accuracy of Math.cbrt, Math.expm1, Math.log1p

[Yaffle]

var error = function (x, value) {
  return Math.abs(1 - x / value) / 2.220446049250313e-16; // error in number of ulps (http://en.wikipedia.org/wiki/Unit_in_the_last_place)
};

Firefox < 48 on Windows:

https://bugzilla.mozilla.org/show_bug.cgi?id=933257
https://bugzilla.mozilla.org/show_bug.cgi?id=897634

error(Math.log1p(0.0006516748007717887), 0.0006514625529548382) > 500;
error(Math.expm1(0.00001), 0.000010000050000166668) > 43000;
error(Math.cbrt(-3.9355663969868275e+307), -3.401489081587483e+102) > 50;
error(Math.asinh(-0.0005), -0.000499999979166669) > 100; // (caused by issues with `expm1` and `log1p` on Windows)
error(Math.atanh(-0.0005), -0.0005000000416666729) > 100; // (caused by issues with `expm1` and `log1p` on Windows)

Firefox

Math.exp(1) !== Math.E

Chrome < 54 on Windows:

https://code.google.com/p/v8/issues/detail?id=3468
https://code.google.com/p/v8/issues/detail?id=3509

error(Math.exp(-707.9999999999998), 3.30755300363916e-308) > 250;
Math.acosh(1e+308) === Infinity;
Math.asinh(1e+300) === Infinity;
Math.atanh(1e-300) === 0;

Konqueror 4.13 (Ubuntu):

https://github.com/KDE/kjs/blob/master/src/kjs/math_object.cpp

   Math.expm1 == undefined && Math.exmp1 != undefined
   Math.LOG10E !== 0.4342944819032518 // 0.43429448190325176
   Math.abs(+0) === 0 && 1 / Math.abs(+0) < 0
   Math.hypot.length === 0
   Math.hypot(1e200, 1e200) === 1 / 0 // overflow and underflow issues

   Math.asinh == undefined;
   Math.clz32 == undefeined;
   Math.imul() == undefined;

buggy Math.round in Safari 10 / Mobile Safari 10 / IE 11 / Edge 13 / Opera 12 / Konqueror 4.13:

Math.round(0.49999999999999994) !== 0

Math.pow issue in Firefox, Chrome, Safari, Edge when the second argument is an integer:

https://code.google.com/p/v8/issues/detail?id=3599
https://bugzilla.mozilla.org/show_bug.cgi?id=618251

error(Math.pow(10, 305), 1e+305) > 2;
error(Math.pow(10, 308), 1e+308) > 2;
error(Math.pow(9767, 77), 1.627839019440664e+307) > 8
error(Math.pow(1 + Math.pow(2, -27), Math.pow(2, 31) - 1), 8886109.924647752) > Math.pow(2, 28)

Chrome Mobile 52 (Android 5.1):

Math.pow(11, Math.pow(11, 9)) === 0;

Mobile Safari 10 (iOS 9.2):

Math.expm1(709.782712893384) === -1;
// note: Math.expm1(709.7827128933839) === 1.7976931348620688e+308

es6-shim:

nothing

core-js:

error(Math.acosh(1 + Number.EPSILON), 2.1073424255447017e-8) > 1e+7;
Math.asinh(1e-300) === 0;
Math.asinh(1e+300) === Infinity;
Math.atanh(1e-300) === 0;
error(Math.cbrt(1e-300), 1e-100) > 50;
Math.cosh(710) === Infinity;
error(Math.expm1(1e-6), 0.0000010000005000001665) > 1e+5;
error(Math.log1p(1e-8), 9.999999950000001e-9) > 1e+7;
+ Math.sinh, Math.tanh (because of expm1)

Traceur AND MDN's polyfills:

