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Guarantee exact tick values.
Adds d3.tickIncrement. Ticks are now computed in integer space and then either multiplied or divided by the tick increment. Fixes #45.
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‎README.md‎

Lines changed: 9 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -266,17 +266,21 @@ Randomizes the order of the specified *array* using the [Fisher–Yates shuffle]
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267267
<a name="ticks" href="#ticks">#</a> d3.<b>ticks</b>(<i>start</i>, <i>stop</i>, <i>count</i>) [<>](https://github.com/d3/d3-array/blob/master/src/ticks.js "Source")
268268

269-
Returns an array of approximately *count* + 1 uniformly-spaced, nicely-rounded values between *start* and *stop* (inclusive). Each value is a power of ten multiplied by 1, 2 or 5. See also [tickStep](#tickStep) and [*linear*.ticks](https://github.com/d3/d3-scale#linear_ticks). Note that due to the limited precision of IEEE 754 floating point, the returned values may not be exact decimals; use [d3-format](https://github.com/d3/d3-format) to format numbers for human consumption.
269+
Returns an array of approximately *count* + 1 uniformly-spaced, nicely-rounded values between *start* and *stop* (inclusive). Each value is a power of ten multiplied by 1, 2 or 5. See also [d3.tickIncrement](#tickIncrement), [d3.tickStep](#tickStep) and [*linear*.ticks](https://github.com/d3/d3-scale/blob/master/README.md#linear_ticks).
270270

271271
Ticks are inclusive in the sense that they may include the specified *start* and *stop* values if (and only if) they are exact, nicely-rounded values consistent with the inferred [step](#tickStep). More formally, each returned tick *t* satisfies *start* ≤ *t* and *t* ≤ *stop*.
272272

273+
<a name="tickIncrement" href="#tickIncrement">#</a> d3.<b>tickIncrement</b>(<i>start</i>, <i>stop</i>, <i>count</i>) [<>](https://github.com/d3/d3-array/blob/master/src/ticks.js#L16 "Source")
274+
275+
Like [d3.tickStep](#tickStep), except requires that *start* is always less than or equal to *step*, and if the tick step for the given *start*, *stop* and *count* would be less than one, returns the negative inverse tick step instead. This method is always guaranteed to return an integer, and is used by [d3.ticks](#ticks) to avoid guarantee that the returned tick values are represented as precisely as possible in IEEE 754 floating point.
276+
273277
<a name="tickStep" href="#tickStep">#</a> d3.<b>tickStep</b>(<i>start</i>, <i>stop</i>, <i>count</i>) [<>](https://github.com/d3/d3-array/blob/master/src/ticks.js#L16 "Source")
274278

275-
Returns the difference between adjacent tick values if the same arguments were passed to [ticks](#ticks): a nicely-rounded value that is a power of ten multiplied by 1, 2 or 5. Note that due to the limited precision of IEEE 754 floating point, the returned value may not be exact decimals; use [d3-format](https://github.com/d3/d3-format) to format numbers for human consumption.
279+
Returns the difference between adjacent tick values if the same arguments were passed to [d3.ticks](#ticks): a nicely-rounded value that is a power of ten multiplied by 1, 2 or 5. Note that due to the limited precision of IEEE 754 floating point, the returned value may not be exact decimals; use [d3-format](https://github.com/d3/d3-format) to format numbers for human consumption.
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277281
<a name="range" href="#range">#</a> d3.<b>range</b>([<i>start</i>, ]<i>stop</i>[, <i>step</i>]) [<>](https://github.com/d3/d3-array/blob/master/src/range.js "Source")
278282

279-
Returns an array containing an arithmetic progression, similar to the Python built-in [range](http://docs.python.org/library/functions.html#range). This method is often used to iterate over a sequence of uniformly-spaced numeric values, such as the indexes of an array or the ticks of a linear scale. (See also [ticks](#ticks) for nicely-rounded values.)
283+
Returns an array containing an arithmetic progression, similar to the Python built-in [range](http://docs.python.org/library/functions.html#range). This method is often used to iterate over a sequence of uniformly-spaced numeric values, such as the indexes of an array or the ticks of a linear scale. (See also [d3.ticks](#ticks) for nicely-rounded values.)
280284

281285
If *step* is omitted, it defaults to 1. If *start* is omitted, it defaults to 0. The *stop* value is exclusive; it is not included in the result. If *step* is positive, the last element is the largest *start* + *i* \* *step* less than *stop*; if *step* is negative, the last element is the smallest *start* + *i* \* *step* greater than *stop*. If the returned array would contain an infinite number of values, an empty range is returned.
282286

