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optional log base for ticks functions #233

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optional log base for ticks functions #233

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a8ac7f9
feat: add optional base to ticks functions
nickofthyme Sep 21, 2021
73a4b62
add better tests around base e and 2
nickofthyme Oct 28, 2021
e983ad4
add base to nice function
nickofthyme Oct 28, 2021
4ecda5f
update docs to reflect addition of new base argument
nickofthyme Oct 28, 2021
13793c9
add more tests for tick with different bases
nickofthyme Oct 28, 2021
a764258
remove it.only in tests
nickofthyme Oct 28, 2021
f700c34
Merge branch 'main' into tick-with-base
nickofthyme Oct 28, 2021
79c5d11
Cleanup markdown styles
nickofthyme Nov 23, 2021
28d1a03
Merge branch 'main' into tick-with-base
nickofthyme Dec 8, 2023
c6ae425
merge with changes to main and use high precision Math methods
nickofthyme Dec 8, 2023
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4 changes: 2 additions & 2 deletions src/nice.js
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@@ -1,9 +1,9 @@
import {tickIncrement} from "./ticks.js";

export default function nice(start, stop, count) {
export default function nice(start, stop, count, log) {
let prestep;
while (true) {
const step = tickIncrement(start, stop, count);
const step = tickIncrement(start, stop, count, log);
if (step === prestep || step === 0 || !isFinite(step)) {
return [start, stop];
} else if (step > 0) {
Expand Down
38 changes: 24 additions & 14 deletions src/ticks.js
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Other implicit dependencies on base 10 (e.g., 2 and 5 are factors of 10, and 10 appears elsewhere in code)

Could you point me to the other parts of the code that would need to be addressed?

Expand Up @@ -2,35 +2,42 @@ const e10 = Math.sqrt(50),
e5 = Math.sqrt(10),
e2 = Math.sqrt(2);

function tickSpec(start, stop, count) {
const logParams = {
binary: [Math.log2, 2],
natural: [(n) => Math.log1p(n - 1), Math.E],
common: [Math.log10, 10],
Comment on lines +6 to +8
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In practice will probably only be used for base = 2 and base = e

Limits the log types to just common, natural and binary.

}
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Tiny loss of precision and performance using Math.log(base) etc. instead of Math.LN10

Does this approach work for you to use the respective precise Math methods for each allowed log type?


function tickSpec(start, stop, count, log = 'common') {
const [logFn, base] = logParams[log] || logParams.common;
const step = (stop - start) / Math.max(0, count),
power = Math.floor(Math.log10(step)),
error = step / Math.pow(10, power),
factor = error >= e10 ? 10 : error >= e5 ? 5 : error >= e2 ? 2 : 1;
power = Math.floor(logFn(step)),
error = step / Math.pow(base, power),
factor = error >= e10 ? 10 : error >= e5 ? 5 : error >= e2 ? 2 : 1;
let i1, i2, inc;
if (power < 0) {
inc = Math.pow(10, -power) / factor;
inc = Math.pow(base, -power) / factor;
i1 = Math.round(start * inc);
i2 = Math.round(stop * inc);
if (i1 / inc < start) ++i1;
if (i2 / inc > stop) --i2;
inc = -inc;
} else {
inc = Math.pow(10, power) * factor;
inc = Math.pow(base, power) * factor;
i1 = Math.round(start / inc);
i2 = Math.round(stop / inc);
if (i1 * inc < start) ++i1;
if (i2 * inc > stop) --i2;
}
if (i2 < i1 && 0.5 <= count && count < 2) return tickSpec(start, stop, count * 2);
if (i2 < i1 && 0.5 <= count && count < 2) return tickSpec(start, stop, count * 2, log);
return [i1, i2, inc];
}

export default function ticks(start, stop, count) {
export default function ticks(start, stop, count, log) {
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Too many positional arguments (start, stop, count, base). Overall, I think I would prefer to keep the existing methods as-is and have base-specific variants of d3.ticks and d3.tickIncrement for base = 2 (binary) and base = e (“natural”).

I'm happy to separate these added methods into separate methods. But I'd like to avoid duplication of code. Would you be ok with having the log option on a base function then exporting a wrapped function for each log type with just start, stop and count args?

