diff --git a/js/src/tests/ecma_6/Math/acosh-approx.js b/js/src/tests/ecma_6/Math/acosh-approx.js
index 630303fe8d5..53800e2ebe6 100644
--- a/js/src/tests/ecma_6/Math/acosh-approx.js
+++ b/js/src/tests/ecma_6/Math/acosh-approx.js
@@ -262,20 +262,22 @@ var cosh_data = [
     [1875817529344, 28.953212876533797]
 ];
 
+var sloppy_tolerance = 1000;  // FIXME
+
 for (var [x, y] of cosh_data)
-    assertNear(Math.acosh(x), y);
+    assertNear(Math.acosh(x), y, sloppy_tolerance);
+
+assertNear(Math.acosh(1e300), 691.4686750787737, sloppy_tolerance);
+assertNear(Math.acosh(1.0000000001), 0.000014142136208675862, sloppy_tolerance);
 
 for (var i = 0; i <= 100; i++) {
     var x = (i - 50) / 5;
     var y = Math.cosh(x);
     var z = Math.acosh(y);
-    assertNear(z, Math.abs(x));
+    assertNear(z, Math.abs(x), sloppy_tolerance);
 }
 
 for (var i = 1; i < 20; i++)
-    assertNear(Math.acosh(Math.cosh(i)), i);
-
-assertNear(Math.acosh(1e300), 691.4686750787737);
-assertNear(Math.acosh(1.0000000001), 0.000014142136208675862);
+    assertNear(Math.acosh(Math.cosh(i)), i, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
diff --git a/js/src/tests/ecma_6/Math/asinh-approx.js b/js/src/tests/ecma_6/Math/asinh-approx.js
index bfb8eb47935..daf60ed8dbb 100644
--- a/js/src/tests/ecma_6/Math/asinh-approx.js
+++ b/js/src/tests/ecma_6/Math/asinh-approx.js
@@ -282,24 +282,24 @@ var sinh_data = [
     [1581915832320, 28.78280496108106]
 ];
 
+var sloppy_tolerance = 1000;  // FIXME
+
 for (var [x, y] of sinh_data)
-    assertNear(Math.asinh(x), y);
+    assertNear(Math.asinh(x), y, sloppy_tolerance);
+
+assertNear(Math.asinh(1e300), 691.4686750787737, sloppy_tolerance);
+assertNear(Math.asinh(1e-300), 1e-300, sloppy_tolerance);
+assertNear(Math.asinh(1e-5), 0.000009999999999833334, sloppy_tolerance);
+assertNear(Math.asinh(0.3), 0.29567304756342244, sloppy_tolerance);
+assertNear(Math.asinh(1), 0.881373587019543, sloppy_tolerance);
 
 for (var i = 0; i <= 80; i++) {
     var x = (i - 40) / 4;
-    var y = Math.sinh(x);
-    var z = Math.asinh(y);
-    assertNear(z, x);
+    assertNear(Math.asinh(Math.sinh(x)), x, sloppy_tolerance);
 }
 
 for (var i = -20; i < 20; i++)
-  assertNear(Math.asinh(Math.sinh(i)), i);
-
-assertNear(Math.asinh(1e300), 691.4686750787737);
-assertNear(Math.asinh(1e-300), 1e-300);
-assertNear(Math.asinh(1e-5), 0.000009999999999833334);
-assertNear(Math.asinh(0.3), 0.29567304756342244);
-assertNear(Math.asinh(1), 0.881373587019543);
+    assertNear(Math.asinh(Math.sinh(i)), i, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
 
diff --git a/js/src/tests/ecma_6/Math/atanh-approx.js b/js/src/tests/ecma_6/Math/atanh-approx.js
index 41d4fd1ca95..998db4e7f2c 100644
--- a/js/src/tests/ecma_6/Math/atanh-approx.js
+++ b/js/src/tests/ecma_6/Math/atanh-approx.js
@@ -1,6 +1,3 @@
-for (var i = -1; i < 1; i += 0.05)
-    assertNear(Math.atanh(Math.tanh(i)), i);
-
 var tanh_data = [
     [-0.9999983310699463, -6.998237084679027],
     [-0.9999978542327881, -6.87257975132917],
@@ -269,10 +266,15 @@ var tanh_data = [
     [1e-10, 1e-10],
 ];
 
