Computing definite integral over sinus in range [0;pi] (with terrible precision :3 ~2.0097% deviation from analytic solution)

crapy_integ.hs

-- basically this in wolfram alpha terms
-- 2*(Integrate[sin\(40)x\(41),{x,0,Divide[pi,4]}]+Integrate[sin\(40)x\(41),{x,Divide[pi,4],Divide[pi,2]}])

-- https://www.desmos.com/calculator/1fokxfyphz
-- https://www.wolframalpha.com/input?i2d=true&i=Integrate%5Bsin%5C%2840%29x%5C%2841%29%2C%7Bx%2C0%2Cpi%7D%5D
-- https://www.wolframalpha.com/input?i2d=true&i=2*%5C%2840%29Integrate%5Bsin%5C%2840%29x%5C%2841%29%2C%7Bx%2C0%2CDivide%5Bpi%2C4%5D%7D%5D%2BIntegrate%5Bsin%5C%2840%29x%5C%2841%29%2C%7Bx%2CDivide%5Bpi%2C4%5D%2CDivide%5Bpi%2C2%5D%7D%5D%5C%2841%29

import Text.Printf (printf)

main = putStrLn $ res

integRes = 
    -- left semi could be better honestly , but idgaf cus rate of change is minimal there
    let leftSemiSegment = ((pi/4) * (sin $ pi/4)) / 2
        rightSemiSegment = foldr (+) 0 $ take 16 (areaSubdivs :: [Double])
    in 2 * (leftSemiSegment + rightSemiSegment)

res = 
    let value = integRes
        integResFmtStr = ((printf "%.16f" value) :: String)
        errorRate = (100/2) * (2-value)
        errorRateFmt = ((printf "%.4f" errorRate) :: String)
    in "Expected value is 2, algo has computed: " <> integResFmtStr <> "\n"
        <> "Error rate is " <> errorRateFmt <> "%"
        
terms = (( map (printf "%.16f") $ take 16 (areaSubdivs :: [Double])) :: [String])

areaSubdivs = asc (pi/4) 4
    where
        asc c d =
            let ch = sin c
                nt = d*2
                n = c + pi/nt
                nh = sin n
                hdel = nh - ch
                ldel = n - c
            in ch*ldel+(hdel*ldel/2) : asc n nt