What's the difference between java.lang.Math and java.lang.StrictMath?

Question

Obviously java.lang.StrictMath contains additional functions (hyperbolics etc.) which java.lang.Math doesn't, but is there a difference in the functions which are found in both libraries?

Answer 1

The Javadoc for the Math class provides some information on the differences between the two classes:

Unlike some of the numeric methods of class StrictMath, all implementations of the equivalent functions of class Math are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required.

By default many of the Math methods simply call the equivalent method in StrictMath for their implementation. Code generators are encouraged to use platform-specific native libraries or microprocessor instructions, where available, to provide higher-performance implementations of Math methods. Such higher-performance implementations still must conform to the specification for Math.

Therefore, the Math class lays out some rules about what certain operations should do, but they do not demand that the exact same results be returned in all implementations of the libraries.

This allows for specific implementations of the libraries to return similiar, but not the exact same result if, for example, the Math.cos class is called. This would allow for platform-specific implementations (such as using x86 floating point and, say, SPARC floating point) which may return different results.

(Refer to the Software Implementations section of the Sine article in Wikipedia for some examples of platform-specific implementations.)

However, with StrictMath, the results returned by different implementations must return the same result. This would be desirable for instances where the reproducibility of results on different platforms are required.

Answer 2

@ntoskrnl As somebody who is working with JVM internals, I would like to second your opinion that "intrinsics don't necessarily behave the same way as StrictMath methods". To find out (or prove) it, we can just write a simple test.

Take Math.pow for example, examining the Java code for java.lang.Math.pow(double a, double b), we will see:

 public static double pow(double a, double b) {
    return StrictMath.pow(a, b); // default impl. delegates to StrictMath
}

But the JVM is free to implement it with intrinsics or runtime calls, thus the returning result can be different from what we would expect from StrictMath.pow.

And the following code shows this calling Math.pow() against StrictMath.pow()

//Strict.java, testing StrictMath.pow against Math.pow
import java.util.Random;
public class Strict {
    static double testIt(double x, double y) {
        return Math.pow(x, y);
    }
    public static void main(String[] args) throws Exception{
        final double[] vs = new double[100];
        final double[] xs = new double[100];
        final double[] ys = new double[100];
        final Random random = new Random();

        // compute StrictMath.pow results;
        for (int i = 0; i<100; i++) {
            xs[i] = random.nextDouble();
            ys[i] = random.nextDouble();
            vs[i] = StrictMath.pow(xs[i], ys[i]);
        }
        boolean printed_compiled = false;
        boolean ever_diff = false;
        long len = 1000000;
        long start;
        long elapsed;
        while (true) {
            start = System.currentTimeMillis();
            double blackhole = 0;
            for (int i = 0; i < len; i++) {
                int idx = i % 100;
                double res = testIt(xs[idx], ys[idx]);
                if (i >= 0 && i<100) {
                    //presumably interpreted
                    if (vs[idx] != res && (!Double.isNaN(res) || !Double.isNaN(vs[idx]))) {
                        System.out.println(idx + ":\tInterpreted:" + xs[idx] + "^" + ys[idx] + "=" + res);
                        System.out.println(idx + ":\tStrict pow : " + xs[idx] + "^" + ys[idx] + "=" + vs[idx] + "\n");
                    }
                }
                if (i >= 250000 && i<250100 && !printed_compiled) {
                    //presumably compiled at this time
                    if (vs[idx] != res && (!Double.isNaN(res) || !Double.isNaN(vs[idx]))) {
                        System.out.println(idx + ":\tcompiled   :" + xs[idx] + "^" + ys[idx] + "=" + res);
                        System.out.println(idx + ":\tStrict pow :" + xs[idx] + "^" + ys[idx] + "=" + vs[idx] + "\n");
                        ever_diff = true;
                    }
                }
            }
            elapsed = System.currentTimeMillis() - start;
            System.out.println(elapsed + " ms ");
            if (!printed_compiled && ever_diff) {
                printed_compiled = true;
                return;
            }

        }
    }
}

I ran this test with OpenJDK 8u5-b31 and got the result below:

10: Interpreted:0.1845936372497491^0.01608930867480518=0.9731817015518033
10: Strict pow : 0.1845936372497491^0.01608930867480518=0.9731817015518032

41: Interpreted:0.7281259501809544^0.9414406865385655=0.7417808233050295
41: Strict pow : 0.7281259501809544^0.9414406865385655=0.7417808233050294

49: Interpreted:0.0727813262968815^0.09866028976654662=0.7721942440239148
49: Strict pow : 0.0727813262968815^0.09866028976654662=0.7721942440239149