-for (var [x, y] of tanh_data)
-    assertNear(Math.atanh(x), y);
+var sloppy_tolerance = 10;  // FIXME
 
-assertNear(Math.atanh(+3 / 5), +Math.log(2));
-assertNear(Math.atanh(-3 / 5), -Math.log(2));
+for (var [x, y] of tanh_data)
+    assertNear(Math.atanh(x), y, sloppy_tolerance);
+
+assertNear(Math.atanh(+3 / 5), +Math.log(2), sloppy_tolerance);
+assertNear(Math.atanh(-3 / 5), -Math.log(2), sloppy_tolerance);
+
+for (var i = -1; i < 1; i += 0.05)
+    assertNear(Math.atanh(Math.tanh(i)), i, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
diff --git a/js/src/tests/ecma_6/Math/cbrt-approx.js b/js/src/tests/ecma_6/Math/cbrt-approx.js
index b5511753756..2c7f26fafc2 100644
--- a/js/src/tests/ecma_6/Math/cbrt-approx.js
+++ b/js/src/tests/ecma_6/Math/cbrt-approx.js
@@ -1,8 +1,10 @@
 assertEq(Math.cbrt(1), 1);
 assertEq(Math.cbrt(-1), -1);
 
-assertNear(Math.cbrt(1e-300), 1e-100);
-assertNear(Math.cbrt(-1e-300), -1e-100);
+var sloppy_tolerance = 200;  // FIXME
+
+assertNear(Math.cbrt(1e-300), 1e-100, sloppy_tolerance);
+assertNear(Math.cbrt(-1e-300), -1e-100, sloppy_tolerance);
 
 var cbrt_data = [
     [ Math.E, 1.3956124250860895 ], 
@@ -12,7 +14,6 @@ var cbrt_data = [
 ];
 
 for (var [x, y] of cbrt_data)
-    assertNear(Math.cbrt(x), y);
+    assertNear(Math.cbrt(x), y, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
-
diff --git a/js/src/tests/ecma_6/Math/cosh-approx.js b/js/src/tests/ecma_6/Math/cosh-approx.js
index acf18668df9..b701bb1df98 100644
--- a/js/src/tests/ecma_6/Math/cosh-approx.js
+++ b/js/src/tests/ecma_6/Math/cosh-approx.js
@@ -1,9 +1,9 @@
-for (var i = -20; i < 20; i++)
-    assertNear(Math.cosh(i), (Math.exp(i) + Math.exp(-i)) / 2);
+var sloppy_tolerance = 100;
 
-assertNear(Math.cosh(1e5), Infinity);
-assertNear(Math.cosh(1e-30), 1);
-assertNear(Math.cosh(1e-10), 1);
+assertEq(Math.cosh(1000), Infinity);
+assertEq(Math.cosh(Number.MAX_VALUE), Infinity);
+assertNear(Math.cosh(1e-30), 1, sloppy_tolerance);
+assertNear(Math.cosh(1e-10), 1, sloppy_tolerance);
 
 var cosh_data = [
     [0.0016914556651292944, 1.0000014305114746],
@@ -270,6 +270,9 @@ var cosh_data = [
 ];
 
 for (var [x, y] of cosh_data)
-    assertNear(Math.cosh(x), y);
+    assertNear(Math.cosh(x), y, sloppy_tolerance);
+
+for (var i = -20; i < 20; i++)
+    assertNear(Math.cosh(i), (Math.exp(i) + Math.exp(-i)) / 2, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
diff --git a/js/src/tests/ecma_6/Math/hypot-approx.js b/js/src/tests/ecma_6/Math/hypot-approx.js
index 9960d55d2ea..45c22afb9e5 100644
--- a/js/src/tests/ecma_6/Math/hypot-approx.js
+++ b/js/src/tests/ecma_6/Math/hypot-approx.js
@@ -1,17 +1,16 @@
 // |reftest| skip
 // Math.hypot is disabled pending the resolution of spec issues (bug 896264).
 
