Average Error: 2.1 → 3.1
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.3635533604117678 \cdot 10^{-104} \lor \neg \left(y \le 1.551706070831678 \cdot 10^{-271}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left({z}^{y} \cdot {a}^{t}\right)}{e^{b} \cdot {a}^{1.0}}}{y}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if y < -2.3635533604117678e-104 or 1.551706070831678e-271 < y

    1. Initial program 1.5

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

    3. Applied associate-/l*1.3

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

    if -2.3635533604117678e-104 < y < 1.551706070831678e-271

    1. Initial program 4.7

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

    2. Taylor expanded around inf 4.7

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

    3. Simplified10.5

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

    5. Applied pow-neg10.5

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

    6. Applied un-div-inv10.5

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

    7. Applied exp-neg10.5

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

    8. Applied un-div-inv10.5

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

    9. Applied frac-times10.5

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{z}^{y} \cdot {a}^{t}}{e^{b} \cdot {a}^{1.0}}}}{y}\]

    10. Applied associate-*r/10.5

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.3635533604117678 \cdot 10^{-104} \lor \neg \left(y \le 1.551706070831678 \cdot 10^{-271}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left({z}^{y} \cdot {a}^{t}\right)}{e^{b} \cdot {a}^{1.0}}}{y}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.03.10.02.0-57.1%
herbie shell --seed 2018339 
(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))