3337ded5-239e-4205-b271-2425e531ea2e 33 6 9 decimal

&[DATE] &[TIME] - &[FILENAME]

HyperSine HyperExponential was derived from f(p,x)

Generatrix Conjugate v Conjugate v Re v Im 1 i * - (* + :

p x f 12 / i * 1 - (p π / x *^1 - (p π / x *^Conjugate - (* - :

i 1 - (p π / x *^i 1 - (p π / x *^Im * + 1 - (p π / x *^Re - (* 2 / -

p 2 :

General form

p x f p x π 2 p * / + f 2 1 sys

t0 1 time :

Unary Sin(x)/Cos(x)

1 - ( p π / x * ^ Im 1 - ( p π / x * ^ Re 2 1 sys

1 time t0 - 0.25 sec *

Recast/Explicit the Unary system p x I 1 - (p π / x *^: p x J i 1 - (p π / x *^Im * 1 - (p π / x *^Re - : p x S i p x I p x J + (* 2 / - : 31 line

p 1.25 : User spread/shrink Sine/Cosine ≡

Φ 1 : user sprad/shrink Sine/Cosine + phase shift (Phi) ≡

UNARY Sin/Cos wrt 'p'

p x S p x π 2 p * / + ( S 2 1 sys

UNARY Sin/Cos wrt 'p' & 'Φ'

p x Φ - ( S p x Φ - ( π 2 p * / + ( S Φ 2 Φ 2 - 2 2 mat 3 1 sys

============= Congruence of the forms =============

User system
Triplet [p,Φ,x] p 1.25 : Φ 0.5 : x 0.875 : 31 line

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure ...

Reduced equivalent forms [identity]

p x Φ Sin p x Φ - (S : p x Φ Cos p x Φ - (π 2 p * / + (S : 21 line

p x Φ sin p x Φ - (* sin : p x Φ cos p x Φ - (* cos : 21 line

Sanity check the forms

p x Φ Sin p x Φ Cos 21 mat 0.451771471491684 0.892133699366994 21 mat

p x Φ sin p x Φ cos 21 mat 0.451771471491684 0.892133699366994 21 mat

p u sine 12 / i * i p * u * exp - i p u * abs (2^p u * / * - exp + (* : p u cosine 12 / i p * u * exp * 12 / i p u * abs (2^p u * / * - exp * + : 21 line

p 1 :

residual imaginary freaks the plot. Use the Real arguments

p x sine Re p x cosine Re 2 1 sys

Numerical: sin(x), cos(x) identities

p 1 : Φ π : 21 line

x p Φ sin x p Φ cos 21 mat i i p * x Φ - (* exp * Re - i p * x Φ - (* exp Re 21 mat ≡

X_Y does not support complex arguments

i i p * x Φ - ( * exp * Re - i p * x Φ - ( * exp Re 1 p x Φ - ( * sin * 1 p x Φ - ( * cos * Φ 2 Φ 2 - 2 2 mat Φ 0 o 11 black 1 5 mat 6 1 sys

Compare user algo vs kernel algo

t0 1 time :

0 i p * x Φ - ( * exp Re 2 1 sys

1 time t0 - 0.21 sec *

t0 1 time :

0 p x Φ - ( * cos 2 1 sys

0.25 0.05 / 5

1 time t0 - 0.04 sec *

t0 1 time :

Curiosity: sensibly slower than Re(exp(i*p*(x-Φ))) ... timing is not much relevant !

0 i x Φ - ( * exp Re 2 1 sys

1 time t0 - 0.22 sec *

Note that many numerical approximations, Smath does not calculate. Rather, it fetches those functions from Microsoft "machine code", thus hyperfast, as proven above.

Euler x cos x sin 21 mat ≡ e i x *^Re e i x *^Im 21 mat ≡ e i x *^e i x * (-^+ 2 / e i x *^e i x * (-^- 2 i * / 21 mat ≡

x Exp e x^confrac x 16 convert maple :

1.23456789 Exp 1.23456789 exp 21 mat 3.4368930839217 3.4368930839217 21 mat