Error

Derivation

1. Split input into 2 regimes

2. if y < -0.42156715725308463 or 0.0006797883881164825 < y

   1. Initial program 0.0

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \frac{x \cdot e^{\left(\color{blue}{-1 \cdot \left(\log \left(\frac{1}{z}\right) \cdot y\right)} + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

   3. Simplified0.0

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

   if -0.42156715725308463 < y < 0.0006797883881164825

   1. Initial program 3.6

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

   2. Taylor expanded around inf 3.6

      \[\leadsto \frac{x \cdot e^{\left(\color{blue}{-1 \cdot \left(\log \left(\frac{1}{z}\right) \cdot y\right)} + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

   3. Simplified3.6

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

   4. Taylor expanded around inf 3.6

      \[\leadsto \frac{x \cdot \color{blue}{e^{1.0 \cdot \log \left(\frac{1}{a}\right) - \left(\log \left(\frac{1}{z}\right) \cdot y + \left(b + t \cdot \log \left(\frac{1}{a}\right)\right)\right)}}}{y}\]

   5. Simplified2.2

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}{y}\]
   6. Using strategy rm

   7. Applied clear-num2.2

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot \left(\left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}}\]
   8. Using strategy rm

   9. Applied pow-neg2.2

      \[\leadsto \frac{1}{\frac{y}{x \cdot \left(\left(\color{blue}{\frac{1}{{a}^{1.0}}} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}\]

   10. Applied associate-*l/2.2

       \[\leadsto \frac{1}{\frac{y}{x \cdot \left(\color{blue}{\frac{1 \cdot {z}^{y}}{{a}^{1.0}}} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}\]

   11. Applied associate-*l/2.2

       \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{\frac{\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}{{a}^{1.0}}}}}\]

   12. Applied associate-*r/2.2

       \[\leadsto \frac{1}{\frac{y}{\color{blue}{\frac{x \cdot \left(\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}{{a}^{1.0}}}}}\]

   13. Applied associate-/r/2.4

       \[\leadsto \frac{1}{\color{blue}{\frac{y}{x \cdot \left(\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)} \cdot {a}^{1.0}}}\]

   14. Applied associate-/r*2.5

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{y}{x \cdot \left(\left(1 \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}}{{a}^{1.0}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -0.42156715725308463 \lor \neg \left(y \le 0.0006797883881164825\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{y}{x \cdot \left({z}^{y} \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)}}}{{a}^{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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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