/
roundoffVisualization.html
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/
roundoffVisualization.html
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<html>
<head>
<title>Curious case of disappearing roundoff errors</title>
</head>
<script type="x-shader/x-vertex" id="passThruVS">
attribute vec2 coord;
varying vec2 pos;
void main(void) {
pos = coord;
gl_Position = vec4(coord, 0, 1.0);
gl_PointSize = 1.0;
}
</script>
<script type="x-shader/x-fragment" id="renderFrac">
precision highp float;
varying vec2 pos;
vec2 twoSum(float a, float b) {
float s = a + b;
float av = s - b;
float bv = s - a;
float e = (a - av)+(b - bv);
return vec2(s, e);
}
void main(void) {
vec2 sum = twoSum(pos.x, pos.y * 2e-8);
gl_FragColor = vec4(0, fract(sum.y*1e8), fract(abs(sum.x)), 1);
}
</script>
<script type="x-shader/x-fragment" id="renderFracONE">
precision highp float;
varying vec2 pos;
uniform vec2 ONE;
vec2 twoSum(float a, float b) {
float s = a + b;
float av = s * ONE.x - b * ONE.y;
float bv = s * ONE.y - a * ONE.x;
float e = (a - av) + (b - bv);
return vec2(s, e);
}
void main(void) {
vec2 sum = twoSum(pos.x, pos.y * 2e-8);
gl_FragColor = vec4(0, fract(sum.y*1e8), fract(abs(sum.x)), 1);
}
</script>
<script type="text/javascript">
class GLContext {
constructor(canvas) {
if (canvas == null)
throw 'Can not find canvas';
const ctx = canvas.getContext('webgl');
if (ctx == null)
throw 'Can not get WebGL context...';
ctx.viewport(0, 0, canvas.clientWidth, canvas.clientHeight);
ctx.clearColor(0, 0, 0, 1.0);
ctx.clear(ctx.COLOR_BUFFER_BIT);
this._ctx = ctx;
this.PRIMITIVES = {
POINTS: ctx.POINTS,
LINE_STRIP: ctx.LINE_STRIP,
LINES: ctx.LINES,
TRIANGLE_STRIP: ctx.TRIANGLE_STRIP,
TRIANGLES: ctx.TRIANGLES,
};
}
getGLContext() {
if (!this._ctx) throw 'Missing context';
return this._ctx;
}
compileShader(type, source) {
const glCtx = this.getGLContext();
var rc = glCtx.createShader(type);
glCtx.shaderSource(rc, source);
glCtx.compileShader(rc);
if (!glCtx.getShaderParameter(rc, glCtx.COMPILE_STATUS)) {
throw glCtx.getShaderInfoLog(rc) + ' in ' + source;
}
return rc;
}
compileShaderById(id) {
const glCtx = this.getGLContext();
const element = document.getElementById(id);
if (element == null) {
throw 'Can not find shader by element id ' + id;
}
const source = element.innerHTML;
if (element.type === 'x-shader/x-vertex') {
return this.compileShader(glCtx.VERTEX_SHADER, source);
}
if (element.type === 'x-shader/x-fragment') {
return this.compileShader(glCtx.FRAGMENT_SHADER, source);
}
throw 'Unknown shader type ' + element.type;
}
linkProgram(shaders) {
const glCtx = this.getGLContext();
const rc = glCtx.createProgram();
for(var cnt = 0; cnt < shaders.length; cnt++) {
if (!shaders[cnt]) {
throw 'Can not link program with null shaders';
}
glCtx.attachShader(rc, shaders[cnt]);
}
glCtx.linkProgram(rc);
if (!glCtx.getProgramParameter(rc, glCtx.LINK_STATUS)) {
throw glCtx.getProgramInfoLog(rc);
}
return rc;
}
setUniform2f(name, a, b) {
const glCtx = this.getGLContext();
const prog = glCtx.getParameter(glCtx.CURRENT_PROGRAM);
const idx = glCtx.getUniformLocation(prog, name);
glCtx.uniform2f(idx, a, b);
}
draw2DArray(type, name, arr) {
const glCtx = this.getGLContext();
const prog = glCtx.getParameter(glCtx.CURRENT_PROGRAM);
const idx = glCtx.getAttribLocation(prog, name);
var buf = glCtx.createBuffer();
glCtx.bindBuffer(glCtx.ARRAY_BUFFER, buf);
