Average Error: 1.9 → 2.2
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le -2.072856137200743 \cdot 10^{+18} \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le -219.3720441657439\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{t}\right) \cdot {z}^{y}}{\left(e^{b} \cdot {a}^{1.0}\right) \cdot y}\\ \end{array}\]

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)) < -2.072856137200743e+18 or -219.3720441657439 < (* (- t 1.0) (log a))

    1. Initial program 0.7

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

    if -2.072856137200743e+18 < (* (- t 1.0) (log a)) < -219.3720441657439

    1. Initial program 6.1

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified11.3

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

    5. Applied pow-neg11.3

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

    6. Applied associate-*r/11.3

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

    7. Applied exp-neg11.3

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

    8. Applied un-div-inv11.3

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

    9. Applied frac-times11.3

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

    10. Applied associate-*r/11.2

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

    11. Applied associate-/l/7.2

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

    12. Simplified7.2

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t - 1.0\right) \le -2.072856137200743 \cdot 10^{+18} \lor \neg \left(\log a \cdot \left(t - 1.0\right) \le -219.3720441657439\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{t}\right) \cdot {z}^{y}}{\left(e^{b} \cdot {a}^{1.0}\right) \cdot y}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.92.20.01.9-13.8%
herbie shell --seed 2018290 
(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))