即使是伟大的数学家也会犯错误

我们知道数学是一门精确的科学。我们可以对互动数学学习软件 GeoGebra 说同样的话吗？让我们用PVS-Studio来分析一下项目源码吧！

介绍

你还记得你在大学是如何学习计算机科学的吗？所有这些矩阵和向量乘法、多项式方程、插值、外推……如果我们在真实的项目中而不是在另一个实验室报告中查看这些可怕的公式会怎么样？如果我们在这样的代码库中挖掘问题怎么办？我建议运行 PVS-Studio 并掸掉数学教科书的灰尘。为什么是教科书？让我告诉你。

数学挑战

检查此类程序源代码的主要挑战之一是了解正在发生的事情。在查看分析器报告时，我们对警告是否表明真正的问题存在疑问。

让我们看一下以下片段：

@Override
public void compute() {
  ....
  if (cumulative != null && cumulative.getBoolean()) {
    ....
  } else {
    ....
    branchAtoMode = fv.wrap().subtract(a).multiplyR(2)
      .divide(bEn.subtract(a).multiply(modeEn.subtract(a)));
    branchModeToB = fv.wrap().subtract(b).multiplyR(2)                
      .divide(bEn.subtract(a).multiply(modeEn.subtract(b)));
      rightBranch = new MyDouble(kernel, 0);
  }
  ....
}

我们收到以下 PVS-Studio 警告：

V6072 发现两个相似的代码片段。也许，这是一个拼写错误，应该使用“b”变量而不是“a”。 AlgoTriangleDF.java 145、AlgoTriangleDF.java 146、AlgoTriangleDF.java 147、AlgoTriangleDF.java 148

真的是打字错误吗？经过快速研究，一旦我们找到了正确的公式，我们就可以说一切都写对了。

代码片段评估三角分布，即该分布的概率密度函数（PDF）。我们找到了公式：

现在让我们看一下代码。

这里，* fv* 是一个函数变量。 wrap 返回 wrapper，然后执行必要的数学运算。有趣的是，有 multiply 和 multiplyR 方法。在第二种方法中，R 代表 右 并交换操作数，因为乘法并不总是可交换的。

因此，第二个表达式结果写入branchAToMode，第四个表达式写入branchModeToB。

我们还注意到，在 branchModeToB 中，分子和分母的符号已更改。我们得到以下表达式：

表情值没有改变。

因此，我们刷新了数学知识来理解我们收到的一些警告。识别代码中是否存在真正的错误并不难，但很难理解它应该是什么。

错误

丢失代码段

让我们从简单开始，看看下面的方法：

private void updateSide(int index, int newBottomPointsLength) {
  ....
  GeoSegmentND[] s = new GeoSegmentND[4];
  GeoSegmentND segmentBottom = outputSegmentsBottom.getElement(index);
  s[0] = segmentBottom;
  s[1] = segmentSide1;
  s[2] = segmentSide2;
  s[2] = segmentSide3;
  polygon.setSegments(s);
  polygon.calcArea();
}

我们发现有人忘记将 s[2] 替换为 s[3]。最后一行的效果非常出色。这是传说中的、非常常见的复制粘贴错误。结果，第四个数组项丢失，并且为 null!

V6033 具有相同键“2”的项目已被更改。 AlgoPolyhedronNetPrism.java 376、AlgoPolyhedronNetPrism.java 377

价值观呢？

现在尝试在以下代码片段中发现问题：

static synchronized HashMap<String, String> getGeogebraMap() {
  ....
  geogebraMap.put("−", "-");
  geogebraMap.put("⊥", "# ");
  geogebraMap.put("∼", "~ ");
  geogebraMap.put("′", "# ");
  geogebraMap.put("≤", Unicode.LESS_EQUAL + "");
  geogebraMap.put("&ge;", Unicode.GREATER_EQUAL + "");
  geogebraMap.put("&infin;", Unicode.INFINITY + "");
  ....
  geogebraMap.put("∏", "# ");
  geogebraMap.put("&Product;", "# ");
  geogebraMap.put("〉", "# ");
  geogebraMap.put("&rangle;", "# ");
  geogebraMap.put("&rarr;", "# ");
  geogebraMap.put("&rArr;", "# ");
  geogebraMap.put("&RightAngleBracket;", "# ");
  geogebraMap.put("&rightarrow;", "# ");
  geogebraMap.put("&Rightarrow;", "# ");
  geogebraMap.put("&RightArrow;", "# ");
  geogebraMap.put("&sdot;", "* ");
  geogebraMap.put("∼", "# ");
  geogebraMap.put("∝", "# ");
  geogebraMap.put("&Proportional;", "# ");
  geogebraMap.put("&propto;", "# ");
  geogebraMap.put("⊂", "# ");
  ....
  return geogebraMap;
}