So exporting ticks, ticksNatural and ticksBinary, along with their respective tickIncrement functions.

stop = +stop, start = +start, count = +count;
if (!(count > 0)) return [];
if (start === stop) return [start];
const reverse = stop < start, [i1, i2, inc] = reverse ? tickSpec(stop, start, count) : tickSpec(start, stop, count);
const reverse = stop < start, [i1, i2, inc] = reverse ? tickSpec(stop, start, count, log) : tickSpec(start, stop, count, log);
if (!(i2 >= i1)) return [];
const n = i2 - i1 + 1, ticks = new Array(n);
if (reverse) {
Expand All @@ -43,13 +50,16 @@ export default function ticks(start, stop, count) {
return ticks;
}

export function tickIncrement(start, stop, count) {
stop = +stop, start = +start, count = +count;
return tickSpec(start, stop, count)[2];
export function tickIncrement(start, stop, count, log) {
(stop = +stop), (start = +start), (count = +count);
return tickSpec(start, stop, count, log)[2];
}

export function tickStep(start, stop, count) {
export function tickStep(start, stop, count, log) {
stop = +stop, start = +start, count = +count;
const reverse = stop < start, inc = reverse ? tickIncrement(stop, start, count) : tickIncrement(start, stop, count);
const reverse = stop < start,
inc = reverse
? tickIncrement(stop, start, count, log)
: tickIncrement(start, stop, count, log);
return (reverse ? -1 : 1) * (inc < 0 ? 1 / -inc : inc);
}
21 changes: 21 additions & 0 deletions test/nice-test.js
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Expand Up @@ -44,3 +44,24 @@ it("nice(start, stop, count) returns the expected values", () => {
assert.deepStrictEqual(nice(132, 876, 5), [0, 1000]);
assert.deepStrictEqual(nice(132, 876, 1), [0, 1000]);
});

it("nice(start, stop, count, base) returns the expected values for binary log", () => {
assert.deepStrictEqual(nice(0.126, 0.51, 32, 'binary'), [0.125, 0.515625]);
assert.deepStrictEqual(nice(0.126, 0.48, 10, 'binary'), [0.125, 0.5]);
assert.deepStrictEqual(nice(0.126, 0.48, 6, 'binary'), [0.125, 0.5]);
assert.deepStrictEqual(nice(0.126, 0.48, 5, 'binary'), [0.125, 0.5]);
assert.deepStrictEqual(nice(0.126, 0.48, 1, 'binary'), [0, 0.5]);
assert.deepStrictEqual(nice(129, 876, 1000, 'binary'), [129, 876]);
assert.deepStrictEqual(nice(129, 876, 100, 'binary'), [128, 880]);
assert.deepStrictEqual(nice(129, 876, 30, 'binary'), [128, 896]);
assert.deepStrictEqual(nice(129, 876, 10, 'binary'), [128, 896]);
assert.deepStrictEqual(nice(129, 876, 6, 'binary'), [128, 896]);
assert.deepStrictEqual(nice(129, 876, 5, 'binary'), [128, 896]);
assert.deepStrictEqual(nice(129, 876, 1, 'binary'), [0, 1024]);
});

it("nice(start, stop, count, base) returns the expected values for natural log", () => {
assert.deepStrictEqual(nice(Math.E + 0.1, Math.E * 3 + 0.1, 2, 'natural'), [0, Math.E * 4]);
assert.deepStrictEqual(nice(-Math.E - 0.1, Math.E + 0.1, 2, 'natural'), [-Math.E * 2, Math.E * 2]);
assert.deepStrictEqual(nice(-Math.E * 10 - 0.5, Math.E * 20 - 0.5, 30, 'natural'), [-Math.E * 11, Math.E * 20]);
});
95 changes: 95 additions & 0 deletions test/tickIncrement-test.js
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Expand Up @@ -60,3 +60,98 @@ it("tickIncrement(start, stop, count) returns approximately count + 1 tickIncrem
assert.strictEqual(tickIncrement(-10, 10, 2), 10);
assert.strictEqual(tickIncrement(-10, 10, 1), 20);
});


