Table 4 Variance times cost (i.e., inefficiency, the smaller the better) for multi-sample estimator for the four examples in Appendix C and the environment map example (× 10⁻³) for balance heuristic using equal count of samples, for the count inversely proportional to the variances of independent estimators [9] and[10], for the provably better, non-balance heuristic, estimator defined in [11], and for the three new estimators defined in this paper

Example 1

Example 2

Example 3

Example 4

Example EM

\(\alpha _{k} \propto \frac {1}{n}\)

102.26

17.24

37.47

98.68

13.41

\(\alpha _{k} \propto \frac {1}{c_{k} v_{k}}\) [9, 10]

40.41

9.28

4.03

300.12

1.31

\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}}\) [11]

89.40

15.44

31.80

83.78

11.31

\(\alpha _{k} \propto \frac {1}{c_{k} m^{2}_{k}}\)

43.06

9.82

3.12

534.37

1.68

\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}}\)

81.43

13.54

28.68

91.01

5.88

\(\alpha _{k} \propto \frac {M_{k,{\text {eq}}}}{\sqrt {c_{k}}}\)

79.73

13.08

25.74

31.77

4.92