@@ -286,7 +290,7 @@ The arguments are not required to be integers; however, the results are more pre
286290
d3.range(0, 1, 0.2) // [0, 0.2, 0.4, 0.6000000000000001, 0.8]
287291
```
288292

289-
This unexpected behavior is due to IEEE 754 double-precision floating point, which defines 0.2 * 3 = 0.6000000000000001. Use [d3-format](https://github.com/d3/d3-format) to format numbers for human consumption with appropriate rounding; see also [linear.tickFormat](https://github.com/d3/d3-scale#linear_tickFormat) in [d3-scale](https://github.com/d3/d3-scale).
293+
This unexpected behavior is due to IEEE 754 double-precision floating point, which defines 0.2 * 3 = 0.6000000000000001. Use [d3-format](https://github.com/d3/d3-format) to format numbers for human consumption with appropriate rounding; see also [linear.tickFormat](https://github.com/d3/d3-scale/blob/master/README.md#linear_tickFormat) in [d3-scale](https://github.com/d3/d3-scale).
290294

291295
Likewise, if the returned array should have a specific length, consider using [array.map](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/map) on an integer range. For example:
292296

@@ -336,7 +340,7 @@ This is similar to mapping your data to values before invoking the histogram gen
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337341
If *domain* is specified, sets the domain accessor to the specified function or array and returns this histogram generator. If *domain* is not specified, returns the current domain accessor, which defaults to [extent](#extent). The histogram domain is defined as an array [*min*, *max*], where *min* is the minimum observable value and *max* is the maximum observable value; both values are inclusive. Any value outside of this domain will be ignored when the histogram is [generated](#_histogram).
338342

339-
For example, if you are using the the histogram in conjunction with a [linear scale](https://github.com/d3/d3-scale#linear-scales) `x`, you might say:
343+
For example, if you are using the the histogram in conjunction with a [linear scale](https://github.com/d3/d3-scale/blob/master/README.md#linear-scales) `x`, you might say:
340344

341345
```js
342346
var histogram = d3.histogram()

‎index.js‎

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -21,7 +21,7 @@ export {default as range} from "./src/range";
2121
export {default as scan} from "./src/scan";
2222
export {default as shuffle} from "./src/shuffle";
2323
export {default as sum} from "./src/sum";
24-
export {default as ticks, tickStep} from "./src/ticks";
24+
export {default as ticks, tickIncrement, tickStep} from "./src/ticks";
2525
export {default as transpose} from "./src/transpose";
2626
export {default as variance} from "./src/variance";
2727
export {default as zip} from "./src/zip";

‎src/ticks.js‎

Lines changed: 34 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -1,16 +1,42 @@
1-
import range from "./range";
2-
31
var e10 = Math.sqrt(50),
42
e5 = Math.sqrt(10),
53
e2 = Math.sqrt(2);
64

75
export default function(start, stop, count) {
8-
var step = tickStep(start, stop, count);
9-
return range(
10-
Math.ceil(start / step) * step,
11-
Math.floor(stop / step) * step + step / 2, // inclusive
12-
step
13-
);
6+
var reverse = stop < start,
7+
i = -1,
8+
n,
9+
ticks,
10+
step;
11+
12+
if (reverse) n = start, start = stop, stop = n;
13+
14+
if ((step = tickIncrement(start, stop, count)) === 0 || !isFinite(step)) return [];
15+
16+
if (step > 0) {
17+
start = Math.ceil(start / step);
18+
stop = Math.floor(stop / step);
19+
ticks = new Array(n = Math.ceil(stop - start + 1));
20+
while (++i < n) ticks[i] = (start + i) * step;
21+
} else {
22+
start = Math.floor(start * step);
23+
stop = Math.ceil(stop * step);
24+
ticks = new Array(n = Math.ceil(start - stop + 1));
25+
while (++i < n) ticks[i] = (start - i) / step;
26+
}
27+
28+
if (reverse) ticks.reverse();
29+
30+
return ticks;
31+
}
32+
33+
export function tickIncrement(start, stop, count) {
34+
var step = (stop - start) / Math.max(0, count),
35+
power = Math.floor(Math.log(step) / Math.LN10),
36+
error = step / Math.pow(10, power);
37+
return power >= 0
38+
? (error >= e10 ? 10 : error >= e5 ? 5 : error >= e2 ? 2 : 1) * Math.pow(10, power)
39+
: -Math.pow(10, -power) / (error >= e10 ? 10 : error >= e5 ? 5 : error >= e2 ? 2 : 1);
1440
}
1541

