The science of the Guardian FSD booster 2.0

TL;DR The Guardian FSD booster increases your FSD's fuel multiplier stat by a factor which would yield the listed range increase to your ship's maximum laden range.

When the Guardian FSD booster was originally made available I started a thread describing how it applied to the known FSD range formula. However the booster was removed from the game shortly after it was added and I never got the chance to unlock one myself. Now that I have unlocked it I have been able to revise the formula and correct the mistakes in the original thread.

[size=+2]Overview[/size]
The revised Guardian FSD booster advertises itself as "increasing the distance travelled per unit of fuel and enabling single jumps of a greater size". The range increase scales with the class of the module. For example a class 3 booster gives an increase of 7.75Ly. Unlike the original implementation, the new booster does not have a penalty to fuel cost, drawing additional power instead.

[size=+1]Experimentation[/size]
I fitted several different ships with various FSDs. Some with engineer modifications and some without. I then performed jumps back and forth between pairs of systems with scoopable stars (so I could refuel to maximum and keep my ship's total mass more-or-less constant for each test) both with and without a Guardian FSD booster activated. I made a note of the fuel cost for the jump with the booster and without it. The experiments were repeated for different classes of booster.

The following table lists a subset of the data I collected.
FSDMassFuelDistanceRegular costBoosted costFactor
2A38.325.9160.080450.047380.588336
2A38.3213.40.4130380.2430050.588616
2A38.3213.40.7979070.4696610.588337
Here the Mass is the total mass of the hull and modules excluding fuel, Fuel is the amount of fuel carried, the two cost values are the fuel costs without the booster and with it activated. Factor is result of dividing the boosted cost by the regular cost.

You can see that factor is almost identical in each column. Further experimentation revealed that for identical combinations of FSD, total mass and fuel the factor would be more-or-less the same regardless of distance jumped. Different combinations of FSD and total mass would have different factors but as long as the only change in conditions was the distance jumped, the factor would not change.

That led me to devise the following hypothesis.

Suppose the ship's normal laden maximum range is r and the fitted Guardian FSD booster's bonus range is b.
When the Guardian FSD booster is active, the fuel multiplier stat in all fuel cost calculation is scaled by an amount which would allow the ship to jump r + b lightyears by expending the FSD's max fuel per jump amount of fuel.​

[size=+1]Formula[/size]
Let's look again at the jump range equation.

range = (optimised mass / total mass) * (fuel cost / multiplier)[color=#cb7e07](1 / fuel power)[/color]

Rearranging for cost:

range = (optimised mass / total mass) * (fuel cost / multiplier)[color=#cb7e07](1 / fuel power)[/color]
range / (optimised mass / total mass) = (fuel cost / multiplier)[color=#cb7e07](1 / fuel power)[/color]
(range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color] = (fuel cost / multiplier)
multiplier * (range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color] = fuel cost

Let bonus range be the amount of additional range available from fitting a particular Guardian FSD booster. If the ship can jump that new range for the same fuel cost by applying a scaling factor to the fuel multiplier, it follows that

fuel cost = factor * multiplier * ((range + bonus range) / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color]

In other words

multiplier * (range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color] = factor * multiplier * ((range + bonus range) / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color]
(range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color] = factor * ((range + bonus range) / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color]
range * total mass = log[sub](fuel power)[/sub] factor * (range + bonus range) * total mass
range = log[sub](fuel power)[/sub] factor * (range + bonus range)
range / (range + bonus range) = log[sub](fuel power)[/sub] factor
(range / (range + bonus range))[color=#cb7e07](fuel power)[/color] = factor

[size=+2]Conclusion[/size]
If max range is the range a ship can jump by expending max fuel per jump then the cost to jump an arbitary distance would be

fuel cost = factor * multiplier * (range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color]
= (max range / (max range + bonus range))[color=#cb7e07](fuel power)[/color] * multiplier * (range / (optimised mass / total mass))[color=#cb7e07](fuel power)[/color]
= multiplier * ((range * max range * total mass) / ((max range + bonus range) * optimised mass))[color=#cb7e07](fuel power)[/color]

[size=+2]Example[/size]
For an example, consider a Diamondback Explorer with a A2 power plant, D4 thrusters, A5 frame shift drive, D3 life support, D3 power distributor, D3 sensors and C4 fuel tank. The total mass of the ship when fully fueled is 307.82T assuming a fully filled reserve fuel tank of 0.52T. Given that an A5 FSD has optimised mass 1050T and max fuel per jump 5T, the ship's laden range is thus:

range = (optimised mass / total mass) * (fuel cost / multiplier)(1 / fuel power)
= (1050 / 307.82) * (5 / 0.012)(1 / 2.45)
= 3.4110 * 416.6667 0.4082
= 3.4110 * 11.7301
= 40.0123

Now suppose the ship is fitted with a class 3 Guardian FSD booster with a mass of 1.3T and a bonus range of 7.75Ly.

