linel: 64$

t[ 0](x) := 1$
t[ 1](x) := x$
t[ 2](x) := t[ 1](x) * x * 2 - t[ 0](x)$
t[ 3](x) := t[ 2](x) * x * 2 - t[ 1](x)$
t[ 4](x) := t[ 3](x) * x * 2 - t[ 2](x)$
t[ 5](x) := t[ 4](x) * x * 2 - t[ 3](x)$
t[ 6](x) := t[ 5](x) * x * 2 - t[ 4](x)$
t[ 7](x) := t[ 6](x) * x * 2 - t[ 5](x)$
t[ 8](x) := t[ 7](x) * x * 2 - t[ 6](x)$
t[ 9](x) := t[ 8](x) * x * 2 - t[ 7](x)$
t[10](x) := t[ 9](x) * x * 2 - t[ 8](x)$
t[11](x) := t[10](x) * x * 2 - t[ 9](x)$
t[12](x) := t[11](x) * x * 2 - t[10](x)$
t[13](x) := t[12](x) * x * 2 - t[11](x)$
t[14](x) := t[13](x) * x * 2 - t[12](x)$
t[15](x) := t[14](x) * x * 2 - t[13](x)$
t[16](x) := t[15](x) * x * 2 - t[14](x)$
t[17](x) := t[16](x) * x * 2 - t[15](x)$
t[18](x) := t[17](x) * x * 2 - t[16](x)$
t[19](x) := t[18](x) * x * 2 - t[17](x)$
t[20](x) := t[19](x) * x * 2 - t[18](x)$

vec: [a, b, c, d, e, f, g, h, i, j]$

poly(x) := sum(vec[y] * t[(y - 1) * 2](x), y, 1, length(vec))$

range_reduce(x) := x / 2^63$

solveme(x) := poly(range_reduce(x))$

factor: 2^63$

result_factor: 2^60;

results: linsolve([
    solveme(factor * 0 / 16) = 1.0 * result_factor,
    solveme(factor * 1 / 16) = sqrt(2 + sqrt(2 + sqrt(2))) / 2.0 * result_factor,
    solveme(factor * 2 / 16) = sqrt(2 + sqrt(2)) / 2.0 * result_factor,
    solveme(factor * 3 / 16) = sqrt(2 + sqrt(2 - sqrt(2))) / 2.0 * result_factor,
    solveme(factor * 4 / 16) = sqrt(2) / 2.0 * result_factor,
    solveme(factor * 5 / 16) = sqrt(2 - sqrt(2 - sqrt(2))) / 2 * result_factor,
    solveme(factor * 6 / 16) = sqrt(2 - sqrt(2)) / 2 * result_factor,
    solveme(factor * 7 / 16) = sqrt(2 - sqrt(2 + sqrt(2))) / 2 * result_factor,
    solveme(factor * 15 / 32) = sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2)))) / 2 * result_factor,
    solveme(factor * 8 / 16) = 0
], vec)$

fpprec: 256$


equation(x) := rhs(sum(results[y] * t[(y - 1) * 2](x), y, 1, length(vec)))$

arccos(y) = bfloat(horner(solve([equation(x)], [x])));



result: bfloat(horner(rhs(sum(results[y] * t[(y - 1) * 2](x), y, 1, length(vec))))), fpprec: 256$
float(result);

a: part(result, 1)$

factor: 1$

result: part(result, 2)$
b: floor(part(result, 2) * a * factor);
b: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
c: floor(part(result, 2) * a * factor);
c: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
d: floor(part(result, 2) * a * factor);
d: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
e: floor(part(result, 2) * a * factor);
e: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
f: floor(part(result, 2) * a * factor);
f: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
g: floor(part(result, 2) * a * factor);
g: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
h: floor(part(result, 2) * a * factor);
h: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$
i: floor(part(result, 2) * a * factor);
i: part(result, 2);

result: part(result, 1)$
result: part(result, 3)$

j: floor(part(result, 1) * a * factor);
j: part(result, 1);
result: part(result, 2)$
result: part(result, 1)$

k: floor(-part(result, 1) * a * factor);
k: part(result, 1);