1642
export function tickStep(start, stop, count) {

‎test/ticks-test.js‎

Lines changed: 60 additions & 64 deletions
Original file line numberDiff line numberDiff line change
@@ -34,73 +34,69 @@ tape("ticks(start, stop, count) returns the empty array if count is infinity", f
3434
});
3535

3636
tape("ticks(start, stop, count) returns approximately count + 1 ticks when start < stop", function(test) {
37-
test.deepEqual(array.ticks( 0, 1, 10).map(round), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
38-
test.deepEqual(array.ticks( 0, 1, 9).map(round), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
39-
test.deepEqual(array.ticks( 0, 1, 8).map(round), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
40-
test.deepEqual(array.ticks( 0, 1, 7).map(round), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
41-
test.deepEqual(array.ticks( 0, 1, 6).map(round), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
42-
test.deepEqual(array.ticks( 0, 1, 5).map(round), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
43-
test.deepEqual(array.ticks( 0, 1, 4).map(round), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
44-
test.deepEqual(array.ticks( 0, 1, 3).map(round), [0.0, 0.5, 1.0]);
45-
test.deepEqual(array.ticks( 0, 1, 2).map(round), [0.0, 0.5, 1.0]);
46-
test.deepEqual(array.ticks( 0, 1, 1).map(round), [0.0, 1.0]);
47-
test.deepEqual(array.ticks( 0, 10, 10).map(round), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
48-
test.deepEqual(array.ticks( 0, 10, 9).map(round), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
49-
test.deepEqual(array.ticks( 0, 10, 8).map(round), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
50-
test.deepEqual(array.ticks( 0, 10, 7).map(round), [0, 2, 4, 6, 8, 10]);
51-
test.deepEqual(array.ticks( 0, 10, 6).map(round), [0, 2, 4, 6, 8, 10]);
52-
test.deepEqual(array.ticks( 0, 10, 5).map(round), [0, 2, 4, 6, 8, 10]);
53-
test.deepEqual(array.ticks( 0, 10, 4).map(round), [0, 2, 4, 6, 8, 10]);
54-
test.deepEqual(array.ticks( 0, 10, 3).map(round), [0, 5, 10]);
55-
test.deepEqual(array.ticks( 0, 10, 2).map(round), [0, 5, 10]);
56-
test.deepEqual(array.ticks( 0, 10, 1).map(round), [0, 10]);
57-
test.deepEqual(array.ticks(-10, 10, 10).map(round), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
58-
test.deepEqual(array.ticks(-10, 10, 9).map(round), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
59-
test.deepEqual(array.ticks(-10, 10, 8).map(round), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
60-
test.deepEqual(array.ticks(-10, 10, 7).map(round), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
61-
test.deepEqual(array.ticks(-10, 10, 6).map(round), [-10, -5, 0, 5, 10]);
62-
test.deepEqual(array.ticks(-10, 10, 5).map(round), [-10, -5, 0, 5, 10]);
63-
test.deepEqual(array.ticks(-10, 10, 4).map(round), [-10, -5, 0, 5, 10]);
64-
test.deepEqual(array.ticks(-10, 10, 3).map(round), [-10, -5, 0, 5, 10]);
65-
test.deepEqual(array.ticks(-10, 10, 2).map(round), [-10, 0, 10]);
66-
test.deepEqual(array.ticks(-10, 10, 1).map(round), [ 0, ]);
37+
test.deepEqual(array.ticks( 0, 1, 10), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
38+
test.deepEqual(array.ticks( 0, 1, 9), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
39+
test.deepEqual(array.ticks( 0, 1, 8), [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]);
40+
test.deepEqual(array.ticks( 0, 1, 7), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
41+
test.deepEqual(array.ticks( 0, 1, 6), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
42+
test.deepEqual(array.ticks( 0, 1, 5), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
43+
test.deepEqual(array.ticks( 0, 1, 4), [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]);
44+
test.deepEqual(array.ticks( 0, 1, 3), [0.0, 0.5, 1.0]);
45+
test.deepEqual(array.ticks( 0, 1, 2), [0.0, 0.5, 1.0]);
46+
test.deepEqual(array.ticks( 0, 1, 1), [0.0, 1.0]);
47+
test.deepEqual(array.ticks( 0, 10, 10), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
48+
test.deepEqual(array.ticks( 0, 10, 9), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
49+
test.deepEqual(array.ticks( 0, 10, 8), [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
50+
test.deepEqual(array.ticks( 0, 10, 7), [0, 2, 4, 6, 8, 10]);
51+
test.deepEqual(array.ticks( 0, 10, 6), [0, 2, 4, 6, 8, 10]);
52+
test.deepEqual(array.ticks( 0, 10, 5), [0, 2, 4, 6, 8, 10]);
53+
test.deepEqual(array.ticks( 0, 10, 4), [0, 2, 4, 6, 8, 10]);
54+
test.deepEqual(array.ticks( 0, 10, 3), [0, 5, 10]);
55+
test.deepEqual(array.ticks( 0, 10, 2), [0, 5, 10]);
56+
test.deepEqual(array.ticks( 0, 10, 1), [0, 10]);
57+
test.deepEqual(array.ticks(-10, 10, 10), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
58+
test.deepEqual(array.ticks(-10, 10, 9), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
59+
test.deepEqual(array.ticks(-10, 10, 8), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
60+
test.deepEqual(array.ticks(-10, 10, 7), [-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10]);
61+
test.deepEqual(array.ticks(-10, 10, 6), [-10, -5, 0, 5, 10]);
62+
test.deepEqual(array.ticks(-10, 10, 5), [-10, -5, 0, 5, 10]);
63+
test.deepEqual(array.ticks(-10, 10, 4), [-10, -5, 0, 5, 10]);
64+
test.deepEqual(array.ticks(-10, 10, 3), [-10, -5, 0, 5, 10]);
65+
test.deepEqual(array.ticks(-10, 10, 2), [-10, 0, 10]);
66+
test.deepEqual(array.ticks(-10, 10, 1), [ 0, ]);
6767
test.end();
6868
});
6969