-for (var i = -20; i < 20; i++)
+for (var i = -20; i < 20; i++) {
     assertEq(Math.hypot(+0, i), Math.abs(i));
-
-for (var i = -20; i < 20; i++)
     assertEq(Math.hypot(-0, i), Math.abs(i));
-
-for (var i = 1, j = 1; i < 2; i += 0.05, j += 0.05)
-    assertNear(Math.hypot(i, j), Math.sqrt(i * i + j * j));
+}
 
 assertNear(Math.hypot(1e300, 1e300), 1.4142135623730952e+300);
 assertNear(Math.hypot(1e-300, 1e-300), 1.414213562373095e-300);
 assertNear(Math.hypot(1e3, 1e-3), 1000.0000000005);
 
+for (var i = 1, j = 1; i < 2; i += 0.05, j += 0.05)
+    assertNear(Math.hypot(i, j), Math.sqrt(i * i + j * j));
+
 reportCompare(0, 0, "ok");
diff --git a/js/src/tests/ecma_6/Math/log10-approx.js b/js/src/tests/ecma_6/Math/log10-approx.js
index d6919efa523..04894485130 100644
--- a/js/src/tests/ecma_6/Math/log10-approx.js
+++ b/js/src/tests/ecma_6/Math/log10-approx.js
@@ -1,9 +1,9 @@
-for (var i = -10; i < 10; i++)
-    assertNear(Math.log10(Math.pow(10, i)), i);
-
 assertNear(Math.log10(2), 0.3010299956639812);
 assertNear(Math.log10(7), 0.8450980400142568);
 assertNear(Math.log10(Math.E), Math.LOG10E);
 
+for (var i = -10; i < 10; i++)
+    assertNear(Math.log10(Math.pow(10, i)), i);
+
 reportCompare(0, 0, 'ok');
 
diff --git a/js/src/tests/ecma_6/Math/shell.js b/js/src/tests/ecma_6/Math/shell.js
index deed2bd9262..d423c0814c6 100644
--- a/js/src/tests/ecma_6/Math/shell.js
+++ b/js/src/tests/ecma_6/Math/shell.js
@@ -2,10 +2,73 @@
 const ONE_PLUS_EPSILON = 1 + Math.pow(2, -52);  // 0.9999999999999999
 const ONE_MINUS_EPSILON = 1 - Math.pow(2, -53);  // 1.0000000000000002
 
-function assertNear(actual, expected) {
-    var error = Math.abs(actual - expected);
+{
+    var fail = function (msg) {
+        var exc = new Error(msg);
+        try {
+            // Try to improve on exc.fileName and .lineNumber; leave exc.stack
+            // alone. We skip two frames: fail() and its caller, an assertX()
+            // function.
+            var frames = exc.stack.trim().split("\n");
+            if (frames.length > 2) {
+                var m = /@([^@:]*):([0-9]+)$/.exec(frames[2]);
+                if (m) {
+                    exc.fileName = m[1];
+                    exc.lineNumber = +m[2];
+                }
+            }
+        } catch (ignore) { throw ignore;}
+        throw exc;
+    };
 
