Error

Derivation

1. Split input into 2 regimes

2. if y < -1.3107533376997555 or 4.508099059790703 < 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. Using strategy rm

   3. Applied add-cbrt-cube0.0

      \[\leadsto \frac{x \cdot \color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}{y}\]

   if -1.3107533376997555 < y < 4.508099059790703

   1. Initial program 3.4

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

   3. Applied add-cbrt-cube12.0

      \[\leadsto \frac{x \cdot \color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}{y}\]

   4. Taylor expanded around inf 3.4

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

   5. Simplified2.1

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

   6. Taylor expanded around -inf 62.8

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

   7. Simplified2.1

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3107533376997555 \lor \neg \left(y \le 4.508099059790703\right):\\ \;\;\;\;\frac{x \cdot \sqrt[3]{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot \left(e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b} \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left({a}^{-1.0} \cdot {z}^{y}\right)\right)}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +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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