After Jos Leys tests using analytical DE on the zero seed Julia I thought I'd try investigation of analytical DE values on a quaternionic Mandelbrot.

I quickly found that indeed the "0.5" is correct which means that the actual DE values in my wip formula for Ultra Fractal are 2* bigger than they should be - though my step distances are OK since the default is the DE divided by two

This of course results in a given DE threshold taking us to a point twice as close to "inside" as the specified threshold distance.

Here are the test results I did using the correctly scaled DE value.

These tests were carried out @640*480 with a magnification of 1.6 and a bailout of x^2+y^2>=1024.

Note that in the text below "worst" means "largest" rather than "most incorrect".

Quaternion z^2+c side view in parallel projection

DE Threshold 1e-6

Worst cases for entire fractal

"True" -> calculated

1.0 -> 0.828693280840216

0.1 -> 0.0756111410237615

0.01 -> 0.00826817743483479

1e-3 -> 0.00623750530621662

1e-4 -> 0.00616464060211196

1e-5 -> 0.00593556668345077

Quaternion z^2+c side view in parallel projection

DE Threshold 1e-6

Average cases for entire fractal

"True" -> calculated

1.0 -> 0.470217800965237

0.1 -> 0.0277512230487669

0.01 -> 0.0027385391736115

1e-3 -> 0.000537393167849202

1e-4 -> 0.000298006036418662

1e-5 -> 0.000183765097401958

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-6

Worst cases for entire fractal

"True" -> calculated

1.0 -> 0.97783685049528

0.1 -> 0.0971841767897501

0.01 -> 0.0099424008139091

1e-3 -> 0.00291127344517454

1e-4 -> 0.0028424928302291

1e-5 -> 0.00283555485322532

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-6

Average cases for entire fractal

"True" -> calculated

1.0 -> 0.268576868120558

0.1 -> 0.0330877709877468

0.01 -> 0.00333153377206449

1e-3 -> 0.000601747860965648

1e-4 -> 0.000318767428809674

1e-5 -> 0.000195952986975526

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-6

Values for the centre ray (i.e. hitting the point on the real axis)

"True" -> calculated

1.0 -> 0.97783685049528

0.1 -> 0.0971841767897501

0.01 -> 0.0099424008139091

1e-3 -> 0.000999129722927703

1e-4 -> 0.00010003123154646

1e-5 -> 9.99071625909761E-006

Quaternion z^2+c side view in parallel projection

DE Threshold 1e-16

Worst cases for entire fractal

"True" -> calculated

1.0 -> 0.828694191861107

0.1 -> 0.0756118350632922

0.01 -> 0.00826812262793161

1e-3 -> 0.00623734563239439

1e-4 -> 0.00616447654586974

1e-5 -> 0.00593548110247801

1e-6 -> 0.002967740551239

1e-7 -> 0.0014838702756195

1e-8 -> 0.000741935137809751

1e-9 -> 0.000370967568904876

1e-10 -> 0.000185483784452438

1e-11 -> 9.27418922262189E-005

1e-12 -> 4.63709461131094E-005

Quaternion z^2+c side view in parallel projection

DE Threshold 1e-16

Average cases for entire fractal

"True" -> calculated

1.0 -> 0.470218381043924

0.1 -> 0.0277514611184807

0.01 -> 0.00273875075822565

1e-3 -> 0.000537604762798528

1e-4 -> 0.000298158908926989

1e-5 -> 0.000185850698115385

1e-6 -> 9.97458957492159E-005

1e-7 -> 5.0596384017043E-005

1e-8 -> 2.53716863807056E-005

1e-9 -> 1.26919229340975E-005

1e-10 -> 6.34662886913257E-006

1e-11 -> 3.17331443456628E-006

1e-12 -> 1.58665721728314E-006

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-16

Worst cases for entire fractal

"True" -> calculated

1.0 -> 0.977836860473067

0.1 -> 0.097184186106234

0.01 -> 0.00994241047465512

1e-3 -> 0.00291124435480221

1e-4 -> 0.00284246358773596

1e-5 -> 0.00283552550759178

1e-6 -> 0.0019832407359585

1e-7 -> 0.00182386184009324

1e-8 -> 0.000911930920046621

1e-9 -> 0.000786028563546067

1e-10 -> 0.000786028308654553

1e-11 -> 0.00078602828316538

1e-12 -> 0.000786028280616523

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-16

Average cases for entire fractal

"True" -> calculated

1.0 -> 0.268571283431845

0.1 -> 0.0330876101698883

0.01 -> 0.00333120301202227

1e-3 -> 0.000602036669078832

1e-4 -> 0.000319200775195032

1e-5 -> 0.000198471399420637

1e-6 -> 0.000106430696869362

1e-7 -> 5.41934392969572E-005

1e-8 -> 2.71886867735502E-005

1e-9 -> 1.36275997817489E-005

1e-10 -> 6.84249791092663E-006

1e-11 -> 3.44994698460904E-006

1e-12 -> 1.75367152223619E-006

Quaternion z^2+c pointed end view in parallel projection

DE Threshold 1e-16

Values for the centre ray (i.e. hitting the point of the Mandy on the real axis)

"True" -> calculated

1.0 -> 0.977836860473067

0.1 -> 0.097184186106234

0.01 -> 0.00994241047465512

1e-3 -> 0.000999139471789839

1e-4 -> 0.000100040977667748

1e-5 -> 1.00004784004224E-005

1e-6 -> 1.00006323931994E-006

1e-7 -> 1.001088304125E-007

1e-8 -> 1.00620844487879E-008

1e-9 -> 1.0638804422352E-009

1e-10 -> 1.61612459695359E-010

1e-11 -> 7.17222631161621E-011

1e-12 -> 3.5861131558081E-011

I think the above shows that the problem of areas with small gradient causing erroneously high DE values is more prevalent than I previously thought even in a Mandelbrot

As I've mentioned previously a potential cure for this is develop a DE algorithm that incorporates the second derivative in some way. For instance the way I derived the standard DE formula was based on using the Newton, a similar derivation could be made say using Householder's formula.

In the last set of figures I think the 1e-10, 1e-11 and 1e-12 results may be affected by the likelyhood that y and z in quaternionic (x,y,z,w) weren't exactly zero due to slight floating point errors in the 90 degree rotation - i.e. we didn't exactly hit the point.

I intend to redo these tests with say a degree 16 quaternion to see if the values confirm Jos Leys' suggestion that no scaling is required based on the power or to show if such scaling is necessary - and if necessary then to try and find what it should be.