numerics - Is NumberForm double rounding numbers?

A number like 0.644696875 is represented internally as 0.6446968749999...:

N[FromDigits[RealDigits[0.644696875, 2], 2], $MachinePrecision]
(* 0.6446968749999999 *)

So if I ask NumberForm to print this number with 8 decimals, I would expect it to be 0.64469687 and not 0.64469688 because the digit after the 87 is a 4 which is less than 5. But it is not what we get with NumberForm:

NumberForm[0.644696875, {8, 8}]
(* 0.64469688 *)

So it looks like we have here two round operations when only one was requested:

First Rounding: From 0.6446968749999999 to 0.644696875

Second Rounding: From 0.644696875 to 0.64469688

I found this while comparing to Python, which doesn't double round. This leads to a result which I believe is correct:

ExternalEvaluate["Python", "'{:.8f}'.format(0.644696875)"]
(* 0.64469687 *)

Also notice that this floating point number is stored in the same way in both systems:

Divide @@ ExternalEvaluate["Python", "0.644696875.as_integer_ratio()"] == FromDigits[RealDigits[0.644696875, 2], 2]
(* True *)

Is Mathematica double rounding? Can this be avoided while still using machine numbers?

Motivation: I'm working on making a hash on an array of floating point numbers. This calculation should be the same from Mathematica and from Python. For this I need to be able to produce the same string representation of numbers in both system. This has proven more challenging than expected.

Update: I think Mathematica is double rounding just like Java does. Please see:

Update2: I asked support about this [CASE:4304365] and they said "It does appear that NumberForm is not behaving properly".

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

Blog

Search This Blog