Average Error: 1.9 → 0.7
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{z}^{y} \cdot \left({a}^{t} \cdot x\right)}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \le 7.816805392612654 \cdot 10^{+266}:\\ \;\;\;\;\frac{{z}^{y} \cdot \left({a}^{t} \cdot x\right)}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}}\\ \end{array}\]

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 (/ (* (* (pow a t) x) (pow z y)) (* y (* (exp b) (pow a 1.0)))) < 7.816805392612654e+266

    1. Initial program 2.6

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

    2. Taylor expanded around inf 2.6

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

    3. Simplified1.7

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

    5. Applied pow-neg1.7

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

    6. Applied associate-*l/1.7

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

    7. Applied exp-neg1.7

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

    8. Applied un-div-inv1.7

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

    9. Applied frac-times1.7

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

    10. Applied associate-*r/1.7

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

    11. Applied associate-/l/0.9

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

    12. Simplified0.9

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

    if 7.816805392612654e+266 < (/ (* (* (pow a t) x) (pow z y)) (* y (* (exp b) (pow a 1.0))))

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{z}^{y} \cdot \left({a}^{t} \cdot x\right)}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \le 7.816805392612654 \cdot 10^{+266}:\\ \;\;\;\;\frac{{z}^{y} \cdot \left({a}^{t} \cdot x\right)}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(\left(t - 1.0\right) \cdot \log a + \log z \cdot y\right) - b}}{y}}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018217 
(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))