Math.acosh(1e+308) === Infinity;
error(Math.acosh(1 + Number.EPSILON), 2.1073424255447017e-8) > 1e+7;
Math.asinh(1e-300) === 0;
Math.asinh(1e+300) === Infinity;
Math.atanh(1e-300) === 0;
error(Math.cbrt(1e-300), 1e-100) > 50;
Math.cosh(710) === Infinity;
Math.sinh(710) === Infinity;
Math.sinh(1e-300) === 0;
Math.tanh(1e-300) === 0;
Math.tanh(1e+300) !== Math.tanh(1e+300);
Math.expm1(1e-300) === 0;
Math.log1p(1e-300) === 0;

Closure:

Math.acosh(1e+308) === Infinity;
error(Math.acosh(1 + Number.EPSILON), 2.1073424255447017e-8) > 1e+7;
Math.asinh(1e-300) === 0;
Math.asinh(1e+300) === Infinity;
error(Math.cbrt(1e-300), 1e-100) > 50;
Math.cosh(710) === Infinity;
Math.sinh(710) === Infinity;
Math.sinh(1e-300) === 0;
Math.tanh(1e-300) === 0;
Math.hypot(0, 0) !== Math.hypot(0, 0)
Math.hypot(1, Infinity) !== Math.hypot(1, Infinity)

[ljharb]

Nothing in the spec defines the implementation or epsilon of the Math functions ( https://people.mozilla.org/~jorendorff/es6-draft.html#sec-function-properties-of-the-math-object ) It specifically says "For other argument values, these functions are intended to compute approximations to the results of familiar mathematical functions, but some latitude is allowed in the choice of approximation algorithms."

What compatibility is necessary to test here?

[kangax]

@Yaffle you should bring it up on es-discuss

[zloirock]

@Yaffle You can propose better implementation.

[zloirock]

@Yaffle I'm serious. I'm not specialist in math. You can propose your implementation math functions for these libraries.

[zloirock]

@Yaffle Thanks for Math.tanh and Math.acosh. I fixed them (but not forced for v8) and Math.hypot.

[ljharb]

Thanks - can you provide some specific examples for each one that I can use as failing tests?

[ljharb]

Math.round is ES5, not ES6, so please do file an es5-shim issue about that one :-) ES6 has Math.fround, which may have similar issues.

[zloirock]

Thanks!

[ljharb]

@Yaffle PRs with the tests to es5-shim/es6-shim, and probably also compat-table, would be very helpful - the implementations are great but the tests are the important part :-)

[ljharb]

@Yaffle thanks!

isExpm1OK, isLog1pOK, isAcoshOK, and isCbrtOK are already true in shimmed Firefox 37, Chrome 41, and Safari 8.

isAsinhOK and isAtanhOK are the only falses, and only in Chrome 41, they pass in FF/Safari.

I've filed an es6-shim issue for those two.

[ljharb]

Ah - new PRs welcome, let's make sure the comments include both the OS and browser version.

[ljharb]

@Yaffle thanks - it's hard to see what the problem is, could you perhaps indicate what the value should be, and what it is in the browser that deviates?

[ljharb]

Thanks, I filed #724 - if you could add comments with other Konqueror issues that'd be great, since I don't have a way to test it.

[kangax]

I would consider removing Konqueror. Seems like an ancient browser with very specific audience. I'm struggling to figure out who would still be using it.

[ljharb]

That's also a fair point - there may not be any value in including it.

[ljharb]

It would be helpful if you could indicate what the correct/best answer should be in all these cases as well

[ljharb]

@Yaffle lol wat? why did you delete all your comments?

[Yaffle]

@ljharb , I have merged everything into the first comment.

