Average Error: 2.0 → 1.2
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{{a}^{t} \cdot {z}^{y}}{{a}^{1.0}}}{y \cdot \frac{e^{b}}{x}} \le +\infty:\\ \;\;\;\;\frac{\frac{\left({z}^{y} \cdot {a}^{t}\right) \cdot x}{{a}^{1.0} \cdot e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (* (pow a t) (pow z y)) (pow a 1.0)) (* y (/ (exp b) x))) < +inf.0

    1. Initial program 2.8

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

    3. Applied sub-neg2.8

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

    4. Applied exp-sum2.8

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

    5. Applied simplify1.8

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

    7. Applied exp-neg1.8

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

    8. Applied pow-sub1.7

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

    9. Applied associate-*r/1.7

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

    10. Applied frac-times1.7

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

    11. Applied associate-*r/1.7

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

    12. Applied simplify1.7

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

    if +inf.0 < (/ (/ (* (pow a t) (pow z y)) (pow a 1.0)) (* y (/ (exp b) x)))

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied exp-prod0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))