The ship's new maximum range is

range = ((1050 / 309.12) * (5 / 0.012)(1 / 2.45)) + 7.75
= (3.3967 * 416.6667 0.4082) + 7.75
= (3.3967 * 11.7301) + 7.75
= 39.8441 + 7.75
= 47.5941

The booster's scaling factor can be calculated from the unboosted max range which is 39.8441.

factor = (max range / (max range + bonus range))(fuel power)
= (39.8441 / 47.5941) 2.45
= 0.83716 2.45
= 0.6469

The fuel cost to jump 30Ly with the booster deactivated would be

fuel cost = multiplier * (range / (optimised mass / total mass))(fuel power)
= 0.012 * (30.0 / (1050 / 309.12)) 2.45
= 0.012 * (30.0 / 3.3967) 2.45
= 0.012 * 8.8319 2.45
= 0.012 * 207.8953
= 2.495

With the booster activated the cost falls

fuel cost = multiplier * ((range * max range * total mass) / ((max range + bonus range) * optimised mass))(fuel power)
= 0.012 * ((30.0 * 39.8441 * 309.12) / (47.5941 * 1050)) 2.45
= 0.012 * (369498.2457 / 49973.805) 2.45
= 0.012 * 7.4094 2.45
= 0.012 * 135.1977
= 1.622
Or alternatively
fuel cost = factor * multiplier * (range / (optimised mass / total mass))(fuel power)
= 0.6469 * 0.012 * (30.0 / (1050 / 309.12)) 2.45
= 0.6469 * 0.012 * (30.0 / 3.3967) 2.45
= 0.6469 * 0.012 * 8.8319 2.45
= 0.6469 * 0.012 * 207.8953
= 0.6469 * 2.495
= 1.622
The jump that took 2.495T without the booster can be achieved with only 1.622T using the booster.

[size=+2]Benefits[/size]
Jumps that are in range of an unboosted drive will be more fuel efficient with a boosted drive.

More interestingly, a drive's maximum range given the same loadout will be increased by the amount listed in the booster's description.

Note that the booster itself has a mass, which is why the examples above describe it as being either activated or deactivated rather than fitted or not fitted. A ship's regular maximum range will be slightly more when it is not carrying the extra mass of the booster.

[size=+2]Idiosyncracies[/size]
Fuel cost
The original implementation of the Guardian FSD booster added a fuel cost penalty. That was bugged and has now been removed entirely.

Supercharging and injection
There was a bug in the implementation of the new FSD booster related to supercharging and injection, which was fixed in patch 3.1.1. Applying an injection or a neutron/white dwarf supercharge will increase your ship's range by a given percentage; 400% for neutron star boosts.

In the outfitting and the galaxy map the formula used to calculate the final range for a neutron boost was 4 * (range + bonus range) but the fuel cost calculations used when you actually jumped used (4 * range) + bonus range. Players plotting a route using neutron boosts or FSD injection would be presented with a jump that they couldn't make, though if they manually chose a closer destination they were able to make the jump.

In 3.1.1 the bug is fixed and a ship's maximum range with a Guardian booster is indeed supercharge bonus * (range + bonus range).

Deep charge vs Mass manager
Because the range increase is a flat benefit applied after other calculations, it does not affect your choice of Deep charge or Mass manager experimental effect.

JonathanBurnage and VerticalBlank demonstrated that the cutoff point for picking Mass manager (optimised mass +4%) vs Deep charge (max fuel per jump +10%) is when

1.04 = 1.1(1 / fuel power)
(1 / fuel power) = log[sub]1.1[/sub] 1.04
(1 / fuel power) = ln 1.04 / ln 1.1
fuel power = ln 1.1 / ln 1.04

Fuel multiplier does not play any part in the calculation. Adding the range increase, eg 7.75, to both sides of the equation will not change the result.
 
Last edited:
TL;DR The Guardian FSD booster increases your FSD's fuel multiplier stat by a factor which would yield the listed range increase to your ship's maximum laden range.