describe("tickIncrement(start, stop, count, base) returns approximately count + 1 tickIncrement when start < stop", () => {
it("common", () => {
assert.strictEqual(tickIncrement( 0, 1, 10), -10);
assert.strictEqual(tickIncrement( 0, 1, 9), -10);
assert.strictEqual(tickIncrement( 0, 1, 8), -10);
assert.strictEqual(tickIncrement( 0, 1, 7), -5);
assert.strictEqual(tickIncrement( 0, 1, 6), -5);
assert.strictEqual(tickIncrement( 0, 1, 5), -5);
assert.strictEqual(tickIncrement( 0, 1, 4), -5);
assert.strictEqual(tickIncrement( 0, 1, 3), -2);
assert.strictEqual(tickIncrement( 0, 1, 2), -2);
assert.strictEqual(tickIncrement( 0, 1, 1), 1);
assert.strictEqual(tickIncrement( 0, 10, 10), 1);
assert.strictEqual(tickIncrement( 0, 10, 9), 1);
assert.strictEqual(tickIncrement( 0, 10, 8), 1);
assert.strictEqual(tickIncrement( 0, 10, 7), 2);
assert.strictEqual(tickIncrement( 0, 10, 6), 2);
assert.strictEqual(tickIncrement( 0, 10, 5), 2);
assert.strictEqual(tickIncrement( 0, 10, 4), 2);
assert.strictEqual(tickIncrement( 0, 10, 3), 5);
assert.strictEqual(tickIncrement( 0, 10, 2), 5);
assert.strictEqual(tickIncrement( 0, 10, 1), 10);
assert.strictEqual(tickIncrement(-10, 10, 10), 2);
assert.strictEqual(tickIncrement(-10, 10, 9), 2);
assert.strictEqual(tickIncrement(-10, 10, 8), 2);
assert.strictEqual(tickIncrement(-10, 10, 7), 2);
assert.strictEqual(tickIncrement(-10, 10, 6), 5);
assert.strictEqual(tickIncrement(-10, 10, 5), 5);
assert.strictEqual(tickIncrement(-10, 10, 4), 5);
assert.strictEqual(tickIncrement(-10, 10, 3), 5);
assert.strictEqual(tickIncrement(-10, 10, 2), 10);
assert.strictEqual(tickIncrement(-10, 10, 1), 20);
});

it("binary", () => {
assert.strictEqual(tickIncrement(0, 1, 10, "binary"), -8);
assert.strictEqual(tickIncrement(0, 1, 9, "binary"), -8);
assert.strictEqual(tickIncrement(0, 1, 8, "binary"), -8);
assert.strictEqual(tickIncrement(0, 1, 7, "binary"), -8);
assert.strictEqual(tickIncrement(0, 1, 6, "binary"), -8);
assert.strictEqual(tickIncrement(0, 1, 5, "binary"), -4);
assert.strictEqual(tickIncrement(0, 1, 4, "binary"), -4);
assert.strictEqual(tickIncrement(0, 1, 3, "binary"), -4);
assert.strictEqual(tickIncrement(0, 1, 2, "binary"), -2);
assert.strictEqual(tickIncrement(0, 1, 1, "binary"), 1);
assert.strictEqual(tickIncrement(0, 10, 10, "binary"), 1);
assert.strictEqual(tickIncrement(0, 10, 9, "binary"), 1);
assert.strictEqual(tickIncrement(0, 10, 8, "binary"), 1);
assert.strictEqual(tickIncrement(0, 10, 7, "binary"), 2);
assert.strictEqual(tickIncrement(0, 10, 6, "binary"), 2);
assert.strictEqual(tickIncrement(0, 10, 5, "binary"), 2);
assert.strictEqual(tickIncrement(0, 10, 4, "binary"), 2);
assert.strictEqual(tickIncrement(0, 10, 3, "binary"), 4);
assert.strictEqual(tickIncrement(0, 10, 2, "binary"), 4);
assert.strictEqual(tickIncrement(0, 10, 1, "binary"), 8);
assert.strictEqual(tickIncrement(-10, 10, 10, "binary"), 2);
assert.strictEqual(tickIncrement(-10, 10, 9, "binary"), 2);
assert.strictEqual(tickIncrement(-10, 10, 8, "binary"), 2);
assert.strictEqual(tickIncrement(-10, 10, 7, "binary"), 4);
assert.strictEqual(tickIncrement(-10, 10, 6, "binary"), 4);
assert.strictEqual(tickIncrement(-10, 10, 5, "binary"), 4);
assert.strictEqual(tickIncrement(-10, 10, 4, "binary"), 4);
assert.strictEqual(tickIncrement(-10, 10, 3, "binary"), 8);
assert.strictEqual(tickIncrement(-10, 10, 2, "binary"), 8);
assert.strictEqual(tickIncrement(-10, 10, 1, "binary"), 16);
});