7070
tape("ticks(start, stop, count) returns the reverse of ticks(stop, start, count)", function(test) {
71-
test.deepEqual(array.ticks( 1, 0, 10).map(round), array.ticks( 0, 1, 10).reverse().map(round));
72-
test.deepEqual(array.ticks( 1, 0, 9).map(round), array.ticks( 0, 1, 9).reverse().map(round));
73-
test.deepEqual(array.ticks( 1, 0, 8).map(round), array.ticks( 0, 1, 8).reverse().map(round));
74-
test.deepEqual(array.ticks( 1, 0, 7).map(round), array.ticks( 0, 1, 7).reverse().map(round));
75-
test.deepEqual(array.ticks( 1, 0, 6).map(round), array.ticks( 0, 1, 6).reverse().map(round));
76-
test.deepEqual(array.ticks( 1, 0, 5).map(round), array.ticks( 0, 1, 5).reverse().map(round));
77-
test.deepEqual(array.ticks( 1, 0, 4).map(round), array.ticks( 0, 1, 4).reverse().map(round));
78-
test.deepEqual(array.ticks( 1, 0, 3).map(round), array.ticks( 0, 1, 3).reverse().map(round));
79-
test.deepEqual(array.ticks( 1, 0, 2).map(round), array.ticks( 0, 1, 2).reverse().map(round));
80-
test.deepEqual(array.ticks( 1, 0, 1).map(round), array.ticks( 0, 1, 1).reverse().map(round));
81-
test.deepEqual(array.ticks(10, 0, 10).map(round), array.ticks( 0, 10, 10).reverse().map(round));
82-
test.deepEqual(array.ticks(10, 0, 9).map(round), array.ticks( 0, 10, 9).reverse().map(round));
83-
test.deepEqual(array.ticks(10, 0, 8).map(round), array.ticks( 0, 10, 8).reverse().map(round));
84-
test.deepEqual(array.ticks(10, 0, 7).map(round), array.ticks( 0, 10, 7).reverse().map(round));
85-
test.deepEqual(array.ticks(10, 0, 6).map(round), array.ticks( 0, 10, 6).reverse().map(round));
86-
test.deepEqual(array.ticks(10, 0, 5).map(round), array.ticks( 0, 10, 5).reverse().map(round));
87-
test.deepEqual(array.ticks(10, 0, 4).map(round), array.ticks( 0, 10, 4).reverse().map(round));
88-
test.deepEqual(array.ticks(10, 0, 3).map(round), array.ticks( 0, 10, 3).reverse().map(round));
89-
test.deepEqual(array.ticks(10, 0, 2).map(round), array.ticks( 0, 10, 2).reverse().map(round));
90-
test.deepEqual(array.ticks(10, 0, 1).map(round), array.ticks( 0, 10, 1).reverse().map(round));
91-
test.deepEqual(array.ticks(10, -10, 10).map(round), array.ticks(-10, 10, 10).reverse().map(round));
92-
test.deepEqual(array.ticks(10, -10, 9).map(round), array.ticks(-10, 10, 9).reverse().map(round));
93-
test.deepEqual(array.ticks(10, -10, 8).map(round), array.ticks(-10, 10, 8).reverse().map(round));
94-
test.deepEqual(array.ticks(10, -10, 7).map(round), array.ticks(-10, 10, 7).reverse().map(round));
95-
test.deepEqual(array.ticks(10, -10, 6).map(round), array.ticks(-10, 10, 6).reverse().map(round));
96-
test.deepEqual(array.ticks(10, -10, 5).map(round), array.ticks(-10, 10, 5).reverse().map(round));
97-
test.deepEqual(array.ticks(10, -10, 4).map(round), array.ticks(-10, 10, 4).reverse().map(round));
98-
