Error Metrics and Mathematical Bounds¶

In exact digital arithmetic, functional correctness is binary: a circuit is either 100% correct or broken.

In approximate and inexact computing, arithmetic error is treated as a continuous, quantifiable design budget. By tolerating bounded numerical errors, designers trade precision for dramatic improvements in Power, Performance, and Area (PPA).

1. Primary Numerical Error Metrics¶

Let \(A\) and \(B\) represent the input operands, and let \(S\) denote the set of all evaluated input combinations (\(|S|\) total test vectors).

Let \(Y_{\text{exact}}(A, B)\) be the mathematical ground truth and \(Y_{\text{approx}}(A, B)\) be the output produced by the approximate hardware.

       Exact Reference:  Y_exact  = f(A, B)
   Approximate Circuit:  Y_approx = f_approx(A, B)
                         ────────────────────────
       Error Distance:   ED(A, B) = |Y_approx - Y_exact|

1. Error Distance (\(ED\))¶

The fundamental point-wise absolute numerical discrepancy for a specific input pair \((A, B)\):

\[ED(A, B) = |Y_{\text{approx}}(A, B) - Y_{\text{exact}}(A, B)|\]

2. Mean Error Distance (\(MED\))¶

The average absolute error distance across the entire operational input space \(S\). \(MED\) measures the general accuracy of the circuit:

\[MED = \frac{1}{|S|} \sum_{(A,B) \in S} |Y_{\text{approx}}(A, B) - Y_{\text{exact}}(A, B)|\]

Key Insight: \(MED\) has physical arithmetic units (e.g., LSBs or integers). For an 8-bit adder, an \(MED = 1.2\) means the approximate output is off by only \(1.2\) counts on average out of \(510\).

3. Mean Relative Error Distance (\(MRED\))¶

The average relative percentage error normalized against the true exact magnitude:

\[MRED = \frac{1}{|S^*|} \sum_{(A,B) \in S^*} \frac{|Y_{\text{approx}}(A, B) - Y_{\text{exact}}(A, B)|}{|Y_{\text{exact}}(A, B)|}\]

(where \(S^* = \{(A,B) \in S : Y_{\text{exact}}(A, B) \neq 0\}\) to avoid division by zero).

Application Rule: For image and video processing, human visual perception is sensitive to relative contrast rather than absolute magnitude. Circuits with \(MRED < 2\%\) are generally visually indistinguishable from exact hardware.

4. Normalized Mean Error Distance (\(NMED\))¶

\(NMED\) normalizes \(MED\) by the maximum possible output range \(D_{\text{max}}\) of the circuit. This allows fair comparison between circuits of different bit-widths (e.g. comparing an 8-bit multiplier to a 16-bit multiplier):

\[NMED = \frac{MED}{D_{\text{max}}}\]

For an unsigned \(N\)-bit multiplier, \(D_{\text{max}} = (2^N - 1)^2\). For an \(N\)-bit adder, \(D_{\text{max}} = 2^{N+1} - 2\).

5. Worst-Case Error (\(WCE\)) / Maximum Error Distance (\(MaxED\))¶

The upper bound on the numerical error that can ever be produced by the circuit under any possible input combination:

\[WCE = \max_{(A,B) \in S} |Y_{\text{approx}}(A, B) - Y_{\text{exact}}(A, B)|\]

Safety Significance: \(WCE\) is critical for safety-critical and bounded algorithms. If an algorithm requires \(|Error| \le 32\), a circuit with \(WCE = 28\) is formally guaranteed to never violate system specifications.

6. Error Rate (\(ER\))¶

The fraction of input test vectors for which the approximate circuit produces a non-zero error:

\[ER = \frac{1}{|S|} \sum_{(A,B) \in S} \mathbb{I}\Big(Y_{\text{approx}}(A, B) \neq Y_{\text{exact}}(A, B)\Big)\]

(where \(\mathbb{I}(\cdot)\) is the indicator function equal to \(1\) if true and \(0\) otherwise).

Counter-Intuitive Insight: A circuit can have a high Error Rate (\(ER = 90\%\)) while still being exceptionally accurate if its Error Distance is tiny (\(ED = 1\)). For example, dropping the single lowest bit produces \(ER = 50\%\) but an unnoticeable \(MRED = 0.01\%\).

7. Mean Error (\(ME\)) / Statistical Bias¶

The signed average error produced by the circuit:

\[ME = \frac{1}{|S|} \sum_{(A,B) \in S} \Big(Y_{\text{approx}}(A, B) - Y_{\text{exact}}(A, B)\Big)\]

     Truncation Multiplier:   ME < 0  (Always underestimates ──> Negative Drift)
     Rounding Multiplier:     ME ≈ 0  (Unbiased ──> Zero Accumulation Drift)

Deep Learning Relevance: In large matrix multiplications (\(\sum_{i=1}^{1024} A_i \cdot B_i\)), if \(ME = 0\), thousands of tiny positive and negative errors cancel each other out, yielding an almost exact accumulated sum!

2. Downstream Application Quality Metrics¶

Numerical metrics alone do not tell the whole story. We evaluate approximate arithmetic across real-world downstream perceptual and machine learning metrics:

🖼️ Peak Signal-to-Noise Ratio (\(PSNR\))¶

Used in computer vision and image processing. It measures the ratio between the maximum possible pixel energy and the Mean Squared Error (\(MSE\)):

\[MSE = \frac{1}{M \times N} \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} \Big(I_{\text{exact}}(x, y) - I_{\text{approx}}(x, y)\Big)^2\]

\[PSNR = 10 \cdot \log_{10}\left( \frac{MAX_I^2}{MSE} \right) = 20 \cdot \log_{10}\left( \frac{255}{\sqrt{MSE}} \right)\]

For 8-bit grayscale images, \(MAX_I = 255\).
\(> 35\text{ dB}\): Visually indistinguishable from exact image.
\(30\text{ dB} - 35\text{ dB}\): High quality, minor imperceptible noise.
\(< 25\text{ dB}\): Visible grain and structural artifacts.

🧠 Structural Similarity Index (\(SSIM\))¶

While \(PSNR\) measures raw pixel squared errors, \(SSIM\) models the human visual system's perception of luminance (\(l\)), contrast (\(c\)), and structure (\(s\)):

\[SSIM(x, y) = [l(x, y)]^\alpha \cdot [c(x, y)]^\beta \cdot [s(x, y)]^\gamma = \frac{(2\mu_x\mu_y + C_1)(2\sigma_{xy} + C_2)}{(\mu_x^2 + \mu_y^2 + C_1)(\sigma_x^2 + \sigma_y^2 + C_2)}\]

Range: \([-1, 1]\), where \(1.0\) indicates identical image structure.
Approximate computing designs typically target \(SSIM \ge 0.95\).

📚 Primary Literature & IEEE Citations¶

Comprehensive Survey on Approximate Arithmetic Metrics:
J. Liang et al., "New Metrics for the Reliability of Approximate and Probabilistic Adders", IEEE Transactions on Computers, IEEE Xplore (DOI: 10.1109/TC.2012.146).
Probabilistic Error Modeling Framework:
V. K. Chippa et al., "Analysis and Characterization of Inherent Application Resilience for Approximate Computing", ACM/IEEE DAC, ACM/IEEE (DOI: 10.1145/2463209.2488873).
SSIM Quality Assessment:
Z. Wang et al., "Image Quality Assessment: From Error Visibility to Structural Similarity", IEEE Transactions on Image Processing, IEEE Xplore (DOI: 10.1109/TIP.2003.819861).