Average Error: 1.9 → 1.8
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le -675.0835959836515 \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le -437.2778104905943\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {z}^{y}\right) \cdot {a}^{t}}{\left(e^{b} \cdot {a}^{1.0}\right) \cdot y}\\ \end{array}\]

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 (* (- t 1.0) (log a)) < -675.0835959836515 or -437.2778104905943 < (* (- t 1.0) (log a))

    1. Initial program 1.3

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

    2. Taylor expanded around 0 1.3

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

    if -675.0835959836515 < (* (- t 1.0) (log a)) < -437.2778104905943

    1. Initial program 7.3

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

    2. Taylor expanded around inf 7.3

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

    3. Simplified12.0

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

    5. Applied neg-sub012.0

      \[\leadsto \frac{x \cdot \left(\left(e^{-b} \cdot {z}^{y}\right) \cdot \left({a}^{t} \cdot {a}^{\color{blue}{\left(0 - 1.0\right)}}\right)\right)}{y}\]

    6. Applied pow-sub12.0

      \[\leadsto \frac{x \cdot \left(\left(e^{-b} \cdot {z}^{y}\right) \cdot \left({a}^{t} \cdot \color{blue}{\frac{{a}^{0}}{{a}^{1.0}}}\right)\right)}{y}\]

    7. Applied associate-*r/12.0

      \[\leadsto \frac{x \cdot \left(\left(e^{-b} \cdot {z}^{y}\right) \cdot \color{blue}{\frac{{a}^{t} \cdot {a}^{0}}{{a}^{1.0}}}\right)}{y}\]

    8. Applied exp-neg12.0

      \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\frac{1}{e^{b}}} \cdot {z}^{y}\right) \cdot \frac{{a}^{t} \cdot {a}^{0}}{{a}^{1.0}}\right)}{y}\]

    9. Applied associate-*l/12.0

      \[\leadsto \frac{x \cdot \left(\color{blue}{\frac{1 \cdot {z}^{y}}{e^{b}}} \cdot \frac{{a}^{t} \cdot {a}^{0}}{{a}^{1.0}}\right)}{y}\]

    10. Applied frac-times12.0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{\left(1 \cdot {z}^{y}\right) \cdot \left({a}^{t} \cdot {a}^{0}\right)}{e^{b} \cdot {a}^{1.0}}}}{y}\]

    11. Applied associate-*r/12.0

      \[\leadsto \frac{\color{blue}{\frac{x \cdot \left(\left(1 \cdot {z}^{y}\right) \cdot \left({a}^{t} \cdot {a}^{0}\right)\right)}{e^{b} \cdot {a}^{1.0}}}}{y}\]

    12. Applied associate-/l/6.3

      \[\leadsto \color{blue}{\frac{x \cdot \left(\left(1 \cdot {z}^{y}\right) \cdot \left({a}^{t} \cdot {a}^{0}\right)\right)}{y \cdot \left(e^{b} \cdot {a}^{1.0}\right)}}\]

    13. Simplified6.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le -675.0835959836515 \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le -437.2778104905943\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {z}^{y}\right) \cdot {a}^{t}}{\left(e^{b} \cdot {a}^{1.0}\right) \cdot y}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018230 
(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))