Error

Derivation

1. Split input into 2 regimes

2. if y < -1.9813937686979932e-95 or 0.00920090525632143 < y

   1. Initial program 0.4

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

   3. Applied associate-/l*0.4

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

   5. Applied *-un-lft-identity0.4

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

   6. Applied exp-prod0.4

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

   7. Simplified0.4

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

   if -1.9813937686979932e-95 < y < 0.00920090525632143

   1. Initial program 3.7

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

   2. Taylor expanded around inf 3.7

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

   3. Simplified2.4

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

   5. Applied add-sqr-sqrt2.4

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

   6. Applied associate-*r*2.4

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

4. Final simplification1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9813937686979932 \cdot 10^{-95} \lor \neg \left(y \le 0.00920090525632143\right):\\ \;\;\;\;\frac{x}{\frac{y}{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left(\sqrt{{z}^{y}} \cdot \left(\sqrt{{z}^{y}} \cdot {a}^{\left(-1.0\right)}\right)\right)\right)}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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

Time bar (total: 43.9s)Debug log