User:Dhatfield/sandbox

$$cps\left( E, N, \mbox{Rw}, k, t_{0}, C, \mbox{fps} \right) \ = \ \frac{\mbox{fps}}{{\left(\mbox{fps} - 1\right)} {\left(\frac{\mbox{fps}}{\mbox{fps} - 1} + \frac{\mbox{Rw}}{{\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^{N} e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)} - 1}\right)}}$$

$$ \frac{dcps}{dRw} \left( E, N, \mbox{Rw}, k, t_{0}, C, \mbox{fps} \right) \ = \ \frac{{\left(\frac{{\left(-\frac{0.5 \, {\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^{N} C k t_{0} \left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{{\left(C - 1\right)}} e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)}}{{\left(-0.5 \, \mbox{Rw} + 1\right)}^{2}} - {\left(\frac{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E {\left(-\mbox{Rw} + 1\right)}^{{\left(\frac{1}{N} - 1\right)}}}{{\left({\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1\right)}^{2} N} - \frac{E {\left(-\mbox{Rw} + 1\right)}^{{\left(\frac{1}{N} - 1\right)}}}{{\left({\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1\right)} N}\right)} N {\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^ e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)}\right)} \mbox{Rw}}{{\left({\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^{N} e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)} - 1\right)}^{2}} - \frac{1}{{\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^{N} e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)} - 1}\right)} \mbox{fps}}{{\left(\mbox{fps} - 1\right)} {\left(\frac{\mbox{fps}}{\mbox{fps} - 1} + \frac{\mbox{Rw}}{{\left(-\frac{{\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} E}{{\left(E - 1\right)} {\left({\left(-\mbox{Rw} + 1\right)}^{\left(\frac{1}{N}\right)} - 1\right)} - 1} + 1\right)}^{N} e^{\left(-\left(\frac{t_{0}}{-0.5 \, \mbox{Rw} + 1}\right)^{C} k\right)} - 1}\right)}^{2}}$$

$$ \left( E, a, b \right) \ {\mapsto} \ \left(R_{w_{c}} = \frac{{\left(-R_{w_{c}} + 1\right)}^ + {\left(-R_{w_{c}} + 1\right)}^{N} a \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(-R_{w_{c}} + 1\right)}^ - E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^{N} - E {\left(-R_{w_{c}} + 1\right)}^ - a \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + \sqrt{-{\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} {\left(-R_{w_{c}} + 1\right)}^{\left(2 \, N\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + {\left(2 \, {\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^\right)} {\left(-R_{w_{c}} + 1\right)}^{N} - {\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + {\left({\left(b - 1\right)} E^{3} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(b - 1\right)} E^{3} - {\left(a + 2 \, b - 2\right)} E^{2} + {\left(a + b - 1\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^\right)} {\left(-R_{w_{c}} + 1\right)}^ + {\left({\left(b - 1\right)} E^{3} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(b - 1\right)} E^{3} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^}}{{\left(-R_{w_{c}} + 1\right)}^ - {\left(E a - a\right)} {\left(-R_{w_{c}} + 1\right)}^{N} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(E a - a\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(-R_{w_{c}} + 1\right)}^ + E^{2} {\left(-R_{w_{c}} + 1\right)}^ - 2 \, E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^{N} - 2 \, E {\left(-R_{w_{c}} + 1\right)}^},\,R_{w_{c}} = \frac{{\left(-R_{w_{c}} + 1\right)}^ + {\left(-R_{w_{c}} + 1\right)}^{N} a \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(-R_{w_{c}} + 1\right)}^ - E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^{N} - E {\left(-R_{w_{c}} + 1\right)}^ - a \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - \sqrt{-{\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} {\left(-R_{w_{c}} + 1\right)}^{\left(2 \, N\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + {\left(2 \, {\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^\right)} {\left(-R_{w_{c}} + 1\right)}^{N} - {\left({\left(a b - a\right)} E^{2} - {\left(a^{2} + a b - a\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + {\left({\left(b - 1\right)} E^{3} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(b - 1\right)} E^{3} - {\left(a + 2 \, b - 2\right)} E^{2} + {\left(a + b - 1\right)} E\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^\right)} {\left(-R_{w_{c}} + 1\right)}^ + {\left({\left(b - 1\right)} E^{3} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(b - 1\right)} E^{3} {\left(-R_{w_{c}} + 1\right)}^ - {\left(a + 2 \, b - 2\right)} E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(a + b - 1\right)} E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^}}{{\left(-R_{w_{c}} + 1\right)}^ - {\left(E a - a\right)} {\left(-R_{w_{c}} + 1\right)}^{N} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ + E^{2} {\left(-R_{w_{c}} + 1\right)}^ + {\left(E a - a\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^ - {\left({\left(-R_{w_{c}} + 1\right)}^ + E^{2} {\left(-R_{w_{c}} + 1\right)}^ - 2 \, E {\left(-R_{w_{c}} + 1\right)}^\right)} \left(\frac{R_{w_{c}}^{2} a - {\left(a - b + 1\right)} R_{w_{c}} - b + 1}{E R_{w_{c}} - R_{w_{c}} + 1}\right)^{N} - 2 \, E {\left(-R_{w_{c}} + 1\right)}^}\right) $$