多么壮丽的景色啊！读起来很愉快，这只是一小部分，因为这个方法从第 66 行开始，到第 404 行结束。分析仪发出 50 个 V6033 类型的警告。让我们快速浏览一下其中一个警告：

V6033 已添加具有相同键“∼”的项目。 MathMLParser.java 229, MathMLParser.java 355

让我们删除多余的片段，看看涉及警告的表达式：

geogebraMap.put("∼", "~ ");
....
geogebraMap.put("∼", "# ");

不过，这很有趣。方法调用之间的间隔是多少？有126行。好吧，祝你好运手动发现这样的错误！

大多数是键和值重复。然而，有一些情况与上面的示例类似，开发人员用不同的值覆盖该值。我们应该使用哪一个？

圆形或椭圆形

@Override
protected boolean updateForItSelf() {
  ....
  if (conic.getType() == GeoConicNDConstants.CONIC_SINGLE_POINT) {
    ....
  } else {
    if (visible != Visible.FRUSTUM_INSIDE) {
      ....
      switch (conic.getType()) {
        case GeoConicNDConstants.CONIC_CIRCLE:
          updateEllipse(brush);                     // <=
          break;
        case GeoConicNDConstants.CONIC_ELLIPSE:
          updateEllipse(brush);                     // <=
          break;
        case GeoConicNDConstants.CONIC_HYPERBOLA:
          updateHyperbola(brush);
          break;
        case GeoConicNDConstants.CONIC_PARABOLA:
          updateParabola(brush);
          break;
        case GeoConicNDConstants.CONIC_DOUBLE_LINE:
          createTmpCoordsIfNeeded();
          brush.segment(tmpCoords1.setAdd3(m, tmpCoords1.setMul3(d, minmax[0])),
              tmpCoords2.setAdd3(m, tmpCoords2.setMul3(d, minmax[1])));
          break;
        case GeoConicNDConstants.CONIC_INTERSECTING_LINES:
        case GeoConicNDConstants.CONIC_PARALLEL_LINES:
          updateLines(brush);
          break;
        default:
          break;
      }
    }
  }
}

椭圆和圆都调用椭圆的方法。事实上，我们可以假设这是可以的，因为圆也是椭圆。但是，该类还有 updateCircle 方法。那应该是什么？让我们更深入地探讨一下。

一切都发生在DrawConic3D类中。以下是椭圆和圆的方法：

protected void updateCircle(PlotterBrush brush) {
  if (visible == Visible.CENTER_OUTSIDE) {
    longitude = brush.calcArcLongitudesNeeded(e1, alpha,
          getView3D().getScale());
    brush.arc(m, ev1, ev2, e1, beta - alpha, 2 * alpha, longitude);
  } else {
    longitude = brush.calcArcLongitudesNeeded(e1, Math.PI,
          getView3D().getScale());
    brush.circle(m, ev1, ev2, e1, longitude);
  }
}
protected void updateEllipse(PlotterBrush brush) {
  if (visible == Visible.CENTER_OUTSIDE) {
    brush.arcEllipse(m, ev1, ev2, e1, e2, beta - alpha, 2 * alpha);
  } else {
    brush.arcEllipse(m, ev1, ev2, e1, e2, 0, 2 * Math.PI);
  }
}

Well... It doesn't give that much confidence. The method bodies are different, but nothing here indicates that we risk displaying unacceptable geometric objects if the method is called incorrectly.

Could there be other clues? A whole single one! The updateCircle method is never used in the project. Meanwhile, updateEllipse is used four times: twice in the first fragment and then twice in DrawConicSection3D, the inheritor class of* DrawConic3D*:

@Override
protected void updateCircle(PlotterBrush brush) {
  updateEllipse(brush);
}

@Override
protected void updateEllipse(PlotterBrush brush) {
  // ....
  } else {
    super.updateEllipse(brush);
  }
}

This updateCircle isn't used, either. So, updateEllipse only has a call in its own override and in the fragment where we first found updateForItSelf. In schematic form, the structure looks like as follows:

On the one hand, it seems that the developers wanted to use the all-purpose updateEllipse method to draw a circle. On the other hand, it's a bit strange that DrawConicSection3D has the updateCircle method that calls updateEllipse. However, updateCircle will never be called.

