# Notes on FSD Fuel consumption and jump range

FSD Fuel consumption ([Elite: Dangerous Wiki](https://elite-dangerous.fandom.com/wiki/Frame_Shift_Drive#Hyperspace_Fuel_Equation)):
$$Fuel = 0.001 \cdot l \cdot \left(\frac{dist \cdot \left(m_{fuel} + m_{ship}\right)}{boost \cdot m_{opt}}\right)^{p}$$

Solving for \\(dist\\) gives the jump range (in Ly) for a given amount of fuel (in tons) as:
$$dist = \frac{boost \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}{m_{fuel} + m_{ship}}$$

Assuming \\(f_{max}\\) tons of available fuel gives us the maximum jump range for a single jump as:
$$dist_{max} = \frac{boost \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot f_{max}}{l}\right)^{\frac{1}{p}}}{f_{max} + m_{ship}}$$

Since the guardian FSD booster increases the maximum jump range by \\(B_g\\) light years we can calculate a correction factor for the fuel consumption as:
$$ e_{fuel} = 0.001 \cdot l \cdot \left(\frac{boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}{B_{g} \cdot \left(m_{fuel} + m_{ship}\right) + boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}\right)^{p}$$

Incorporating \\(e_{fuel}\\) into the distance equation yields
$$dist = \frac{boost \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot e_{fuel} \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}{m_{fuel} + m_{ship}}$$

Expanding \\(e_{fuel}\\) yields
$$dist = \frac{boost \cdot m_{opt} \cdot \left(1.0 \cdot \left(\frac{boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}{B_{g} \cdot \left(m_{fuel} + m_{ship}\right) + boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}\right)^{p} \cdot \min\left(f_{max}, m_{fuel}\right)\right)^{\frac{1}{p}}}{m_{fuel} + m_{ship}}$$

Finally, Expanding \\(dist_{max}\\) yields the full equation as
$$dist = \frac{boost \cdot m_{opt} \cdot \left(\frac{1000000.0 \cdot \left(\frac{boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}{B_{g} \cdot \left(m_{fuel} + m_{ship}\right) + boost^{2} \cdot m_{opt} \cdot \left(\frac{1000.0 \cdot \min\left(f_{max}, m_{fuel}\right)}{l}\right)^{\frac{1}{p}}}\right)^{- p} \cdot \min\left(f_{max}, m_{fuel}\right)}{l^{2}}\right)^{\frac{1}{p}}}{m_{fuel} + m_{ship}}$$

Where:
- \\(Fuel\\) is the fuel needed to jump (in tons)
- \\(l\\) is the linear constant of your FSD (depends on the rating)
- \\(p\\) is the power constant of your FSD (depends on the class)
- \\(m_{ship}\\) is the mass of your ship (including cargo)
- \\(m_{fuel}\\) is the amount of fuel in tons currently stored in your tanks
- \\(m_{opt}\\) is the optimized mass of your FSD (in tons)
- \\(f_{max}\\) is the maximum amount of fuel your FSD can use per jump
- \\(boost\\) is the "boost factor" of your FSD (1.0 when jumping normally, 1.5 when supercharged by a white dwarf, 4.0 for a neutron star, etc)
- \\(dist\\) is the distance you can jump with a given fuel amount
- \\(dist_{max}\\) is the maximum distance you can jump (when \\(m_{fuel}=f_{max}\\))
- \\(B_{g}\\) is the amount of Ly added by your Guardian FSD Booster
- \\(e_{fuel}\\) is the efficiency increase added by the Guardian FSD Booster