Estimate RR from HR in different scenarios,
estimation of confidence interval can be repeated using the same scenario

Scenario 1: Short follow-up time or very low event rate. [1]
Relative Risk (RR) is approximately equal to Hazard Ratio (HR)

Formula: $$RR\approx{HR} $$

Scenario 2: At the end of the follow-up, event rate in the control group is known [2][2]
Hazard Ratio (HR)
Event rate in the control group, %
Relative Risk (RR)
Risk Difference (RD)
Number Needed to Treat/Harm (NNT/NNH)

Formula: $$RR\approx{\frac{1-e^{HR*ln(1-p)}}{p}}$$ $$RD=(RR-1)*p$$

Scenario 3: When the event rate is unknown but estimated between 0.2 and 0.8 for both groups [3,4]
Hazard Ratio (HR)
Relative Risk (RR)

Formula: $$RR\approx{\frac{1-0.5^{\sqrt(HR)}}{1-0.5^{\sqrt(1/HR)}}} $$

Scenario 4: When the event rate is unknown but estimated within a range [w, u] for both groups[3,4]
Hazard Ratio (HR)
Lower bound of event rate, %
Upper bound of event rate, %
Relative Risk (RR)

Formula:
If HR>1， $$RR\approx{(\frac{1-(1-w)^{HR}}{1-(1-u)^{\frac{1}{HR}}}*\frac{u}{w})^{(1/2)}} $$ If HR<1， $$RR\approx{(\frac{1-(1-u)^{HR}}{1-(1-w)^{\frac{1}{HR}}}*\frac{w}{u})^{(1/2)}} $$

Referemce:

[1] Qiao Huang, Ming-Juan Zhao, et al. Discrimination and conversion between hazard ratio and risk ratio as effect measures inprospective studies. CHINESE JOURNAL OF EVIDENCE.BASED MEDICINE, 2020, 20(10): 1221-1225. doi: 10.7507/1672-2531.202005092
[2] Shor E, Roelfs D, Vang ZM. The "Hispanic mortality paradox" revisited: Meta-analysis and meta-regression of life-course differentials in Latin American and Caribbean immigrants' mortality. Soc Sci Med. 2017;186:20-33. doi:10.1016/j.socscimed.2017.05.049
[3] VanderWeele TJ. On a Square-Root transformation of the odds ratio for a common outcome. Epidemiology, 2017, 28(6): e58-e60.
[4] VanderWeele TJ. Optimal approximate conversions of odds ratios and hazard ratios to risk ratios. Biometrics, 2019, (1): 1-7.