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What is the Ultraradical
The ultraradical, or the Bring radical, of a real number \$a\$ is defined as the only real root of the quintic equation \$x^5+x+a=0\$.
Here we use \$UR(\cdot)\$ to denote the ultraradical function. For example, \$UR(-100010)=10\$, since \$10^5+10-100010=0\$.
Challenge
Write a full program or a function, that takes a real number as input, and returns or outputs its ultraradical.
Requirements
No standard loopholes are allowed. The results for the test cases below must be accurate to at least 6 significant digits, but in general the program should calculate the corresponding values for any valid real number inputs.
Test Cases
9 decimal places rounded towards 0 are given for reference. Explanation is added for some of the test cases.
a | UR(a)
---------------------------+---------------------
0 | 0.000 000 000 # 0
1 | -0.754 877 (666) # UR(a) < 0 when a > 0
-1 | 0.754 877 (666) # UR(a) > 0 when a < 0
1.414 213 562 | -0.881 616 (566) # UR(sqrt(2))
-2.718 281 828 | 1.100 93(2 665) # UR(-e)
3.141 592 653 | -1.147 96(5 385) # UR(pi)
-9.515 716 566 | 1.515 71(6 566) # 5th root of 8, fractional parts should match
10 | -1.533 01(2 798)
-100 | 2.499 20(3 570)
1 000 | -3.977 89(9 393)
-100 010 | 10.000 0(00 000) # a = (-10)^5 + (-10)
1 073 741 888 | -64.000 0(00 000) # a = 64^5 + 64
Winning Criteria
The shortest valid submission in every language wins.
9I keep wondering why Mathematica uses
and
instead of↦
andᵀ
– Adám – 2019-09-23T12:19:42.1072@Adám am I supposed to just see squares for the first two, or am I missing some kind of font... – mbrig – 2019-09-24T17:11:15.113
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@mbrig Just squares. That's my point. Mathematica uses characters in the Private Use Areas even though Unicode does have most of them.
– Adám – 2019-09-24T17:39:04.430