Add Chebyshev approximations #18
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Prototype implementation demonstrating jet-compatible approach for transcendental functions (exp, sin, cos, tan) using: - Fixed polynomial coefficients from musl libc - Deterministic argument reduction - Horner's method evaluation Key principle: every FP operation happens in same order with same intermediate values in both Hoon and C jet.
Complete fixed-polynomial math library with: - Half precision (@rh): degree 5 sin, degree 4 cos, degree 3 exp - Single precision (@rs): degree 9 sin, degree 10 cos, degree 4 exp - Double precision (@rd): degree 13 sin, degree 14 cos, degree 5 exp All coefficients from musl libc expressed as hex literals for bit-exact jet compatibility. Each precision level includes: - Polynomial coefficients and reduction constants - Chebyshev polynomial generator (+cheb) - Kernel functions (sindf, cosdf) - Public API (sin, cos, tan, exp) - Helper functions (scalbn, floor-int, rem-pio2)
Complete binary128 (128-bit) implementation with: - Degree 17 sin polynomial (S1-S8) - Degree 16 cos polynomial (C1-C8) - Degree 7 exp polynomial (P1-P7) - Chebyshev polynomial generator (+cheb) - Helper functions (scalbn, floor-int, rem-pio2) Coefficients derived from FreeBSD ld128/k_sinl.c, k_cosl.c. Uses .~~~N syntax for quad precision literals.
Extended math-chebyshev.hoon with: - log: Fixed polynomial (Lg1-Lg4 for @rs, Lg1-Lg7 for @rd) - sqrt: Newton-Raphson with fixed iterations (4 for @rs, 6 for @rd) - atan: Fixed polynomial with range reduction - asin, acos, atan2: Derived from atan - pow-n: Binary exponentiation (fixed 32/64 iterations) - pow: exp(n*log(x)) for non-integer exponents - log-2, log-10: Derived from log Added test suite with: - Individual function tests for @rs and @rd - Identity tests (sin^2+cos^2=1, exp(log(x))=x, etc.) - Chebyshev polynomial verification
Half precision (@rh): - log: Simplified polynomial (single Lg coefficient) - sqrt: Newton-Raphson with 3 iterations - atan: Reduced polynomial with large-argument reduction - asin, acos, atan2, pow-n, pow Quad precision (@rq): - log: 8-coefficient polynomial (Lg1-Lg8) for 113-bit accuracy - sqrt: Newton-Raphson with 8 iterations - atan: 6-coefficient polynomial with range reduction - asin, acos, atan2, pow-n, pow All functions use hex-exact coefficients for jet compatibility.
Adds comprehensive tests for half precision (@rh) and quad precision (@rq): - Helper functions: expect-near-rh, expect-near-rq - Function tests: sin, cos, exp, log, sqrt, atan, pow-n, cheb - Identity tests: sin²+cos²=1, exp(log(x))=x, sqrt(x)²=x Half precision tests use hex literals (0x3c00 = 1.0, etc.) Quad precision tests use .~~~X literal format
Hyperbolic functions added to all precision levels (@rh, @rs, @rd, @rq): - sinh: Hyperbolic sine via (exp(x) - exp(-x)) / 2 - cosh: Hyperbolic cosine via (exp(x) + exp(-x)) / 2 - tanh: Hyperbolic tangent via (e^2x - 1) / (e^2x + 1) - asinh: Inverse hyperbolic sine via ln(x + sqrt(x^2 + 1)) - acosh: Inverse hyperbolic cosine via ln(x + sqrt(x^2 - 1)) - atanh: Inverse hyperbolic tangent via 0.5 * ln((1+x)/(1-x)) Special function: - erf: Error function using Abramowitz & Stegun approximation 7.1.26 Tests include: - Individual function tests for sinh, cosh, tanh, asinh, acosh, atanh - Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 - Inverse identities: asinh(sinh(x))=x, acosh(cosh(x))=x, atanh(tanh(x))=x - erf function tests and symmetry: erf(-x) = -erf(x)
Gamma function: - Uses Lanczos approximation (g=7, n=9 coefficients) - lgamma: Log-gamma function via log(gamma(x)) - Handles special cases: Γ(1)=1, Γ(2)=1 Bessel functions of first kind: - j0: J₀(x) using polynomial for |x|<8, asymptotic otherwise - j1: J₁(x) with proper sign handling Bessel functions of second kind: - y0: Y₀(x) using log-based formula for small x - y1: Y₁(x) with asymptotic expansion for large x Tests include: - Gamma at integer values (factorial property) - Γ(1.5) = √π/2 ~ 1.7724538 - Γ(n+1) = n*Γ(n) identity - J0, J1 at standard test points - Numerical derivative identity: J1(x) ≈ -J0'(x) - Y0, Y1 at standard test points
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Prototype implementation demonstrating jet-compatible approach for transcendental functions (exp, sin, cos, tan) using:
Key principle: every FP operation happens in same order with same intermediate values in both Hoon and C jet.