Average Error: 1.9 → 2.1
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le 158.1336739000043 \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le 593.5793423463056\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\left(e^{-b} \cdot {z}^{y}\right) \cdot {a}^{t}}}}{{a}^{1.0}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (- t 1.0) (log a)) < 158.1336739000043 or 593.5793423463056 < (* (- t 1.0) (log a))

    1. Initial program 2.0

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

    3. Applied associate-/l*1.1

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

    if 158.1336739000043 < (* (- t 1.0) (log a)) < 593.5793423463056

    1. Initial program 1.4

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

    3. Applied associate-/l*5.7

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

    4. Taylor expanded around inf 5.7

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

    5. Simplified12.9

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

    7. Applied pow-neg12.9

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

    8. Applied un-div-inv12.9

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

    9. Applied associate-*r/12.9

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

    10. Applied associate-/r/12.9

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

    11. Applied associate-/r*8.1

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le 158.1336739000043 \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le 593.5793423463056\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\left(e^{-b} \cdot {z}^{y}\right) \cdot {a}^{t}}}}{{a}^{1.0}}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018255 
(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))