it("natural", () => {
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E / 100, 10, 'natural')), -1);
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E / 10, 10, 'natural')), -0.5);
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E / 10, 1, 'natural')), -0.5);
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E, 3, 'natural')), -0.5);
assert.strictEqual(normalizeLn(tickIncrement( -Math.E, Math.E, 2, 'natural')), 1);
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E * 10, 10, 'natural')), 1);
assert.strictEqual(normalizeLn(tickIncrement( 1, Math.E * 10, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickIncrement( -Math.E * 10, Math.E * 10, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickIncrement( Math.E, Math.E * 100, 10, 'natural')), 1);
assert.strictEqual(normalizeLn(tickIncrement( -Math.E * 100, Math.E * 100, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickIncrement( 0, Math.E * 100, 10, 'natural')), 1);
assert.strictEqual(tickIncrement( 0, 10, 10, 'natural'), 1);
assert.strictEqual(tickIncrement( -10, 10, 10, 'natural'), 2);
});
})

export function normalizeLn(value) {
let tries = 0, normValue = value;
while (tries++ < 10 && normValue % 0.5 !== 0) normValue *= Math.E;
if (tries > 10) {
tries = 0, normValue = value
while (tries++ < 10 && normValue % 0.5 !== 0) normValue /= Math.E;
}
return tries > 10 ? NaN : normValue;
}
116 changes: 116 additions & 0 deletions test/tickStep-test.js
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@@ -1,4 +1,5 @@
import assert from "assert";
import {normalizeLn} from "./tickIncrement-test.js";
import {tickStep} from "../src/index.js";

it("tickStep(start, stop, count) returns NaN if any argument is NaN", () => {
Expand Down Expand Up @@ -93,3 +94,118 @@ it("tickStep(start, stop, count) returns -tickStep(stop, start, count)", () => {
assert.strictEqual(tickStep(-10, 10, 2), -tickStep(10, -10, 2));
assert.strictEqual(tickStep(-10, 10, 1), -tickStep(10, -10, 1));
});

it("tickStep(start, stop, count, base) with binary log returns approximately count + 1 tickStep when start < stop", () => {
assert.strictEqual(tickStep( 0, 1, 10, 'binary'), 0.125);
assert.strictEqual(tickStep( 0, 1, 9, 'binary'), 0.125);
assert.strictEqual(tickStep( 0, 1, 8, 'binary'), 0.125);
assert.strictEqual(tickStep( 0, 1, 7, 'binary'), 0.125);
assert.strictEqual(tickStep( 0, 1, 6, 'binary'), 0.125);
assert.strictEqual(tickStep( 0, 1, 5, 'binary'), 0.25);
assert.strictEqual(tickStep( 0, 1, 4, 'binary'), 0.25);
assert.strictEqual(tickStep( 0, 1, 3, 'binary'), 0.25);
assert.strictEqual(tickStep( 0, 1, 2, 'binary'), 0.5);
assert.strictEqual(tickStep( 0, 1, 1, 'binary'), 1);
assert.strictEqual(tickStep( 0, 10, 10, 'binary'), 1);
assert.strictEqual(tickStep( 0, 10, 9, 'binary'), 1);
assert.strictEqual(tickStep( 0, 10, 8, 'binary'), 1);
assert.strictEqual(tickStep( 0, 10, 7, 'binary'), 2);
assert.strictEqual(tickStep( 0, 10, 6, 'binary'), 2);
assert.strictEqual(tickStep( 0, 10, 5, 'binary'), 2);
assert.strictEqual(tickStep( 0, 10, 4, 'binary'), 2);
assert.strictEqual(tickStep( 0, 10, 3, 'binary'), 4);
assert.strictEqual(tickStep( 0, 10, 2, 'binary'), 4);
assert.strictEqual(tickStep( 0, 10, 1, 'binary'), 8);
assert.strictEqual(tickStep(-10, 10, 10, 'binary'), 2);
assert.strictEqual(tickStep(-10, 10, 9, 'binary'), 2);
assert.strictEqual(tickStep(-10, 10, 8, 'binary'), 2);
assert.strictEqual(tickStep(-10, 10, 7, 'binary'), 4);
assert.strictEqual(tickStep(-10, 10, 6, 'binary'), 4);
assert.strictEqual(tickStep(-10, 10, 5, 'binary'), 4);
assert.strictEqual(tickStep(-10, 10, 4, 'binary'), 4);
assert.strictEqual(tickStep(-10, 10, 3, 'binary'), 8);
assert.strictEqual(tickStep(-10, 10, 2, 'binary'), 8);
assert.strictEqual(tickStep(-10, 10, 1, 'binary'), 16);
});