glCtx.bufferData(glCtx.ARRAY_BUFFER, new Float32Array(arr), glCtx.STATIC_DRAW);
glCtx.enableVertexAttribArray(idx);
glCtx.vertexAttribPointer(idx, 2, glCtx.FLOAT, false, 0, 0);
glCtx.drawArrays(type, 0, arr.length/2);
glCtx.deleteBuffer(buf);
}
}
function printMessage(elementName, msg) {
document.getElementById(elementName).innerHTML=
'<div style="color:#FFFFFF;'+
'font-size:120%;margin-top:40%;'+
'margin-left:20%;margin-right:20%">' +
msg + '</div>';
}
function visualizeRoundoffs(canvasName, shaderName, elementName) {
try {
printMessage(elementName, 'Initializing WebGL context...');
const canvas = document.getElementById(canvasName);
const ctx = new GLContext(canvas);
const prog = ctx.linkProgram(['passThruVS', shaderName].map(_ => ctx.compileShaderById(_)));
const gl = ctx.getGLContext();
gl.useProgram(prog);
ctx.setUniform2f('ONE', 1.0, 1.0);
ctx.draw2DArray(gl.TRIANGLE_STRIP, 'coord', [-1, -1, 1, -1, -1, 1, 1, 1]);
const pixelCount = 100;
const buf = new Uint8Array(4 * pixelCount);
gl.readPixels(0, 0, pixelCount, 1, gl.RGBA, gl.UNSIGNED_BYTE, buf);
if (gl.getError())
throw "Failed to read pixels";
// Check for non-zero values in green channel
for(var cnt = 0; cnt < pixelCount; ++cnt) {
if (buf[cnt*4+1] != 0) {
printMessage(elementName, '👍Shader compiler is aware of rounding errors.');
return;
}
}
printMessage(elementName, '💔Shader compiler lives in the world of the infinitely precision floats.');
} catch (e) { printMessage(elementName, 'WebGL context creation failed'); alert(e); }
}
</script>
<body onLoad="visualizeRoundoffs('canvas1', 'renderFrac', 'floatingMsg1');visualizeRoundoffs('canvas2', 'renderFracONE', 'floatingMsg2')">
<div width="1024px">
<div style="position:relative;float:left">
<canvas id="canvas1" width="512px" height="512px"></canvas>
<div id="floatingMsg1" style="position:absolute;top:0px; left:0px; width:512px;height:512px"></div>
</div>
<div style="position:relative;margin-left:512px;">
<canvas id="canvas2" width="512px" height="512px"></canvas>
<div id="floatingMsg2" style="position:absolute;top:0px; left:0px; width:512px;height:512px"></div>
</div>
</div>
<br/>
If all you see in the above left image is a dull black stripe on a blue background, then WebGL fragment shader compiler decided that following two functions are identical:
<div>
<pre style="display:inline-block;padding:10px; border: 1px solid #777; background-color:#eee;vertical-align: middle;">
vec2 sum(float a, float b) {
float s = a + b;
float e = (b - (s-a)) + (a - (s-b));
return vec2(s, e);
}
</pre>
<div style="display:inline;font-size:200%;vertical-align: middle;">↔</div>
<pre style="display:inline-block;padding:10px; border: 1px solid #777; background-color:#eee;vertical-align: middle;">
vec2 sum(float a, float b) {
return vec2(a + b, 0.0);
}
</pre>
</div>
which is clearly not the case, when non-overlapping floats are added together (for example 1.0 and pow(2,-24)).<br>
Image on the right is an attempt to workaround the limitations by using uniform that shader compilers can not optimize away, see <a href="https://github.com/visgl/luma.gl/blame/7aa4f21b956a3b16b1035e71915727bce71071a8/modules/shadertools/src/modules/fp64/fp64-arithmetic.glsl.ts#L7">this</a> explanation from luma.gl.<br/>
When compiler does not assume rounding errors to be zero, resulting picture looks as follows:<br>
<img src="roundoffErrors.png" width="512px" height="512px" />
</body>
</html>