User blog comment:ArtismScrub/Generalizing the fast-growing hierarchy towards positive real heights/@comment-32783837-20180127015421

Computing biroot-balum and goldalum using the new version:

BIROOT-BALUM
f1.4142135623730950488016887242097(10) (approximated to 31 decimal digits)

f0.414213562373095048801688724209710(10)

First Iteration
f0.4142135623730950488016887242097(10)

= 10(1.4142135623730950488016887242097) + 1 - 0.4142135623730950488016887242097

= 14.142135623730950488016887242097 + 0.5857864376269049511983112757903

= 14.727922061357855439215198517887

Second Iteration
14.727922061357855439215198517887(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 20.828427124746190097603377448419 + 0.5857864376269049511983112757903

= 21.41421356237309504880168872421

Third Iteration
21.41421356237309504880168872421(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 30.284271247461900976033774484194 + 0.5857864376269049511983112757903

= 30.870057685088805927232085759984

Fourth Iteration
30.870057685088805927232085759984(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 43.656854249492380195206754896839 + 0.5857864376269049511983112757903

= 44.242640687119285146405066172629

Fifth Iteration
44.242640687119285146405066172629(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 62.568542494923801952067548968388 + 0.5857864376269049511983112757903

= 63.154328932550706903265860244179

Sixth Iteration
63.154328932550706903265860244179(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 89.313708498984760390413509793678 + 0.5857864376269049511983112757903

= 89.899494936611665341611821069468

Seventh Iteration
89.899494936611665341611821069468(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 127.13708498984760390413509793678 + 0.5857864376269049511983112757903

= 127.72287142747450885533340921257

Eighth Iteration
127.72287142747450885533340921257(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 180.62741699796952078082701958736 + 0.5857864376269049511983112757903

= 181.21320343559642573202533086315

Ninth Iteration
181.21320343559642573202533086315(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 256.27416997969520780827019587355 + 0.5857864376269049511983112757903

= 256.85995641732211275946850714934

Tenth Iteration
256.85995641732211275946850714934(1.4142135623730950488016887242097) + 0.5857864376269049511983112757903

= 363.25483399593904156165403917471 + 0.5857864376269049511983112757903

= 363.8406204335659465128523504505 (final result)

GOLDALUM
f1.6180339887498948482045868343656(10) (approximated to 31 decimal digits)

f0.618033988749894848204586834365610(10)

First Iteration
f0.6180339887498948482045868343656(10)

= 10(1.6180339887498948482045868343656) + 1 - 0.6180339887498948482045868343656

= 16.180339887498948482045868343656 + 0.38196601125010515179541316563436

= 16.56230589874905363384128150929

Second Iteration
16.56230589874905363384128150929(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 26.798373876248843330250455178021 + 0.38196601125010515179541316563436

= 27.180339887498948482045868343655

Third Iteration
27.180339887498948482045868343655(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 43.978713763747791812296323521675 + 0.38196601125010515179541316563436

= 44.36067977499789696409173668731

Fourth Iteration
44.36067977499789696409173668731(1.6180339887498948482045868343656)+ 0.38196601125010515179541316563436

= 71.777087639996635142546778699694 + 0.38196601125010515179541316563436

= 72.159053651246740294342191865328

Fifth Iteration
72.159053651246740294342191865328(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 116.75580140374442695484310222137 + 0.38196601125010515179541316563436

= 117.137767414994532106638515387

Sixth Iteration
117.137767414994532106638515387(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 189.53288904374106209738988092105 + 0.38196601125010515179541316563436

= 189.91485505499116724918529408669

Seventh Iteration
189.91485505499116724918529408669(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 307.28869044748548905223298314241 + 0.38196601125010515179541316563436

= 307.67065645873559420402839630804

Eighth Iteration
307.67065645873559420402839630804(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 497.82157949122655114962286406344 + 0.38196601125010515179541316563436 = 498.20354550247665630141827722908

Ninth Iteration
498.20354550247665630141827722908(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 806.11026993871204020185584720583 + 0.38196601125010515179541316563436

= 806.49223594996214535365126037146

Tenth Iteration
806.49223594996214535365126037146(1.6180339887498948482045868343656) + 0.38196601125010515179541316563436

= 1,304.9318494299385913514787112692 + 0.38196601125010515179541316563436

= 1,305.3138154411886965032741244349 (final result)

(note that since i approximated, the result may deviate further and further from the actual result as the function is iterated more and more.)