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Derivation

1. Split input into 2 regimes

2. if t < 8.105866356025745e-274 or 3.6483414341945934e-83 < t

   1. Initial program 1.6

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied associate-/l*1.4

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}\]

   if 8.105866356025745e-274 < t < 3.6483414341945934e-83

   1. Initial program 3.7

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied exp-diff10.4

      \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1.0\right) \cdot \log a}}{e^{b}}}}{y}\]

   4. Applied associate-*r/11.5

      \[\leadsto \frac{\color{blue}{\frac{x \cdot e^{y \cdot \log z + \left(t - 1.0\right) \cdot \log a}}{e^{b}}}}{y}\]

   5. Applied associate-/l/11.5

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \left(t - 1.0\right) \cdot \log a}}{y \cdot e^{b}}}\]

   6. Simplified10.1

      \[\leadsto \frac{\color{blue}{\left({z}^{y} \cdot x\right) \cdot {a}^{\left(t - 1.0\right)}}}{y \cdot e^{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.105866356025745 \cdot 10^{-274} \lor \neg \left(t \le 3.6483414341945934 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log z \cdot y + \log a \cdot \left(t - 1.0\right)\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1.0\right)} \cdot \left({z}^{y} \cdot x\right)}{y \cdot e^{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

Details

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