0 * result.bit5;

    result.bit4 = when_gt(bits0to4, vec4(15.5));
    vec4 bits0to3 = bits0to4 - 16.0 * result.bit4;

    result.bit3 = when_gt(bits0to3, vec4(7.5));
    vec4 bits0to2 = bits0to3 - 8.0 * result.bit3;

    result.bit2 = when_gt(bits0to2, vec4(3.5));
    vec4 bits0to1 = bits0to2 - 4.0 * result.bit2;

    result.bit1 = when_gt(bits0to1, vec4(1.5));
    vec4 bit0 = bits0to1 - 2.0 * result.bit1;

    result.bit0 = when_gt(bit0, vec4(0.5));

    return result;
}

float when_gt (float l, float r) {
  return max(sign(l - r), 0.0);
}




vec4 pack_4_bytes (Bitset8Bits state) {

  vec4 data;

  data = state.bit0
    + 2.0 * state.bit1
    + 4.0 * state.bit2
    + 8.0 * state.bit3
    + 16.0 * state.bit4
    + 32.0 * state.bit5
    + 64.0 * state.bit6
    + 128.0 * state.bit7;

  return data;
}

vec4 brians_float_pack (
    float original_value) {

    // Remove the sign
    float pos_value = abs(original_value);

    float exp_real = floor(log2(pos_value));
    float multiplier = pow(2.0, exp_real);
    float normalized = pos_value / multiplier - 1.0;

    float exp_v = exp_real + 127.0;
    // if exp_v == -Inf -> 0
    // if exp_v == +Inf -> 255
    // if exp_v < -126.0 -> denormalized (remove the "1")
    // otherwise + 127.0;

    Bitset8Bits packed_v;

    packed_v.bit7.a =
        step(sign(original_value) - 1.0, -1.5); // pos

    // Exponent 8 bits

    packed_v.bit6.a = when_gt(exp_v, 127.5);
    float bits0to6 = exp_v - 128.0 * packed_v.bit6.a;

    packed_v.bit5.a = when_gt(bits0to6, 63.5);
    float bits0to5 = bits0to6 - 64.0 * packed_v.bit5.a;

    packed_v.bit4.a = when_gt(bits0to5, 31.5);
    float bits0to4 = bits0to5 - 32.0 * packed_v.bit4.a;

    packed_v.bit3.a = when_gt(bits0to4, 15.5);
    float bits0to3 = bits0to4 - 16.0 * packed_v.bit3.a;

    packed_v.bit2.a = when_gt(bits0to3, 7.5);
    float bits0to2 = bits0to3 - 8.0 * packed_v.bit2.a;

    packed_v.bit1.a = when_gt(bits0to2, 3.5);
    float bits0to1 = bits0to2 - 4.0 * packed_v.bit1.a;

    packed_v.bit0.a = when_gt(bits0to1, 1.5);
    float bit0 = bits0to1 - 2.0 * packed_v.bit0.a;

    packed_v.bit7.b = when_gt(bit0, 0.5);

    // Significand 23 bits

    float factor = 0.5;
    // 0.4999999
    
    // Significand MSB bit 22:
    packed_v.bit6.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit6.b;
    factor = 0.5 * factor;

    packed_v.bit5.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit5.b;
    factor = 0.5 * factor;

    packed_v.bit4.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit4.b;
    factor = 0.5 * factor;

    packed_v.bit3.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit3.b;
    factor = 0.5 * factor;

    packed_v.bit2.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit2.b;
    factor = 0.5 * factor;

    packed_v.bit1.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit1.b;
    factor = 0.5 * factor;

    packed_v.bit0.b =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit0.b;
    factor = 0.5 * factor;


    packed_v.bit7.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit7.g;
    factor = 0.5 * factor;

    packed_v.bit6.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit6.g;
    factor = 0.5 * factor;

    packed_v.bit5.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit5.g;
    factor = 0.5 * factor;

    packed_v.bit4.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit4.g;
    factor = 0.5 * factor;

    packed_v.bit3.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit3.g;
    factor = 0.5 * factor;

    packed_v.bit2.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit2.g;
    factor = 0.5 * factor;

    packed_v.bit1.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit1.g;
    factor = 0.5 * factor;

    packed_v.bit0.g =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit0.g;
    factor = 0.5 * factor;


    packed_v.bit7.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit7.r;
    factor = 0.5 * factor;

    packed_v.bit6.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit6.r;
    factor = 0.5 * factor;

    packed_v.bit5.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit5.r;
    factor = 0.5 * factor;

    packed_v.bit4.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit4.r;
    factor = 0.5 * factor;

    packed_v.bit3.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit3.r;
    factor = 0.5 * factor;

    packed_v.bit2.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit2.r;
    factor = 0.5 * factor;

    packed_v.bit1.r =
        when_gt(normalized, factor - 0.00000005);
    normalized = normalized - factor * packed_v.bit1.r;
    factor = 0.5 * factor;