it("tickStep(start, stop, count, base) with natural log returns approximately count + 1 tickStep when start < stop", () => {
assert.strictEqual(normalizeLn(tickStep( 0, Math.E / 100, 10, 'natural')), 1);
assert.strictEqual(normalizeLn(tickStep( 0, Math.E / 10, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( 0, Math.E / 10, 1, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( 0, Math.E, 3, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( -Math.E, Math.E, 2, 'natural')), 1);
assert.strictEqual(normalizeLn(tickStep( 0, Math.E * 10, 10, 'natural')), 1);
assert.strictEqual(normalizeLn(tickStep( 1, Math.E * 10, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( -Math.E * 10, Math.E * 10, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( Math.E, Math.E * 100, 10, 'natural')), 1);
assert.strictEqual(normalizeLn(tickStep( -Math.E * 100, Math.E * 100, 10, 'natural')), 2);
assert.strictEqual(normalizeLn(tickStep( 0, Math.E * 100, 10, 'natural')), 1);
assert.strictEqual(tickStep( 0, 10, 10, 'natural'), 1);
assert.strictEqual(tickStep( -10, 10, 10, 'natural'), 2);
});

it("tickStep(start, stop, count, base) with binary log returns -tickStep(stop, start, count)", () => {
assert.strictEqual(tickStep( 0, 1, 10, 'binary'), -tickStep( 1, 0, 10, 'binary'));
assert.strictEqual(tickStep( 0, 1, 9, 'binary'), -tickStep( 1, 0, 9, 'binary'));
assert.strictEqual(tickStep( 0, 1, 8, 'binary'), -tickStep( 1, 0, 8, 'binary'));
assert.strictEqual(tickStep( 0, 1, 7, 'binary'), -tickStep( 1, 0, 7, 'binary'));
assert.strictEqual(tickStep( 0, 1, 6, 'binary'), -tickStep( 1, 0, 6, 'binary'));
assert.strictEqual(tickStep( 0, 1, 5, 'binary'), -tickStep( 1, 0, 5, 'binary'));
assert.strictEqual(tickStep( 0, 1, 4, 'binary'), -tickStep( 1, 0, 4, 'binary'));
assert.strictEqual(tickStep( 0, 1, 3, 'binary'), -tickStep( 1, 0, 3, 'binary'));
assert.strictEqual(tickStep( 0, 1, 2, 'binary'), -tickStep( 1, 0, 2, 'binary'));
assert.strictEqual(tickStep( 0, 1, 1, 'binary'), -tickStep( 1, 0, 1, 'binary'));
assert.strictEqual(tickStep( 0, 10, 10, 'binary'), -tickStep(10, 0, 10, 'binary'));
assert.strictEqual(tickStep( 0, 10, 9, 'binary'), -tickStep(10, 0, 9, 'binary'));
assert.strictEqual(tickStep( 0, 10, 8, 'binary'), -tickStep(10, 0, 8, 'binary'));
assert.strictEqual(tickStep( 0, 10, 7, 'binary'), -tickStep(10, 0, 7, 'binary'));
assert.strictEqual(tickStep( 0, 10, 6, 'binary'), -tickStep(10, 0, 6, 'binary'));
assert.strictEqual(tickStep( 0, 10, 5, 'binary'), -tickStep(10, 0, 5, 'binary'));
assert.strictEqual(tickStep( 0, 10, 4, 'binary'), -tickStep(10, 0, 4, 'binary'));
assert.strictEqual(tickStep( 0, 10, 3, 'binary'), -tickStep(10, 0, 3, 'binary'));
assert.strictEqual(tickStep( 0, 10, 2, 'binary'), -tickStep(10, 0, 2, 'binary'));
assert.strictEqual(tickStep( 0, 10, 1, 'binary'), -tickStep(10, 0, 1, 'binary'));
assert.strictEqual(tickStep(-10, 10, 10, 'binary'), -tickStep(10, -10, 10, 'binary'));
assert.strictEqual(tickStep(-10, 10, 9, 'binary'), -tickStep(10, -10, 9, 'binary'));
assert.strictEqual(tickStep(-10, 10, 8, 'binary'), -tickStep(10, -10, 8, 'binary'));
assert.strictEqual(tickStep(-10, 10, 7, 'binary'), -tickStep(10, -10, 7, 'binary'));
assert.strictEqual(tickStep(-10, 10, 6, 'binary'), -tickStep(10, -10, 6, 'binary'));
assert.strictEqual(tickStep(-10, 10, 5, 'binary'), -tickStep(10, -10, 5, 'binary'));
assert.strictEqual(tickStep(-10, 10, 4, 'binary'), -tickStep(10, -10, 4, 'binary'));
assert.strictEqual(tickStep(-10, 10, 3, 'binary'), -tickStep(10, -10, 3, 'binary'));
assert.strictEqual(tickStep(-10, 10, 2, 'binary'), -tickStep(10, -10, 2, 'binary'));
assert.strictEqual(tickStep(-10, 10, 1, 'binary'), -tickStep(10, -10, 1, 'binary'));
});