test.deepEqual(array.ticks(10, -10, 3).map(round), array.ticks(-10, 10, 3).reverse().map(round));
99-
test.deepEqual(array.ticks(10, -10, 2).map(round), array.ticks(-10, 10, 2).reverse().map(round));
100-
test.deepEqual(array.ticks(10, -10, 1).map(round), array.ticks(-10, 10, 1).reverse().map(round));
71+
test.deepEqual(array.ticks( 1, 0, 10), array.ticks( 0, 1, 10).reverse());
72+
test.deepEqual(array.ticks( 1, 0, 9), array.ticks( 0, 1, 9).reverse());
73+
test.deepEqual(array.ticks( 1, 0, 8), array.ticks( 0, 1, 8).reverse());
74+
test.deepEqual(array.ticks( 1, 0, 7), array.ticks( 0, 1, 7).reverse());
75+
test.deepEqual(array.ticks( 1, 0, 6), array.ticks( 0, 1, 6).reverse());
76+
test.deepEqual(array.ticks( 1, 0, 5), array.ticks( 0, 1, 5).reverse());
77+
test.deepEqual(array.ticks( 1, 0, 4), array.ticks( 0, 1, 4).reverse());
78+
test.deepEqual(array.ticks( 1, 0, 3), array.ticks( 0, 1, 3).reverse());
79+
test.deepEqual(array.ticks( 1, 0, 2), array.ticks( 0, 1, 2).reverse());
80+
test.deepEqual(array.ticks( 1, 0, 1), array.ticks( 0, 1, 1).reverse());
81+
test.deepEqual(array.ticks(10, 0, 10), array.ticks( 0, 10, 10).reverse());
82+
test.deepEqual(array.ticks(10, 0, 9), array.ticks( 0, 10, 9).reverse());
83+
test.deepEqual(array.ticks(10, 0, 8), array.ticks( 0, 10, 8).reverse());
84+
test.deepEqual(array.ticks(10, 0, 7), array.ticks( 0, 10, 7).reverse());
85+
test.deepEqual(array.ticks(10, 0, 6), array.ticks( 0, 10, 6).reverse());
86+
test.deepEqual(array.ticks(10, 0, 5), array.ticks( 0, 10, 5).reverse());
87+
test.deepEqual(array.ticks(10, 0, 4), array.ticks( 0, 10, 4).reverse());
88+
test.deepEqual(array.ticks(10, 0, 3), array.ticks( 0, 10, 3).reverse());
89+
test.deepEqual(array.ticks(10, 0, 2), array.ticks( 0, 10, 2).reverse());
90+
test.deepEqual(array.ticks(10, 0, 1), array.ticks( 0, 10, 1).reverse());
91+
test.deepEqual(array.ticks(10, -10, 10), array.ticks(-10, 10, 10).reverse());
92+
test.deepEqual(array.ticks(10, -10, 9), array.ticks(-10, 10, 9).reverse());
93+
test.deepEqual(array.ticks(10, -10, 8), array.ticks(-10, 10, 8).reverse());
94+
test.deepEqual(array.ticks(10, -10, 7), array.ticks(-10, 10, 7).reverse());
95+
test.deepEqual(array.ticks(10, -10, 6), array.ticks(-10, 10, 6).reverse());
96+
test.deepEqual(array.ticks(10, -10, 5), array.ticks(-10, 10, 5).reverse());
97+
test.deepEqual(array.ticks(10, -10, 4), array.ticks(-10, 10, 4).reverse());
98+
test.deepEqual(array.ticks(10, -10, 3), array.ticks(-10, 10, 3).reverse());
99+
test.deepEqual(array.ticks(10, -10, 2), array.ticks(-10, 10, 2).reverse());
100+
test.deepEqual(array.ticks(10, -10, 1), array.ticks(-10, 10, 1).reverse());
101101
test.end();
102102
});
103-
104-
function round(x) {
105-
return Math.round(x * 1e12) / 1e12;
106-
}

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