Truncatedternarymultipliersulperrorcalculation

截断三进制乘法器ulp误差计算

对于截断三进制乘法器，它的误差是由舍入引起的。而这种误差可以通过计算ulp（unit in the last place）来衡量。对于一个给定的截断位数，我们可以计算出一组可能的输入值，然后计算它们在乘法器中的误差，并以ulp为单位进行表示。

下面是一个示例代码，用于计算6位二进制数的截断三进制乘法器的ulp误差：

import numpy as np

def truncated_ternary_multiplier(x, y):
    b = 3  # base of the number system
    n = 6  # number of bits
    
    # convert binary inputs to ternary form
    x_tern = np.zeros(n, dtype=np.int8)
    y_tern = np.zeros(n, dtype=np.int8)
    for i in range(n):
        x_tern[i] = x[i]*2 - x[i-1]
        y_tern[i] = y[i]*2 - y[i-1]

    # compute partial products
    p = np.zeros((2*n-1, n), dtype=np.int8)
    for i in range(n):
        for j in range(n):
            s = x_tern[i]*y_tern[j]
            p[i+j,i] = s//b
            p[i+j+1,i] = s%b

    # compute partial sums
    c = np.zeros(n, dtype=np.int8)
    for i in range(2*n-1):
        c += p[i,:]
        c = np.concatenate(([0], c[:-1]))

    # convert result to binary form
    z = np.zeros(n, dtype=np.int8)
    for i in range(n):
        if c[i] == 1:
            z[i] = 1
        elif c[i] == -1:
            z[i] = 0
            c[i+1] -= 1
    return z

# compute ulp error for truncated ternary multiplier
def compute_ulp_error(n):
    eps = np.finfo(float).eps  # machine epsilon
    x = np.random.randint(2, size=n)
    y = np.random.randint(2, size=n)
    z = truncated_ternary_multiplier(x, y)
    z_float = np.packbits(z)
    z_float /= np.power(2.0, np.arange(n)[::-1])
    x_float = np.packbits(x) / np.power(2.0, np.arange(n)[::-1])
    y_float = np.packbits(y) / np.power(2.0, np.arange(n)[::-1])
    true_z_float = x_float * y_float
    error = true_z_float - z_float
    ulp_error = error / eps
    max_ulp_error = np.max(np.abs(ulp_error))
    return max_ulp_error

# compute maximum ulp error over a set of trials
num_trials = 1000
max_ulp_error = 0
for i in range(num_trials):
    error = compute_ulp_error(6)
    if error > max_ulp_error:
        max_ulp_error = error

print("Maximum ulp error over {} trials: {}".format(num_trials, max_ulp_error))

注意，在本代码示例中，我们使用的是numpy库的函数来实现各种数学运算。同时，对于