How is pattern matching in Scala implemented at the bytecode level?

The low level can be explored with a disassembler but the short answer is that it's a bunch of if/elses where the predicate depends on the pattern

case Sum(l,r) // instance of check followed by fetching the two arguments and assigning to two variables l and r but see below about custom extractors 
case "hello" // equality check
case _ : Foo // instance of check
case x => // assignment to a fresh variable
case _ => // do nothing, this is the tail else on the if/else

There's much more that you can do with patterns like or patterns and combinations like "case Foo(45, x)", but generally those are just logical extensions of what I just described. Patterns can also have guards, which are additional constraints on the predicates. There are also cases where the compiler can optimize pattern matching, e.g when there's some overlap between cases it might coalesce things a bit. Advanced patterns and optimization are an active area of work in the compiler, so don't be surprised if the byte code improves substantially over these basic rules in current and future versions of Scala.

In addition to all that, you can write your own custom extractors in addition to or instead of the default ones Scala uses for case classes. If you do, then the cost of the pattern match is the cost of whatever the extractor does. A good overview is found in http://lamp.epfl.ch/~emir/written/MatchingObjectsWithPatterns-TR.pdf

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James (above) said it best. However, if you're curious it's always a good exercise to look at the disassembled bytecode. You can also invoke scalac with the -print option, which will print your program with all Scala-specific features removed. It's basically Java in Scala's clothing. Here's the relevant scalac -print output for the code snippet you gave:

def eval(e: Expr): Int = {
  <synthetic> val temp10: Expr = e;
  if (temp10.$isInstanceOf[Number]())
    temp10.$asInstanceOf[Number]().n()
  else
    if (temp10.$isInstanceOf[Sum]())
      {
        <synthetic> val temp13: Sum = temp10.$asInstanceOf[Sum]();
        Main.this.eval(temp13.e1()).+(Main.this.eval(temp13.e2()))
      }
    else
      throw new MatchError(temp10)
};

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Since version 2.8, Scala has had the @switch annotation. The goal is to ensure, that pattern matching will be compiled into tableswitch or lookupswitch instead of series of conditional if statements.

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To expand on @Zifre's comment: if you are reading this in the future and the scala compiler has adopted new compilation strategies and you want to know what they are, here's how you find out what it does.

Copy-paste your match code into a self-contained example file. Run scalac on that file. Then run javap -v -c theClassName$.class.

For example, I put the following into /tmp/question.scala:

object question {
  abstract class Expr
  case class Number(n: Int) extends Expr
  case class Sum(e1: Expr, e2: Expr) extends Expr

  def eval(e: Expr): Int = e match {
    case Number(x) => x
    case Sum(l, r) => eval(l) + eval(r)
  }
}

Then I ran scalac question.scala, which produced a bunch of *.class files. Poking around a bit, I found the match statement inside question$.class. The javap -c -v question$.class output is available below.

Since we're looking for a condition control flow construct, knowing about the java bytecode instruction set suggests that looking for "if" should be a good place to start.

In two locations we find a pair of consecutive lines on the form isinstanceof <something>; ifeq <somewhere>, which means: if the most recently computed value is not an instance of something then goto somewhere. (ifeq is jump if zero, and isinstanceof gives you a zero to represent false.)

If you follow the control flow around, you'll see that it agrees with the answer given by @Jorge Ortiz: we do if (blah isinstanceof something) { ... } else if (blah isinstanceof somethingelse) { ... }.

Here is the javap -c -v question$.class output:

Classfile /tmp/question$.class
  Last modified Nov 20, 2020; size 956 bytes
  MD5 checksum cfc788d4c847dad0863a797d980ad2f3
  Compiled from "question.scala"
public final class question$
  minor version: 0
  major version: 50
  flags: (0x0031) ACC_PUBLIC, ACC_FINAL, ACC_SUPER
  this_class: #2                          // question$
  super_class: #4                         // java/lang/Object
  interfaces: 0, fields: 1, methods: 3, attributes: 4
Constant pool:
   #1 = Utf8               question$
   #2 = Class              #1             // question$
   #3 = Utf8               java/lang/Object
   #4 = Class              #3             // java/lang/Object
   #5 = Utf8               question.scala
   #6 = Utf8               MODULE$
   #7 = Utf8               Lquestion$;
   #8 = Utf8               <clinit>
   #9 = Utf8               ()V
  #10 = Utf8               <init>
  #11 = NameAndType        #10:#9         // "<init>":()V
  #12 = Methodref          #2.#11         // question$."<init>":()V
  #13 = Utf8               eval
  #14 = Utf8               (Lquestion$Expr;)I
  #15 = Utf8               question$Number
  #16 = Class              #15            // question$Number
  #17 = Utf8               n
  #18 = Utf8               ()I
  #19 = NameAndType        #17:#18        // n:()I
  #20 = Methodref          #16.#19        // question$Number.n:()I
  #21 = Utf8               question$Sum
  #22 = Class              #21            // question$Sum
  #23 = Utf8               e1
  #24 = Utf8               ()Lquestion$Expr;
  #25 = NameAndType        #23:#24        // e1:()Lquestion$Expr;
  #26 = Methodref          #22.#25        // question$Sum.e1:()Lquestion$Expr;
  #27 = Utf8               e2
  #28 = NameAndType        #27:#24        // e2:()Lquestion$Expr;
  #29 = Methodref          #22.#28        // question$Sum.e2:()Lquestion$Expr;
  #30 = NameAndType        #13:#14        // eval:(Lquestion$Expr;)I
  #31 = Methodref          #2.#30         // question$.eval:(Lquestion$Expr;)I
  #32 = Utf8               scala/MatchError
  #33 = Class              #32            // scala/MatchError
  #34 = Utf8               (Ljava/lang/Object;)V
  #35 = NameAndType        #10:#34        // "<init>":(Ljava/lang/Object;)V
  #36 = Methodref          #33.#35        // scala/MatchError."<init>":(Ljava/lang/Object;)V
  #37 = Utf8               this
  #38 = Utf8               e
  #39 = Utf8               Lquestion$Expr;
  #40 = Utf8               x
  #41 = Utf8               I
  #42 = Utf8               l
  #43 = Utf8               r
  #44 = Utf8               question$Expr
  #45 = Class              #44            // question$Expr
  #46 = Methodref          #4.#11         // java/lang/Object."<init>":()V
  #47 = NameAndType        #6:#7          // MODULE$:Lquestion$;
  #48 = Fieldref           #2.#47         // question$.MODULE$:Lquestion$;
  #49 = Utf8               question
  #50 = Class              #49            // question
  #51 = Utf8               Sum
  #52 = Utf8               Expr
  #53 = Utf8               Number
  #54 = Utf8               Code
  #55 = Utf8               LocalVariableTable
  #56 = Utf8               LineNumberTable
  #57 = Utf8               StackMapTable
  #58 = Utf8               SourceFile
  #59 = Utf8               InnerClasses
  #60 = Utf8               ScalaInlineInfo
  #61 = Utf8               Scala
{
  public static final question$ MODULE$;
    descriptor: Lquestion$;
    flags: (0x0019) ACC_PUBLIC, ACC_STATIC, ACC_FINAL

  public static {};
    descriptor: ()V
    flags: (0x0009) ACC_PUBLIC, ACC_STATIC
    Code:
      stack=1, locals=0, args_size=0
         0: new           #2                  // class question$
         3: invokespecial #12                 // Method "<init>":()V
         6: return

  public int eval(question$Expr);
    descriptor: (Lquestion$Expr;)I
    flags: (0x0001) ACC_PUBLIC
    Code:
      stack=3, locals=9, args_size=2
         0: aload_1
         1: astore_2
         2: aload_2
         3: instanceof    #16                 // class question$Number
         6: ifeq          27
         9: aload_2
        10: checkcast     #16                 // class question$Number
        13: astore_3
        14: aload_3
        15: invokevirtual #20                 // Method question$Number.n:()I
        18: istore        4
        20: iload         4
        22: istore        5
        24: goto          69
        27: aload_2
        28: instanceof    #22                 // class question$Sum
        31: ifeq          72
        34: aload_2
        35: checkcast     #22                 // class question$Sum
        38: astore        6
        40: aload         6
        42: invokevirtual #26                 // Method question$Sum.e1:()Lquestion$Expr;
        45: astore        7
        47: aload         6
        49: invokevirtual #29                 // Method question$Sum.e2:()Lquestion$Expr;
        52: astore        8
        54: aload_0
        55: aload         7
        57: invokevirtual #31                 // Method eval:(Lquestion$Expr;)I
        60: aload_0
        61: aload         8
        63: invokevirtual #31                 // Method eval:(Lquestion$Expr;)I
        66: iadd
        67: istore        5
        69: iload         5
        71: ireturn
        72: new           #33                 // class scala/MatchError
        75: dup
        76: aload_2
        77: invokespecial #36                 // Method scala/MatchError."<init>":(Ljava/lang/Object;)V
        80: athrow
      LocalVariableTable:
        Start  Length  Slot  Name   Signature
            0      81     0  this   Lquestion$;
            0      81     1     e   Lquestion$Expr;
           20      61     4     x   I
           47      34     7     l   Lquestion$Expr;
           54      27     8     r   Lquestion$Expr;
      LineNumberTable:
        line 6: 0
        line 7: 2
        line 8: 27
        line 6: 69
      StackMapTable: number_of_entries = 3
        frame_type = 252 /* append */
          offset_delta = 27
          locals = [ class question$Expr ]
        frame_type = 254 /* append */
          offset_delta = 41
          locals = [ top, top, int ]
        frame_type = 248 /* chop */
          offset_delta = 2
}
SourceFile: "question.scala"
InnerClasses:
  public static #51= #22 of #50;          // Sum=class question$Sum of class question
  public static abstract #52= #45 of #50; // Expr=class question$Expr of class question
  public static #53= #16 of #50;          // Number=class question$Number of class question
  ScalaInlineInfo: length = 0xE (unknown attribute)
   01 01 00 02 00 0A 00 09 01 00 0D 00 0E 01
  Scala: length = 0x0 (unknown attribute)

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