    // LSB bit 0
    packed_v.bit0.r =
        when_gt(normalized, factor - 0.00000005);

    vec4 result = pack_4_bytes(packed_v);

    return result;
}

Answer 2

一种简单的方法是首先同意您支持的浮点范围，然后在打包之前将其重新映射到 [0...1) 范围。

const MIN = -100;
const MAX = 100;

function packRemap(v){
    return (v - MIN) / (MAX - MIN);
}

function unpackRemap(p){
    return MIN + p * (MAX - MIN);
}

Answer 3

嗯，float 是一个 32 位数字 (23 bits for mantissa + 1 bit implicitly, 8 bits for exponent and 1 bit for sign)，RGBA8 纹理的纹素也是 32 位的。因此我们只需要一个编码方案，它可以用 JS（或任何其他语言）打包并用 GLSL 解包（给定 os GLSL ES 1.0 的限制，例如缺少按位操作）。这是我的建议（在 C++ 中）：

#include <cstdint>
#include <iostream>
#include <cmath>

// for storing result of encoding
struct rgba {
    uint8_t r, g, b, a;
};

rgba float2rgba(float x) {
    union {
        float xc;
        uint32_t xi;
    };

    // let's "convert" our float number to uint32_t so we can mess with it's bits
    xc = x;

    // in v we'll pack sign bit and mantissa, that would be exactly 24 bits
    int_least32_t v =
        // sign bit
        (xi >> 31 & 1) |
        // mantissa
        ((xi & 0x7fffff) << 1);

    rgba r;

    // then we just split into bytes and store them in RGB channels
    r.r = v / 0x10000;
    r.g = (v % 0x10000) / 0x100;
    r.b = v % 0x100;

    // and we'll put the exponent to alpha channel
    r.a = xi >> 23 & 0xff;

    return r;
}

float rgba2float(rgba r) {
    // let's "rebuild" mantissa and sign bit first
    uint32_t v = (r.b / 2) + r.g * 0x80 + r.r * 0x8000;

    return
        // let's apply sign (it's in least significant bit of v)
        (r.b % 2 ? -1.f : 1.f) *
        // and reconstruct the number itself
        (1.f + v * pow(2.f, -23.f)) * pow(2.f, static_cast<unsigned>(r.a) - 127);
}

int main() {
    const float a = -1.34320e32f;
    rgba r = float2rgba(a);
    std::cout <<
        a << '\n' <<
        static_cast<unsigned>(r.r) << ',' <<
        static_cast<unsigned>(r.g) << ',' <<
        static_cast<unsigned>(r.b) << ',' <<
        static_cast<unsigned>(r.a) << '\n' <<
        rgba2float(r) << std::endl;
}

输出：

-1.3432e+32
167,214,213,233
-1.3432e+32

Answer 4

由于我找不到任何可以解决我的问题的方法，我已经组装了这个解决方案：

function fract(x){ 
  return x - Math.floor(x);
};

function packFloat(x) {
  var s = x > 0 ? 1 : -1;
  var e = Math.floor(Math.log2(s*x));
  var m = s*x/Math.pow(2, e);
  return [
    Math.floor(fract((m-1)*256*256)*256),
    Math.floor(fract((m-1)*256)*256),
    Math.floor(fract((m-1)*1)*256),
    ((e+63) + (x>0?128:0))];
}

function unpackFloat(v){
  var s = v[3] >= 128 ? 1 : -1;
  var e = v[3] - (v[3] >= 128 ? 128 : 0) - 63;
  var m = 1 + v[0]/256/256/256 + v[1]/256/256 + v[2]/256;
  return s * Math.pow(2, e) * m;
};

for (var i=0; i<10; ++i){
  var num = (Math.random()*2.0-1.0)*1000;
  console.log(num, packFloat(num), unpackFloat(packFloat(num)));
}

它将浮点数来回转换为 4 个字节。与其他解决方案相反，它不限于小范围或预定义范围，并且能够表示形状上的任何数字s * m * 2^e，其中s = -1 or 1、m = 1 til 2（精度为 24 位），和e = -63 to 64。将其移植到 GLSL 很简单，因为它只使用常见的浮点运算。

Answer 5

我不确定我是否理解这个问题。

为什么不只使用浮点纹理？

var ext = gl.getExtension("OES_texture_float");
if (!ext) {
   // sorry no floating point support)
}

至于将数据放入纹理中，您只需使用Float32Array。

var data = new Float32Array([0.123456, Math.sqrt(2), ...]);
gl.texImage2D(gl.TARGET_2D, 0, gl.RGBA, width, height, 0, 
              gl.RGBA, gl.FLOAT, data);

大多数硬件都支持从浮点纹理读取。较少支持渲染到浮点纹理。见WebGL iOS render to floating point texture

我还要指出，您可以从 JavaScript 中的浮点数中获取字节

var arrayOf10Floats = new Float32Array(10);
var arrayOf40bytes = new Uint8Array(arrayOf10Floats.buffer);

这两个数组共享相同的内存。它们都只是底层ArrayBuffer 的ArrayBufferViews。