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Derivation

1. Split input into 3 regimes

2. if t < -5.35729014039294e-28

   1. Initial program 2.9

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

   2. Taylor expanded around inf 6.9

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified3.4

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(0.8333333333333334 \cdot c - \left(b - c\right) \cdot a\right)}}}\]

   if -5.35729014039294e-28 < t < -4.263262868647511e-303

   1. Initial program 6.2

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

   2. Taylor expanded around -inf 6.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]
   3. Using strategy rm

   4. Applied flip-+8.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}}{a - \frac{5.0}{6.0}}} - \frac{0.6666666666666666}{t}\right)\right)}}\]

   5. Applied frac-sub8.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666}{\left(a - \frac{5.0}{6.0}\right) \cdot t}}\right)}}\]

   6. Applied associate-*r/8.7

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)}{\left(a - \frac{5.0}{6.0}\right) \cdot t}}\right)}}\]

   7. Applied frac-sub6.4

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) \cdot t - \left(a - \frac{5.0}{6.0}\right) \cdot 0.6666666666666666\right)\right)}{t \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot t\right)}}}}\]

   if -4.263262868647511e-303 < t

   1. Initial program 3.3

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

   2. Taylor expanded around -inf 3.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.35729014039294 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{y \cdot e^{2.0 \cdot \left(c \cdot 0.8333333333333334 - \left(b - c\right) \cdot a\right)} + x}\\ \mathbf{elif}\;t \le -4.263262868647511 \cdot 10^{-303}:\\ \;\;\;\;\frac{x}{y \cdot e^{2.0 \cdot \frac{\left(t \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(\sqrt{t + a} \cdot z\right) - \left(\left(t \cdot \left(a \cdot a - \frac{5.0}{6.0} \cdot \frac{5.0}{6.0}\right) - 0.6666666666666666 \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot \left(b - c\right)\right) \cdot t}{\left(t \cdot \left(a - \frac{5.0}{6.0}\right)\right) \cdot t}} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{\sqrt{t + a} \cdot z}{t} - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

Details

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