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Derivation

1. Split input into 2 regimes

2. if a < -7.724721290750151e-15 or 6.951921396237591e+158 < a

   1. Initial program 6.0

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

   2. Taylor expanded around -inf 6.0

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

   3. Taylor expanded around inf 11.3

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

   4. Simplified8.3

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

   if -7.724721290750151e-15 < a < 6.951921396237591e+158

   1. Initial program 2.6

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

   2. Taylor expanded around -inf 2.6

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

   3. Taylor expanded around 0 11.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\left(\left(0.6666666666666666 \cdot \frac{c}{t} + 0.8333333333333334 \cdot b\right) - 0.6666666666666666 \cdot \frac{b}{t}\right)}\right)}}\]

   4. Simplified10.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\left(0.8333333333333334 \cdot b + \left(c - b\right) \cdot \frac{0.6666666666666666}{t}\right)}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.724721290750151 \cdot 10^{-15} \lor \neg \left(a \le 6.951921396237591 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{x}{e^{\left(c \cdot 0.8333333333333334 - a \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{2.0 \cdot \left(\frac{\sqrt{a + t} \cdot z}{t} - \left(b \cdot 0.8333333333333334 + \frac{0.6666666666666666}{t} \cdot \left(c - b\right)\right)\right)} + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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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