When the Guardian FSD booster was originally made available I started a thread describing how it applied to the known FSD range formula. However the booster was removed from the game shortly after it was added and I never got the chance to unlock one myself. Now that I have unlocked it I have been able to revise the formula and correct the mistakes in the original thread.

[SIZE=+2]Overview[/SIZE]
The revised Guardian FSD booster advertises itself as "increasing the distance travelled per unit of fuel and enabling single jumps of a greater size". The range increase scales with the class of the module. For example a class 3 booster gives an increase of 7.75Ly. Unlike the original implementation, the new booster does not have a penalty to fuel cost, drawing additional power instead.

[SIZE=+1]Experimentation[/SIZE]
I fitted several different ships with various FSDs. Some with engineer modifications and some without. I then performed jumps back and forth between pairs of systems with scoopable stars (so I could refuel to maximum and keep my ship's total mass more-or-less constant for each test) both with and without a Guardian FSD booster activated. I made a note of the fuel cost for the jump with the booster and without it. The experiments were repeated for different classes of booster.

The following table lists a subset of the data I collected.
FSDMassFuelDistanceRegular costBoosted costFactor
2A38.325.9160.080450.047380.588336
2A38.3213.40.4130380.2430050.588616
2A38.3213.40.080450.4696610.588337
Here the Mass is the total mass of the hull and modules excluding fuel, Fuel is the amount of fuel carried, the two cost values are the fuel costs without the booster and with it activated. Factor is result of dividing the boosted cost by the regular cost.

You can see that factor is almost identical in each column. Further experimentation revealed that for identical combinations of FSD, total mass and fuel the factor would be more-or-less the same regardless of distance jumped. Different combinations of FSD and total mass would have different factors but as long as the only change in conditions was the distance jumped, the factor would not change.

That led me to devise the following hypothesis.

Suppose the ship's normal laden maximum range is r and the fitted Guardian FSD booster's bonus range is b.
When the Guardian FSD booster is active, the fuel multiplier stat in all fuel cost calculation is scaled by an amount which would allow the ship to jump r + b lightyears by expending the FSD's max fuel per jump amount of fuel.​

[SIZE=+1]Formula[/SIZE]
Let's look again at the jump range equation.

range = (optimised mass / total mass) * (fuel cost / multiplier)[COLOR=#cb7e07](1 / fuel power)[/COLOR]

Rearranging for cost:

range = (optimised mass / total mass) * (fuel cost / multiplier)[COLOR=#cb7e07](1 / fuel power)[/COLOR]
range / (optimised mass / total mass) = (fuel cost / multiplier)[COLOR=#cb7e07](1 / fuel power)[/COLOR]
(range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR] = (fuel cost / multiplier)
multiplier * (range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR] = fuel cost

Let bonus range be the amount of additional range available from fitting a particular Guardian FSD booster. If the ship can jump that new range for the same fuel cost by applying a scaling factor to the fuel multiplier, it follows that

fuel cost = factor * multiplier * ((range + bonus range) / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR]

In other words

multiplier * (range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR] = factor * multiplier * ((range + bonus range) / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR]
(range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR] = factor * ((range + bonus range) / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR]
range * total mass = log[SUB](fuel power)[/SUB] factor * (range + bonus range) * total mass
range = log[SUB](fuel power)[/SUB] factor * (range + bonus range)
range / (range + bonus range) = log[SUB](fuel power)[/SUB] factor
(range / (range + bonus range))[COLOR=#cb7e07](fuel power)[/COLOR] = factor

[SIZE=+2]Conclusion[/SIZE]
If max range is the range a ship can jump by expending max fuel per jump then the cost to jump an arbitary distance would be

fuel cost = factor * multiplier * (range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR]
= (max range / (max range + bonus range))[COLOR=#cb7e07](fuel power)[/COLOR] * multiplier * (range / (optimised mass / total mass))[COLOR=#cb7e07](fuel power)[/COLOR]
= multiplier * ((range * max range * total mass) / ((max range + bonus range) * optimised mass))[COLOR=#cb7e07](fuel power)[/COLOR]

[SIZE=+2]Example[/SIZE]
For an example, consider a Diamondback Explorer with a A2 power plant, D4 thrusters, A5 frame shift drive, D3 life support, D3 power distributor, D3 sensors and C4 fuel tank. The total mass of the ship when fully fueled is 307.82T assuming a fully filled reserve fuel tank of 0.52T. Given that an A5 FSD has optimised mass 1050T and max fuel per jump 5T, the ship's laden range is thus:

range = (optimised mass / total mass) * (fuel cost / multiplier)(1 / fuel power)
= (1050 / 307.82) * (5 / 0.012)(1 / 2.45)
= 3.4110 * 416.6667 0.4082
= 3.4110 * 11.7301
= 40.0123

Now suppose the ship is fitted with a class 3 Guardian FSD booster with a mass of 1.3T and a bonus range of 7.75Ly.

