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Derivation

1. Initial program 3.9

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

3. Applied associate-/l*3.3

   \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
4. Using strategy rm

5. Applied flip-+4.8

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

6. Applied frac-sub12.3

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

7. Applied associate-*r/12.5

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

8. Applied frac-sub20.2

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

9. Simplified17.8

   \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(z \cdot \left(3.0 \cdot t\right)\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)\right) \cdot \frac{t \cdot \left(b - c\right)}{\sqrt{a + t}}}}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]
10. Using strategy rm

11. Applied *-un-lft-identity17.8

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\color{blue}{1 \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(z \cdot \left(3.0 \cdot t\right)\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)\right) \cdot \frac{t \cdot \left(b - c\right)}{\sqrt{a + t}}\right)}}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)\right)}}}\]

12. Applied times-frac12.6

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\frac{1}{\frac{t}{\sqrt{t + a}}} \cdot \frac{\left(a - \frac{5.0}{6.0}\right) \cdot \left(z \cdot \left(3.0 \cdot t\right)\right) - \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a + \frac{5.0}{6.0}\right) - 2.0\right)\right) \cdot \frac{t \cdot \left(b - c\right)}{\sqrt{a + t}}}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(t \cdot 3.0\right)}\right)}}}\]

13. Simplified12.6

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

14. Simplified1.2

    \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{\sqrt{t + a}}{t} \cdot \color{blue}{\left(1 \cdot \left(\frac{z}{1} - \frac{\left(\frac{5.0}{6.0} + a\right) \cdot \left(t \cdot 3.0\right) - 2.0}{\frac{t \cdot 3.0}{\frac{t \cdot \left(b - c\right)}{\sqrt{a + t}}}}\right)\right)}\right)}}\]

15. Final simplification1.2

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

Reproduce

herbie shell --seed 2018360 
(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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