[Yaffle]

  Math.log2 = function (x) {
    x = Number(x);
    return Math.log(x) * Math.LOG2E;
  };
  Math.log10 = function (x) {
    x = Number(x);
    return Math.log(x) * Math.LOG10E;
  };

  Math.hypot = function (x, y) {
    // https://bugzilla.mozilla.org/show_bug.cgi?id=896264#c28
    var max = 0;
    var s = 0;
    for (var i = 0; i < arguments.length; i += 1) {
      var z = Number(arguments[i]);
      var a = Math.abs(z);
      if (a === 1 / 0) {
        return a;
      }
      if (a !== 0) {
        if (a > max) {
          s *= (max / a) * (max / a);
          max = a;
        }
        s += (a / max) * (a / max);
      }
    }
    return max * Math.sqrt(s);
  };
  Math.cbrt = function (x) {
    x = Number(x);
    // http://en.wikipedia.org/wiki/Cube_root#Numerical_methods
    if (x !== x || x === 0 || x === -1 / 0 || x === +1 / 0) {
      return x;
    }
    var s = x < 0 ? -1 : +1;
    var a = Math.abs(x);
    var y0 = Math.exp(Math.log(a) / 3);
    return s * (a / (y0 * y0) + 2 * y0) / 3;
  };

  Math.log1p = function (x) {
    x = Number(x);
    // https://bugs.ecmascript.org/show_bug.cgi?id=309#c16
    if (x !== x || x === 0 || x === +1 / 0) {
      return x;
    }
    var x0 = 1 + x;
    var y0 = Math.log(x0);
    return x0 > 0 ? y0 + (1 - x0 + x) / x0 : y0;
  };
  Math.expm1 = function (x) {
    x = Number(x);
    // https://en.wikipedia.org/wiki/Newton%27s_method
    if (x !== x || x === 0) {
      return x;
    }
    var y0 = Math.exp(x);
    var x0 = Math.log(y0);
    return y0 - 1 + (x0 === -1 / 0 || x0 === +1 / 0 ? 0 : (x - x0) * y0);
  };

  Math.acosh = function (x) {
    x = Number(x);
    if (x < 1) {
      return 0 / 0;
    }
    if (x < 2) {
      return Math.log1p(x - 1 + Math.sqrt(1 - 1 / (x * x)) * x);
    }
    return Math.log1p(x / 2 + Math.sqrt(1 - 1 / (x * x)) * x / 2 - 1) + 1 / Math.LOG2E;
  };
  Math.asinh = function (x) {
    x = Number(x);
    if (x !== x || x === 0) {
      return x;
    }
    var s = x < 0 ? -1 : +1;
    var a = Math.abs(x);
    if (a < 1) {
      return s * Math.log1p(a + a * a / (Math.sqrt(a * a + 1) + 1));
    }
    return s * (Math.log1p(a / 2 + Math.sqrt(1 + 1 / (a * a)) * a / 2 - 1) + 1 / Math.LOG2E);
  };
  Math.atanh = function (x) {
    x = Number(x);
    if (x !== x || x === 0) {
      return x;
    }
    var s = x < 0 ? -1 : +1;
    var a = Math.abs(x);
    return s * Math.log1p(2 * a / (1 - a)) / 2;
  };
  Math.cosh = function (x) {
    x = Number(x);
    if (x !== x) {
      return x;
    }
    var a = Math.abs(x);
    var t = Math.expm1(a - 1) + 1;
    return (t + 1 / (t * Math.E * Math.E)) * (Math.E / 2);
  };
  Math.sinh = function (x) {
    x = Number(x);
    if (x !== x || x === 0) {
      return x;
    }
    var s = x < 0 ? -1 : +1;
    var a = Math.abs(x);
    if (a < 1) {
      var u = Math.expm1(a);
      return s * u * (1 + 1 / (u + 1)) / 2;
    }
    var t = Math.expm1(a - 1) + 1;
    return s * (t - 1 / (t * Math.E * Math.E)) * (Math.E / 2);
  };
  Math.tanh = function (x) {
    x = Number(x);
    if (x !== x || x === 0) {
      return x;
    }
    var s = x < 0 ? -1 : +1;
    var a = Math.abs(x);
    if (a < 1) {
      var u = Math.expm1(2 * a);
      return s * u / (u + 2);
    }
    var t = Math.expm1(2 * a) + 1;
    return s / (1 + 2 / (t - 1));
  };