Average Error: 2.0 → 0.7
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{{z}^{y}}{e^{b}} \cdot \frac{{a}^{t}}{{a}^{1.0}}}{\frac{y}{x}} \le 5.21196274515651 \cdot 10^{+272}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{e^{b}} \cdot \frac{{a}^{t}}{{a}^{1.0}}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 z y) (exp b)) (/ (pow a t) (pow a 1.0))) (/ y x)) < 5.21196274515651e+272

    1. Initial program 2.9

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

    3. Applied add-sqr-sqrt2.9

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

    4. Applied simplify2.7

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

    5. Applied simplify2.0

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

    6. Taylor expanded around -inf 62.9

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

    7. Applied simplify0.9

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

    9. Applied pow-sub0.9

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

    if 5.21196274515651e+272 < (/ (* (/ (pow z y) (exp b)) (/ (pow a t) (pow a 1.0))) (/ y x))

    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 *-un-lft-identity0.2

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

    4. Applied exp-prod0.2

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

    5. Applied simplify0.2

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(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))