by Kevin Harlow
June 5, 2003
There is a whole bunch of boring algebra below. The whole purpose of which was to produce an equation for WHIP in terms of only SO/PA, BB/PA, HR/PA, and $H. If you're not a masochist then you'll skip the derivation that follows and only pay attention to the following equation:
WHIP ~= 3 * [BB' + HR' + H$ * (1 - SO' - HR' - BB')] / [SO' + (1 - H$) * (1 - SO' - HR' - BB')]
WHIP = (BB + H) / IP
(WHIP/3) = (BB + H) / O
H = BIPH + HR
O = SO + BIPO
WHIP/3 = BB/O + HR/O + BIPH/O
BIPH/O = BIPH / (SO + BIPO)
BIPH/O = 1 / [(SO + BIPO) / BIPH]
BIPH/O = 1 / (SO/BIPH + BIPO/BIPH)
BIPH/O = 1 / [SO/BIPH - 1 + (BIPO+BIPH)/BIPH]
BIPH/O = 1 / [(SO/BIPO)*(BIPO/BIPH) - 1 + 1/$H]
BIPH/O = 1 / [(SO/BIPO + 1)*(1/$H) - (1 + SO/BIPO)]
BIPH/O = 1 / [(1/$H) - 1) * (1 + SO/BIPO)]
SO + BIPO = O
BIPO = O - SO
BIPO/SO = O/SO - 1
SO/BIPO = 1 / (O/SO - 1)
SO/BIPO + 1 = SO / (O - SO) + 1
SO/BIPO + 1 = (SO + O - SO) / (O - SO)
SO/BIPO + 1 = O / (O-SO)
BIPH/O = [$H/(1-$H)] * [(O - SO)/O]
WHIP/3 = BB/O + HR/O + [$H/(1-$H)] * [(O - SO)/O]
O = (1-OBP) * PA
WHIP/3 = [BB/PA + HR/PA + [$H/(1-$H)] * (O/PA - SO/PA)] / (1-OBP)
BB' = BB/PA
HR' = HR/PA
SO' = SO/PA
O' = O/PA
WHIP/3 = [BB' + HR' + ($H/(1-$H)) * (O' - SO')] / (1-OBP)
(1 - OBP) = SO' + (1 - $H) * (1 - SO' - HR' - BB')
WHIP/3 = [BB' + HR' + ($H/(1-$H)) * (O' - SO')] / [SO' + (1-$H)*(1-SO'-HR'-BB')]
(O-SO)/PA = (AB-H-SO)
(O' - SO') = [(1-OBP) - SO']
WHIP/3 = [BB' + HR' + ($H/(1-$H)) * ((1-OBP) - SO')] / [SO' + (1-$H)*(1-SO'-HR'-BB')]
WHIP/3 = [BB'+HR'+($H/(1-$H))*((1-$H)*(1-SO'-HR'-BB'))] / [SO'+(1-$H)*(1-SO'-HR'-BB')]
WHIP/3 = [BB' + HR' + $H * (1 - SO' - HR' - BB')] / [SO' + (1 - $H) * (1 - SO' - HR' - BB')]
If you are still paying attention thru all this stuff, you may want to see how the derivatives work out.
DEN = SO' + (1 - $H) * (1 - SO' - HR' - BB')
d(WHIP/3)/d(BB') = (1-$H)/DEN + (1-$H)/DEN^2
d(WHIP/3)/d(BB') = (1+DEN)*(1-$H)/DEN^2
d(WHIP/3)/d(HR') = (1-$H)/DEN + (1-$H)/DEN^2
d(WHIP/3)/d(HR') = (1+DEN)*(1-$H)/DEN^2
d(WHIP/3)/d(SO) = $H/DEN - $H/DEN^2
d(WHIP/3)/d(SO) = $H*(DEN-1)/DEN^2
d(WHIP/3)/d($H) = (1-SO'-HR'-BB')/DEN + (1-SO'-HR'-BB')/DEN^2
d(WHIP/3)/d($H) = (1+DEN)*(1-SO'-HR'-BB')/DEN^2
d(WHIP/3)/d(HR') / d(WHIP/3)/d(BB') = 1
d(WHIP/3)/d(SO') / d(WHIP/3)/d(BB') = [(DEN-1)*$H/DEN^2] / [(DEN+1)*(1-$H)/DEN^2]
d(WHIP/3)/d(SO') / d(WHIP/3)/d(BB') = [(DEN-1)/(DEN+1)] * [$H/(1-$H)]
d(WHIP/3)/d($H) / d(WHIP/3)/d(BB') = (1-SO'-HR'-BB') / (1-$H)
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