it("tickStep(start, stop, count, base) with natural log returns -tickStep(stop, start, count)", () => {
assert.strictEqual(tickStep( 0, 1, 10, 'natural'), -tickStep( 1, 0, 10, 'natural'));
assert.strictEqual(tickStep( 0, 1, 9, 'natural'), -tickStep( 1, 0, 9, 'natural'));
assert.strictEqual(tickStep( 0, 1, 8, 'natural'), -tickStep( 1, 0, 8, 'natural'));
assert.strictEqual(tickStep( 0, 1, 7, 'natural'), -tickStep( 1, 0, 7, 'natural'));
assert.strictEqual(tickStep( 0, 1, 6, 'natural'), -tickStep( 1, 0, 6, 'natural'));
assert.strictEqual(tickStep( 0, 1, 5, 'natural'), -tickStep( 1, 0, 5, 'natural'));
assert.strictEqual(tickStep( 0, 1, 4, 'natural'), -tickStep( 1, 0, 4, 'natural'));
assert.strictEqual(tickStep( 0, 1, 3, 'natural'), -tickStep( 1, 0, 3, 'natural'));
assert.strictEqual(tickStep( 0, 1, 2, 'natural'), -tickStep( 1, 0, 2, 'natural'));
assert.strictEqual(tickStep( 0, 1, 1, 'natural'), -tickStep( 1, 0, 1, 'natural'));
assert.strictEqual(tickStep( 0, 10, 10, 'natural'), -tickStep(10, 0, 10, 'natural'));
assert.strictEqual(tickStep( 0, 10, 9, 'natural'), -tickStep(10, 0, 9, 'natural'));
assert.strictEqual(tickStep( 0, 10, 8, 'natural'), -tickStep(10, 0, 8, 'natural'));
assert.strictEqual(tickStep( 0, 10, 7, 'natural'), -tickStep(10, 0, 7, 'natural'));
assert.strictEqual(tickStep( 0, 10, 6, 'natural'), -tickStep(10, 0, 6, 'natural'));
assert.strictEqual(tickStep( 0, 10, 5, 'natural'), -tickStep(10, 0, 5, 'natural'));
assert.strictEqual(tickStep( 0, 10, 4, 'natural'), -tickStep(10, 0, 4, 'natural'));
assert.strictEqual(tickStep( 0, 10, 3, 'natural'), -tickStep(10, 0, 3, 'natural'));
assert.strictEqual(tickStep( 0, 10, 2, 'natural'), -tickStep(10, 0, 2, 'natural'));
assert.strictEqual(tickStep( 0, 10, 1, 'natural'), -tickStep(10, 0, 1, 'natural'));
assert.strictEqual(tickStep(-10, 10, 10, 'natural'), -tickStep(10, -10, 10, 'natural'));
assert.strictEqual(tickStep(-10, 10, 9, 'natural'), -tickStep(10, -10, 9, 'natural'));
assert.strictEqual(tickStep(-10, 10, 8, 'natural'), -tickStep(10, -10, 8, 'natural'));
assert.strictEqual(tickStep(-10, 10, 7, 'natural'), -tickStep(10, -10, 7, 'natural'));
assert.strictEqual(tickStep(-10, 10, 6, 'natural'), -tickStep(10, -10, 6, 'natural'));
assert.strictEqual(tickStep(-10, 10, 5, 'natural'), -tickStep(10, -10, 5, 'natural'));
assert.strictEqual(tickStep(-10, 10, 4, 'natural'), -tickStep(10, -10, 4, 'natural'));
assert.strictEqual(tickStep(-10, 10, 3, 'natural'), -tickStep(10, -10, 3, 'natural'));
assert.strictEqual(tickStep(-10, 10, 2, 'natural'), -tickStep(10, -10, 2, 'natural'));
assert.strictEqual(tickStep(-10, 10, 1, 'natural'), -tickStep(10, -10, 1, 'natural'));
});