Scientific ALgorithms in hOON

wip as of ~2023.3.24

Transcendental and algebraic functions in native Hoon (and later jets).

It's time to supersede lazytrig.

Arms in ++rs and other @r-servicing cores will be provided with necessary transcendental and trigonometric functions.

As a first pass, we implement operators for @rs, @rd, @rq, and @rh. After Lagoon is merged, we implement @lvs, @lms, @lvd, and @lmd.

• @rs
• @rd
• @rq
• @rh
• @rsc
• @rdc
• @rqc
• @rhc
• @lvs
• @lvd
• @lvq
• @lvh
• @lms
• @lmd
• @lmq
• @lmh

This requires us to finally resolves the treatment of complex numbers in Hoon (which I've here noted as @rsc etc.).

• Possible four-letter alternative names: sake, salt, saxe, soon, star
• Possible two-letter core names: sa, sl

Checkmarks are a first pass in @rs single-precision floating-point implementation:

Constants

• pi
• e
• i (complex only)
• phi
• sqrt2

Comparison

• isclose
• allclose (over a (list @r))
• isint
• isreal (complex only)
• isimag (complex only)

Algebraic

• sgn (also ++sig for compatibility)
• abs
• sqrt
• cbrt
• arg
• conj (complex conjugate) (complex only)
• pow
• pown (faster integer pow)
• log
• log10
• log2
• exp
• binomial (Binomial coefficient)

Trigonometric

• sin
• cos
• tan
• csc
• sec
• cot
• arcsin
• arccos
• arctan
• arccsc
• arcsec
• arccot
• crd (chord)
• siv (versine)

Hyperbolic

• sinh
• cosh
• tanh
• csch
• sech
• coth
• arcsinh
• arccosh
• arctanh
• arccsch
• arcsech
• arccoth

Analytical

• besselj
• bessely
• besseli
• besselk
• ai
• bi
• gamma
• elliptic
• zeta

Operations

• round (regular banker's roundaing)
  • round-decimal (rounding to specific accuracy, e.g. .100.01)
• linspace (++gulf which evenly spans a range in floating-point)
• iota (APL-style ++gulf in floating-point)
• diff (placeholder for adaptive algorithm in future)
  • difffinite (finite difference)
  • diffpade (Padé approximation)
• integrate (placeholder for adaptive algorithm as door in future)
  • inttrapez (trapezoid rule)
  • intsimpson (Simpson's rule)
• newton (locate a function zero using Newton's method)

The library will pass through ++rs behaviors for ++add and so forth for ease of use. Thus Saloon's ++rs can act as a drop-in core for almost all Hoon ++rs arms (except ++exp and ++ma).

References

• Milton Abramowitz & Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 1964–2010.
• Forman Acton, Numerical Methods that (Usually) Work, 1ed. 1997.
• Bartosz Ciechanowski, “Float Exposed” (webapp)
• David Goldberg, “What Every Computer Scientist Should Know About Floating-Point Arithmetic”
• Parviz Moin, Fundamentals of Engineering Numerical Analysis. 2ed. 2001.