Published
Edited
Jun 6, 2022
3 stars
// very slow but uses 200-digit decimal arithmetic
precise_tangent = (a, b) => {
let [ax, ay] = Dec(a), [bx, by] = Dec(b);
let {mul, add, sub, div} = Dec;
return +div(
sub(mul(ax, by), mul(ay, bx)),
add(mul(ax, bx), mul(ay, by))
)
}
naive_tangent = ([ax, ay], [bx, by]) =>
(ax*by - ay*bx) /
(ax*bx + ay*by)
careful_tangent = ([ax, ay], [bx, by]) =>
((ax + bx)*(by - ay) - (ay + by)*(bx - ax)) /
((ax - by)*(bx - ay) + (ay + bx)*(by + ax))
trig_tangent = ([ax, ay], [bx, by]) =>
Math.tan(Math.atan2(by, bx) - Math.atan2(ay, ax))
normalize = ([x, y]) =>
[x/Math.hypot(x, y), y/Math.hypot(x, y)]
eps = 2**-52
u1 = normalize([Math.PI, Math.E])
u2 = normalize([Math.PI+2*eps, Math.E])
precise_tangent(u1, u2)
naive_tangent(u1, u2)
careful_tangent(u1, u2)
trig_tangent(u1, u2)
(naive_tangent(u1, u2) - precise_tangent(u1, u2)) / precise_tangent(u1, u2) // relative error
(careful_tangent(u1, u2) - precise_tangent(u1, u2)) / precise_tangent(u1, u2) // relative error
(trig_tangent(u1, u2) - precise_tangent(u1, u2)) / precise_tangent(u1, u2) // relative error
u3 = [Math.cos(1.2), Math.sin(1.2)]
u4 = [Math.cos(1.2 - Math.PI/2), Math.sin(1.2 - Math.PI/2)]
naive_tangent(u3, u4)
careful_tangent(u3, u4)
precise_tangent(u3, u4)
trig_tangent(u3, u4)
(naive_tangent(u3, u4) - precise_tangent(u3, u4)) / precise_tangent(u3, u4) // relative error
(careful_tangent(u3, u4) - precise_tangent(u3, u4)) / precise_tangent(u3, u4) // relative error
(trig_tangent(u3, u4) - precise_tangent(u3, u4)) / precise_tangent(u3, u4) // relative error
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