The ship's new maximum range is

range = ((1050 / 309.12) * (5 / 0.012)(1 / 2.45)) + 7.75
= (3.3967 * 416.6667 0.4082) + 7.75
= (3.3967 * 11.7301) + 7.75
= 39.8441 + 7.75
= 47.5941

The booster's scaling factor can be calculated from the unboosted max range which is 39.8441.

factor = (max range / (max range + bonus range))(fuel power)
= (39.8441 / 47.5941) 2.45
= 0.83716 2.45
= 0.6469

The fuel cost to jump 30Ly with the booster deactivated would be

fuel cost = multiplier * (range / (optimised mass / total mass))(fuel power)
= 0.012 * (30.0 / (1050 / 309.12)) 2.45
= 0.012 * (30.0 / 3.3967) 2.45
= 0.012 * 8.8319 2.45
= 0.012 * 207.8953
= 2.495

With the booster activated the cost falls

fuel cost = multiplier * ((range * max range * total mass) / ((max range + bonus range) * optimised mass))(fuel power)
= 0.012 * ((30.0 * 39.8441 * 309.12) / (47.5941 * 1050)) 2.45
= 0.012 * (369498.2457 / 49973.805) 2.45
= 0.012 * 7.4094 2.45
= 0.012 * 135.1977
= 1.622
Or alternatively
fuel cost = factor * multiplier * (range / (optimised mass / total mass))(fuel power)
= 0.6469 * 0.012 * (30.0 / (1050 / 309.12)) 2.45
= 0.6469 * 0.012 * (30.0 / 3.3967) 2.45
= 0.6469 * 0.012 * 8.8319 2.45
= 0.6469 * 0.012 * 207.8953
= 0.6469 * 2.495
= 1.622
The jump that took 2.495T without the booster can be achieved with only 1.622T using the booster.

[SIZE=+2]Benefits[/SIZE]
Jumps that are in range of an unboosted drive will be more fuel efficient with a boosted drive.

More interestingly, a drive's maximum range given the same loadout will be increased by the amount listed in the booster's description.

Note that the booster itself has a mass, which is why the examples above describe it as being either activated or deactivated rather than fitted or not fitted. A ship's regular maximum range will be slightly more when it is not carrying the extra mass of the booster.

[SIZE=+2]Idiosyncracies[/SIZE]
Fuel cost
The original implementation of the Guardian FSD booster added a fuel cost penalty. That was bugged and has now been removed entirely.

Supercharging and injection
There is a bug in the implementation of the new FSD booster related to supercharging and injection. Applying an injection or a neutron/white dwarf supercharge will increase your ship's range by a given percentage; 400% for neutron star boosts. My initial assumption when writing the first thread was that the flat range boost from the Guardian module would be applied after the supercharge.

In other words I assumed that a ship's neutron-supercharged range would be (4 * range) + bonus range. Players who tested the booster before it was removed saw that actually the supercharge bonus was applied last for a total range of 4 * (range + bonus range).

In the initial release of patch 3.1 the range shown in outfitting and the galaxy map is the greater distance 4 * (range + bonus range) but when you actually try to jump you will find that the code uses (4 * range) + bonus range and you will get the "exceeds max fuel per jump" warning when trying to jump further.

In edts we modelled supercharging as a modifier to optimised mass because it seemed more elegant than slapping a scaling factor at the end of the formula, so our calculations tallied with what is in the game now but not what was present before, and are still in disagreement with the galaxy map.

Deep charge vs Mass manager
Because the range increase is a flat benefit applied after other calculations, it does not affect your choice of Deep charge or Mass manager experimental effect.

JonathanBurnage and VerticalBlank demonstrated that the cutoff point for picking Mass manager (optimised mass +4%) vs Deep charge (max fuel per jump +10%) is when

1.04 = 1.1(1 / fuel power)
(1 / fuel power) = log[SUB]1.1[/SUB] 1.04
(1 / fuel power) = ln 1.04 / ln 1.1
fuel power = ln 1.1 / ln 1.04

Fuel multiplier does not play any part in the calculation. Adding the range increase, eg 7.75, to both sides of the equation will not change the result.
Truly oustanding that you’ve done all the calculations for this but waaaaaaay too much for my tiny little brain on a Sunday afternoon.
 