It's hard to guess what the fixed code may look like if the error is actually in the code. For example, if updateCircle needs to call updateEllipse in DrawConicSection3D, but DrawConic3D needs a more optimized algorithm for the circle, the fixed scheme might look like this:

So, it seems that developers once wrote updateCircle and then lost it, and we may have found its intended "home". Looks like we have discovered the ruins of the refactoring after which the developers forgot about the "homeless" method. In any case, it's worth rewriting this code to make it clearer so that we don't end up with so many questions.

All these questions have arisen because of the PVS-Studio warning. That's the warning, by the way:

V6067 Two or more case-branches perform the same actions. DrawConic3D.java 212, DrawConic3D.java 215

Order of missing object

private void updateOrdering(GeoElement geo, ObjectMovement movement) {
  ....
  switch (movement) {
    ....
    case FRONT:
      ....
      if (index == firstIndex) {
        if (index != 0) { 
          geo.setOrdering(orderingDepthMidpoint(index));
        }
        else { 
          geo.setOrdering(drawingOrder.get(index - 1).getOrdering() - 1);
        }
      }
    ....
  }
  ....
}

We get the following warning:

V6025 Index 'index - 1' is out of bounds. LayerManager.java 393

This is curious because, in the else block, the index variable is guaranteed to get the value 0. So, we pass -1 as an argument to the get method. What's the result? We catch an IndexOutOfBoundsException.

Triangles

@Override
protected int getGLType(Type type) {
  switch (type) {
  case TRIANGLE_STRIP:
    return GL.GL_TRIANGLE_STRIP;
  case TRIANGLE_FAN:
    return GL.GL_TRIANGLE_STRIP;     // <=
  case TRIANGLES:
    return GL.GL_TRIANGLES;
  case LINE_LOOP:
    return GL.GL_LINE_LOOP;
  case LINE_STRIP:
    return GL.GL_LINE_STRIP;
  }

  return 0;
}

The code is new, but the error is already well-known. It's quite obvious that GL.GL_TRIANGLE_STRIP should be GL.GL_TRIANGLE_FAN instead*.* The methods may be similar in some ways, but the results are different. You can read about it under the spoiler.

V6067 Two or more case-branches perform the same actions. RendererImplShadersD.java 243, RendererImplShadersD.java 245

To describe a series of triangles, we need to save the coordinates of the three vertices of each triangle. Thus, given N triangles, we need the saved 3N vertices. If we describe a polyhedral object using a polygon mesh, it's important to know if the triangles are connected. If they are, we can use the Triangle Strip or the Triangle Fan to describe the set of triangles using N + 2 vertices.

We note that the Triangle Fan has been removed in Direct3D 10. In OpenGL 4.6, this primitive still exists.

The Triangle Fan uses one center vertex as common, as well as the last vertex and the new vertex. Look at the following example:

To describe it, we'd need the entry (A, B, C, D, E, F, G). There are five triangles and seven vertices in the entry.

The Triangle Strip uses the last two vertices and a new one. For instance, we can create the image below using the same sequence of vertices:

Therefore, if we use the wrong primitive, we'll get dramatically different results.

Overwritten values

public static void addToJsObject(
JsPropertyMap<Object> jsMap, Map<String, Object> map) {
  for (Entry<String, Object> entry : map.entrySet()) {
    Object object = entry.getValue();
      if (object instanceof Integer) {
        jsMap.set(entry.getKey(), unbox((Integer) object));
      } if (object instanceof String[]) {                          // <=
        JsArray<String> clean = JsArray.of();
        for (String s: (String[]) object) {
          clean.push(s);
        }
        jsMap.set(entry.getKey(), clean);
    } else {                                                       // <=
      jsMap.set(entry.getKey(), object);                           // <=
    }
  }
}

If we don't look closely, we may not notice the issue right away. However, let's speed up and just check the analyzer message.

V6086 Suspicious code formatting. 'else' keyword is probably missing. ScriptManagerW.java 209

Finally, a more interesting bug is here. The object instanceof String[] check will occur regardless of the result of object instanceof Integer. It may not be a big deal, but there's also the else block that will be executed after a failed check for String[]. As the result, the jsMap value by entry.getKey() will be overwritten if the object was Integer.

