Average Error: 2.0 → 0.5
Time: 2.5m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*} \le 1.3260133781902547 \cdot 10^{-305} \lor \neg \left(e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*} \le 5.720735444698693 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{\left(-t\right)}}{{\left(\frac{1}{z}\right)}^{y}}}{\frac{y \cdot {a}^{1.0}}{\frac{x}{e^{b}}}}\\ \end{array}\]

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 (exp (fma (log a) (- t 1.0) (- b))) < 1.3260133781902547e-305 or 5.720735444698693e+105 < (exp (fma (log a) (- t 1.0) (- b)))

    1. Initial program 0.3

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

    if 1.3260133781902547e-305 < (exp (fma (log a) (- t 1.0) (- b))) < 5.720735444698693e+105

    1. Initial program 8.0

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

    3. Applied sub-neg8.0

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

    4. Applied exp-sum8.1

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

    5. Applied simplify5.6

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

    7. Applied exp-neg5.6

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

    8. Applied pow-sub5.6

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

    9. Applied associate-*r/5.6

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

    10. Applied frac-times5.6

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

    11. Applied associate-*r/5.5

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

    12. Applied simplify5.5

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

    13. Taylor expanded around inf 5.6

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

    14. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{a}\right)}^{\left(-t\right)}}{{\left(\frac{1}{z}\right)}^{y}}}{\frac{y \cdot {a}^{1.0}}{\frac{x}{e^{b}}}}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*} \le 1.3260133781902547 \cdot 10^{-305} \lor \neg \left(e^{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*} \le 5.720735444698693 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{e^{\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{\left(-t\right)}}{{\left(\frac{1}{z}\right)}^{y}}}{\frac{y \cdot {a}^{1.0}}{\frac{x}{e^{b}}}}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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))