-    if (error > 1e-300 && error > Math.abs(actual) * 1e-12)
-        throw 'Assertion failed: got "' + actual + '", expected "' + expected + '" (rel error = ' + (error / Math.abs(actual)) + ')';
+    var ENDIAN;  // 0 for little-endian, 1 for big-endian.
+
+    // Return the difference between the IEEE 754 bit-patterns for a and b.
+    //
+    // This is meaningful when a and b are both finite and have the same
+    // sign. Then the following hold:
+    //
+    //   * If a === b, then diff(a, b) === 0.
+    //
+    //   * If a !== b, then diff(a, b) === 1 + the number of representable values
+    //                                         between a and b.
+    //
+    var f = new Float64Array([0, 0]);
+    var u = new Uint32Array(f.buffer);
+    var diff = function (a, b) {
+        f[0] = a;
+        f[1] = b;
+        //print(u[1].toString(16) + u[0].toString(16) + " " + u[3].toString(16) + u[2].toString(16));
+        return Math.abs((u[3-ENDIAN] - u[1-ENDIAN]) * 0x100000000 + u[2+ENDIAN] - u[0+ENDIAN]);
+    };
+
+    // Set ENDIAN to the platform's endianness.
+    ENDIAN = 0;  // try little-endian first
+    if (diff(2, 4) === 0x100000)  // exact wrong answer we'll get on a big-endian platform
+        ENDIAN = 1;
+    assertEq(diff(2,4), 0x10000000000000);
+    assertEq(diff(0, Number.MIN_VALUE), 1);
+    assertEq(diff(1, ONE_PLUS_EPSILON), 1);
+    assertEq(diff(1, ONE_MINUS_EPSILON), 1);
+
+    var assertNear = function assertNear(a, b, tolerance=1) {
+        if (!Number.isFinite(b)) {
+            fail("second argument to assertNear (expected value) must be a finite number");
+        } else if (Number.isNaN(a)) {
+            fail("got NaN, expected a number near " + b);
+        } else if (!Number.isFinite(a)) {
+            if (b * Math.sign(a) < Number.MAX_VALUE)
+                fail("got " + a + ", expected a number near " + b);
+        } else {
+            // When the two arguments do not have the same sign bit, diff()
+            // returns some huge number. So if b is positive or negative 0,
+            // make target the zero that has the same sign bit as a.
+            var target = b === 0 ? a * 0 : b;
+            var err = diff(a, target);
+            if (err > tolerance) {
+                fail("got " + a + ", expected a number near " + b +
+                     " (relative error: " + err + ")");
+            }
+        }
+    };
 }
-
diff --git a/js/src/tests/ecma_6/Math/sinh-approx.js b/js/src/tests/ecma_6/Math/sinh-approx.js
index 415110d92ea..91eda049acb 100644
--- a/js/src/tests/ecma_6/Math/sinh-approx.js
+++ b/js/src/tests/ecma_6/Math/sinh-approx.js
@@ -1,9 +1,11 @@
+var sloppy_tolerance = 100;
 for (var i = -20; i < 20; i++)
-    assertNear(Math.sinh(i), (Math.exp(i) - Math.exp(-i)) / 2);
+    assertNear(Math.sinh(i), (Math.exp(i) - Math.exp(-i)) / 2, sloppy_tolerance);
 
-assertNear(Math.sinh(1e5), Infinity);
-assertNear(Math.sinh(1e-30), 1e-30);
-assertNear(Math.sinh(1e-10), 1e-10);
+assertEq(Math.sinh(1000), Infinity);
+assertEq(Math.sinh(Number.MAX_VALUE), Infinity);
+assertNear(Math.sinh(1e-30), 1e-30, sloppy_tolerance);
+assertNear(Math.sinh(1e-10), 1e-10, sloppy_tolerance);
 
 var sinh_data = [
     [-6.902103625349695, -497.1816406250001],
@@ -290,7 +292,7 @@ var sinh_data = [
 ];
 
 for (var [x, y] of sinh_data)
-    assertNear(Math.sinh(x), y);
+    assertNear(Math.sinh(x), y, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");
 
diff --git a/js/src/tests/ecma_6/Math/tanh-approx.js b/js/src/tests/ecma_6/Math/tanh-approx.js
index dd9bf0e91cc..b41427c30c3 100644
--- a/js/src/tests/ecma_6/Math/tanh-approx.js
+++ b/js/src/tests/ecma_6/Math/tanh-approx.js
@@ -1,5 +1,10 @@
-for (var i = -20; i < 20; i++)
-    assertNear(Math.tanh(i), (Math.exp(i) - Math.exp(-i)) / (Math.exp(i) + Math.exp(-i)));
+var sloppy_tolerance = 4;
+
+for (var i = -20; i < 20; i++) {
+    assertNear(Math.tanh(i),
+               (Math.exp(i) - Math.exp(-i)) / (Math.exp(i) + Math.exp(-i)),
+               sloppy_tolerance);
+}
 
 assertEq(Math.tanh(1e300), 1);
 
@@ -272,6 +277,6 @@ var tanh_data = [
 ];
 
 for (var [x, y] of tanh_data)
-    assertNear(Math.tanh(y), x);
+    assertNear(Math.tanh(y), x, sloppy_tolerance);
 
 reportCompare(0, 0, "ok");