There's another similar case, but the potential consequences are a little harder to understand:

public void bkspCharacter(EditorState editorState) {
  int currentOffset = editorState.getCurrentOffsetOrSelection();
  if (currentOffset > 0) {
    MathComponent prev = editorState.getCurrentField()
        .getArgument(currentOffset - 1);
    if (prev instanceof MathArray) {
      MathArray parent = (MathArray) prev;
      extendBrackets(parent, editorState);
    } if (prev instanceof MathFunction) {                          // <=
      bkspLastFunctionArg((MathFunction) prev, editorState);
    } else {                                                       // <=
      deleteSingleArg(editorState);
    }
  } else {
    RemoveContainer.withBackspace(editorState);
  }
}

V6086 Suspicious code formatting. 'else' keyword is probably missing. InputController.java 854

Almost correct

Do you often have to write many repetitious checks? Are these checks prone to typos? Let's look at the following method:

public boolean contains(BoundingBox other) {
  return !(isNull() || other.isNull()) && other.minx >= minx
      && other.maxy <= maxx && other.miny >= miny
      && other.maxy <= maxy;
}

V6045 Possible misprint in expression 'other.maxy <= maxx'. BoundingBox.java 139

We find a typo, we fix it. The answer is simple, there should be other.maxx <= maxx.

Mixed up numbers

public PointDt[] calcVoronoiCell(TriangleDt triangle, PointDt p) {
  ....
  // find the neighbor triangle
  if (!halfplane.next_12().isHalfplane()) {
    neighbor = halfplane.next_12();
  } else if (!halfplane.next_23().isHalfplane()) {
    neighbor = halfplane.next_23();
  } else if (!halfplane.next_23().isHalfplane()) { // <=
    neighbor = halfplane.next_31();
  } else {
    Log.error("problem in Delaunay_Triangulation");
    // TODO fix added by GeoGebra
    // should we do something else?
    return null;
  }
  ....
}

Let's see the warning:

V6003 The use of 'if (A) {...} else if (A) {...}' pattern was detected. There is a probability of logical error presence. DelaunayTriangulation.java 678, DelaunayTriangulation.java 680

We don't even need to figure out what happens to half-planes, because it's clear that the !halfplane.next_31().isHalfplane() check is missing.

Wrong operation

public static ExpressionNode get(
    ExpressionValue left, ExpressionValue right, 
    Operation operation, FunctionVariable fv, 
    Kernel kernel0
) {
  ....
  switch (operation) {
    ....
    case VEC_FUNCTION:
      break;
    case ZETA:
      break;
    case DIRAC:
      return new ExpressionNode(kernel0, left, Operation.DIRAC, fv);
    case HEAVISIDE:
      return new ExpressionNode(kernel0, left, Operation.DIRAC, fv);
    default:
      break;
  }
  ....
}

It seems that, in the second case, Operation.DIRAC should probably be replaced with Operation.HEAVISIDE. Or not? This method is called to obtain the derivative. After researching, we understand what HEAVISIDE *is—the use of *Operation.DIRAC for it is correct. What about the Dirac derivative? This one is a bit tricky to understand. We may trust the developers and suppress the warning. Still, it'd be better if they'd left any explanatory comment as they had done in some cases before.

case FACTORIAL:
  // x! -> psi(x+1) * x!
  return new ExpressionNode(kernel0, left.wrap().plus(1),
      Operation.PSI, null)
          .multiply(new ExpressionNode(kernel0, left,
              Operation.FACTORIAL, null))
          .multiply(left.derivative(fv, kernel0));

case GAMMA:
  // gamma(x) -> gamma(x) psi(x)
  return new ExpressionNode(kernel0, left, Operation.PSI, null)
      .multiply(new ExpressionNode(kernel0, left, Operation.GAMMA,
          null))
      .multiply(left.derivative(fv, kernel0));

And we were motivated to conduct such research by the message of already favorite diagnostic V6067:

V6067 Two or more case-branches perform the same actions. Derivative.java 442, Derivative.java 444

On the one hand, this is an interesting case because we could have a false positive here. On the other, the warning would be useful either way, as it often highlights a genuine error. Regardless of the results, we need to check such code and either fix the error or write explanatory comments. Then the warning can be suppressed.