70: Interpreted:0.6574309575966407^0.759887845481148=0.7270872740201638
70: Strict pow : 0.6574309575966407^0.759887845481148=0.7270872740201637

82: Interpreted:0.08662340816125613^0.4216580281197062=0.3564883826345057
82: Strict pow : 0.08662340816125613^0.4216580281197062=0.3564883826345058

92: Interpreted:0.20224488115245098^0.7158182878844233=0.31851834311978916
92: Strict pow : 0.20224488115245098^0.7158182878844233=0.3185183431197892

10: compiled   :0.1845936372497491^0.01608930867480518=0.9731817015518033
10: Strict pow :0.1845936372497491^0.01608930867480518=0.9731817015518032

41: compiled   :0.7281259501809544^0.9414406865385655=0.7417808233050295
41: Strict pow :0.7281259501809544^0.9414406865385655=0.7417808233050294

49: compiled   :0.0727813262968815^0.09866028976654662=0.7721942440239148
49: Strict pow :0.0727813262968815^0.09866028976654662=0.7721942440239149

70: compiled   :0.6574309575966407^0.759887845481148=0.7270872740201638
70: Strict pow :0.6574309575966407^0.759887845481148=0.7270872740201637

82: compiled   :0.08662340816125613^0.4216580281197062=0.3564883826345057
82: Strict pow :0.08662340816125613^0.4216580281197062=0.3564883826345058

92: compiled   :0.20224488115245098^0.7158182878844233=0.31851834311978916
92: Strict pow :0.20224488115245098^0.7158182878844233=0.3185183431197892

290 ms 

Please note that Random is used to generate the x and y values, so your mileage will vary from run to run. But good news is that at least the results of compiled version of Math.pow match those of interpreted version of Math.pow. (Off topic: even this consistency was only enforced in 2012 with a series of bug fixes from OpenJDK side.)

The reason?

Well, it's because OpenJDK uses intrinsics and runtime functions to implement Math.pow (and other math functions), instead of just executing the Java code. The main purpose is to take advantage of x87 instructions so that performance for the computation can be boosted. As a result, StrictMath.pow is never called from Math.pow at runtime (for the OpenJDK version that we just used, to be precise).

And this arragement is totally legitimate according to the Javadoc of Math class (also quoted by @coobird above):

The class Math contains methods for performing basic numeric operations such as the elementary exponential, logarithm, square root, and trigonometric functions.

Unlike some of the numeric methods of class StrictMath, all implementations of the equivalent functions of class Math are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required.

By default many of the Math methods simply call the equivalent method in StrictMath for their implementation. Code generators are encouraged to use platform-specific native libraries or microprocessor instructions, where available, to provide higher-performance implementations of Math methods. Such higher-performance implementations still must conform to the specification for Math.

And the conclusion? Well, for languages with dynamic code generation such as Java, please make sure what you see from the 'static' code matches what is executed at runtime. Your eyes can sometimes really mislead you.

Answer 3

Did you check the source code? Many methods in java.lang.Math are delegated to java.lang.StrictMath.

Example:

public static double cos(double a) {
    return StrictMath.cos(a); // default impl. delegates to StrictMath
}

Answer 4

Quoting java.lang.Math:

Accuracy of the floating-point Math methods is measured in terms of ulps, units in the last place.

...

If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is correctly rounded. A correctly rounded method is generally the best a floating-point approximation can be; however, it is impractical for many floating-point methods to be correctly rounded.

And then we see under Math.pow(..), for example:

The computed result must be within 1 ulp of the exact result.

Now, what is the ulp? As expected, java.lang.Math.ulp(1.0) gives 2.220446049250313e-16, which is 2^-52. (Also Math.ulp(8) gives the same value as Math.ulp(10) and Math.ulp(15), but not Math.ulp(16).) In other words, we are talking about the last bit of the mantissa.

So, the result returned by java.lang.Math.pow(..) may be wrong in the last of the 52 bits of the mantissa, as we can confirm in Tony Guan's answer.

It would be nice to dig up some concrete 1 ulp and 0.5 ulp code to compare. I'll speculate that quite a lot of extra work is required to get that last bit correct for the same reason that if we know two numbers A and B rounded to 52 significant figures and we wish to know A×B correct to 52 significant figures, with correct rounding, then actually we need to know a few extra bits of A and B to get the last bit of A×B right. But that means we shouldn't round intermediate results A and B by forcing them into doubles, we need, effectively, a wider type for intermediate results. (In what I've seen, most implementations of mathematical functions rely heavily on multiplications with hard-coded precomputed coefficients, so if they need to be wider than double, there's a big efficiency hit.)