The current info on the drive booster found on the wiki page seems to indicate that the range increase is a flat ammount, not the result of an algorithm.
It is a flat amount. The number crunching is about the fuel cost for jumps that are less than that maximum range.

If you turn the booster off you will see your jump range fall by the number listed against the booster. However it is a property of the fuel cost formula that the higher the ship's maximum range the less fuel it will use for shorter jumps. This is true regardless of how the maximum range is achieved; be it through engineering, adding a Guardian booster or lightening other parts of the ship.

It's why I am always frustrated when people ask "is jump range important for exploration" and the inevitable reply comes "no because real explorers make short jumps to scan every planetary body in the area". Not because people are trying to be exploration police and force their views on "how to be explorers" on others - anyone can see how arrogant and rude that is - but because it is objectively true that having a higher range is beneficial. Even if you do make shorter jumps than that maximum range they will cost you less fuel, enabling you to go longer without scooping or traverse regions of space where scooping isn't possible.

For most people, the interesting thing about the Guardian booster is going to be the flat range increase. But if you care about fuel consumption then it's worth looking at getting one for that reason too.
 
It is a flat amount. The number crunching is about the fuel cost for jumps that are less than that maximum range.

If you turn the booster off you will see your jump range fall by the number listed against the booster. However it is a property of the fuel cost formula that the higher the ship's maximum range the less fuel it will use for shorter jumps. This is true regardless of how the maximum range is achieved; be it through engineering, adding a Guardian booster or lightening other parts of the ship.

It's why I am always frustrated when people ask "is jump range important for exploration" and the inevitable reply comes "no because real explorers make short jumps to scan every planetary body in the area". Not because people are trying to be exploration police and force their views on "how to be explorers" on others - anyone can see how arrogant and rude that is - but because it is objectively true that having a higher range is beneficial. Even if you do make shorter jumps than that maximum range they will cost you less fuel, enabling you to go longer without scooping or traverse regions of space where scooping isn't possible.

For most people, the interesting thing about the Guardian booster is going to be the flat range increase. But if you care about fuel consumption then it's worth looking at getting one for that reason too.
Even better when they figure out how to make it work with jumponium and WD/NS boosts.
 
Great work there. It really shows how the geometric law for fuel consumption vs. distance make a massive impact...

I really wish they would create a "max fuel per jump" slider or alternatively a "max distance per jump" slider so that I could tailor my fuel consumption on the crappy Diamondback explorer.
 
Epic work sir, glad to see that your enthusiasm for deconstructing the science of ED ship performance continues unabated!

o7

P.S. this clearly deserves a place in my "Best of Forum" compendium of all accumulated knowledge and wisdom, will add shortly.
 
Jumps that are in range of an unboosted drive will be more fuel efficient with a boosted drive.
Can somewhat confirm, although without solid proof or calculations as I just only observed it.

Yesterday I bought T-9, D-rated everything, G5 6A FSD, 5D shields, 4H guardian booster, rest is cargo racks with 740t total space. Had 3 cargo missions (~250t) and been performing jump within my unboosted range. Fuel gauge for a ~26 LYs jump showed very little fuel consumption which felt really strange. Not that long time ago I was doing some exploration in a Cutter with 5H guardian booster. Max range jump (~40 LYs) consumed about 25% of my fuel. Boost supported T-9 range is 41 LYs, jumping a distance near min range value costed in fuel almost equally to single digit distance jump.

Been wondering why is that and is it some booster effect. Your post solves the mystery.
 
Truly oustanding that you’ve done all the calculations for this but waaaaaaay too much for my tiny little brain on a Sunday afternoon.
I'm staring in dis-belief and absolute admiration that someone has not only the intelligence but the will power to work this stuff out. Very impressive.....I don't do maths but I can make a mean spaghetti bolognese. Oh yes!! I have skills too. Have some rep!
 
FSDMassFuelDistanceRegular costBoosted costFactor
2A38.325.9160.080450.047380.588336
2A38.3213.40.4130380.2430050.588616
2A38.3213.40.080450.4696610.588337
Should the "Regular cost" in the third row be 0.8 rather than 0.08?

Great work - I'm going to go get me one of these...

I'm sure I used to be able to math like that...
 
Top Bottom