Vector or point?

private static GeoElement doGetTemplate(Construction cons,
    GeoClass listElement) {
  switch (listElement) {
  case POINT:
    return new GeoPoint(cons);
  case POINT3D:
    return (GeoElement) cons.getKernel().getGeoFactory().newPoint(3, cons);
  case VECTOR:
    return new GeoVector(cons);
  case VECTOR3D:
    return (GeoElement) cons.getKernel().getGeoFactory().newPoint(3, cons);
  }
  return new GeoPoint(cons);
}

The warning isn't hard to guess:

V6067 Two or more case-branches perform the same actions. PointNDFold.java 38, PointNDFold.java 43

Instead of cons.getKernel().getGeoFactory().newPoint(3, cons) in the second case, cons.getKernel().getGeoFactory().newVector(3, cons) may have been intended. If we want to make sure of it, we need to go deeper again. Well, let's dive into it.

So,* getGeoFactory() returns GeoFactory, let's look at the *newPoint *and newVector* methods:

public GeoPointND newPoint(int dimension, Construction cons) {
  return new GeoPoint(cons);
}
public GeoVectorND newVector(int dimension, Construction cons) {
  return new GeoVector(cons);
}

It looks a bit strange. The dimension argument isn't used at all. What's going on here? Inheritance, of course! Let's find the GeoFactory3D inheritor class and see what happens in it.

@Override
public GeoVectorND newVector(int dimension, Construction cons) {
  if (dimension == 3) {
    return new GeoVector3D(cons);
  }
  return new GeoVector(cons);
}
@Override
public GeoPointND newPoint(int dimension, Construction cons) {
  return dimension == 3 ? new GeoPoint3D(cons) : new GeoPoint(cons);
}

Excellent! Now we can admire the creation of four different objects in all their glory. We return to the code fragment with the possible error. For POINT and POINT3D, the objects of the GeoPoint and GeoPoint3D classes are created. GeoVector is created for VECTOR, but poor GeoVector3D seems to have been abandoned.

Sure, it's a bit strange to use a factory method pattern in two cases and call constructors directly in the remaining two. Is this a leftover from the refactoring process, or is it some kind of temporary solution that'll stick around until the end of time? In my opinion, if the responsibility for creating objects has been handed over to another class, it'd be better to fully delegate that responsibility.

Missing quadrillions

@Override
final public void update() {
  ....
  switch (angle.getAngleStyle()) {
    ....
    case EuclidianStyleConstants.RIGHT_ANGLE_STYLE_L:
      // Belgian offset |_
      if (square == null) {
        square = AwtFactory.getPrototype().newGeneralPath();
      } else {
        square.reset();
      }
      length = arcSize * 0.7071067811865;                   // <=
      double offset = length * 0.4;
      square.moveTo(
          coords[0] + length * Math.cos(angSt)
              + offset * Math.cos(angSt)
              + offset * Math.cos(angSt + Kernel.PI_HALF),
          coords[1]
              - length * Math.sin(angSt)
                  * view.getScaleRatio()
              - offset * Math.sin(angSt)
              - offset * Math.sin(angSt + Kernel.PI_HALF));
      ....
      break;
    }
}

Where's the warning? There it is:

V6107 The constant 0.7071067811865 is being utilized. The resulting value could be inaccurate. Consider using Math.sqrt(0.5). DrawAngle.java 303

我们发现了此代码片段的四个实例。更准确的值为0.7071067811865476。挑战一下自己，尝试背熟它，呵呵。最后三位数字被省略。这很关键吗？准确性可能足够了，但使用预定义的常量或 Math.sqrt(0.5)（如本例所示）不仅有助于恢复丢失的数字，还可以消除拼写错误的风险。请注意代码可读性将如何增强。也许当看到0.707……有人立刻明白这是一个什么样的神奇数字。然而，有时我们可以做一些不那么神奇的事情来增强代码的可读性。

结论

我们小小的数学之旅结束了。正如我们所看到的，即使在解决了许多几何挑战之后，人们仍然可能在代码中犯一些简单的错误！如果代码是新鲜的，那么开发人员仍然可以牢记他们过去的意图，那就太好了。在这种情况下，人们可能会明白需要做什么改变（如果有的话），但时间长了之后可能就不那么容易认识到解决这些问题的必要性了。

因此，事实证明，分析器在编写代码时非常高效，而不是像我那样运行一次。尤其是现在，当我们拥有像 PVS-Studio 分析仪这样的工具时。

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