Average Error: 1.8 → 1.2
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{a}^{\left(t - 1.0\right)}}{e^{b}} \le 2.3452044325780695 \cdot 10^{-268}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \mathbf{if}\;\frac{{a}^{\left(t - 1.0\right)}}{e^{b}} \le 1.6237556946325117 \cdot 10^{+244}:\\ \;\;\;\;\frac{\left(x \cdot {a}^{t}\right) \cdot \frac{{z}^{y}}{y}}{{a}^{1.0} \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 1.0)) (exp b)) < 2.3452044325780695e-268 or 1.6237556946325117e+244 < (/ (pow a (- t 1.0)) (exp b))

    1. Initial program 0.4

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

    3. Applied *-un-lft-identity0.4

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

    4. Applied exp-prod0.4

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

    5. Applied simplify0.4

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

    if 2.3452044325780695e-268 < (/ (pow a (- t 1.0)) (exp b)) < 1.6237556946325117e+244

    1. Initial program 6.4

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

    3. Applied sub-neg6.4

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

    4. Applied exp-sum6.4

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

    5. Applied simplify4.1

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

    7. Applied div-inv4.1

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

    9. Applied exp-neg4.1

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

    10. Applied pow-sub4.0

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

    11. Applied associate-*r/4.0

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

    12. Applied frac-times4.0

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

    13. Applied associate-*r/4.0

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

    14. Applied associate-*l/3.7

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

    15. Applied simplify3.7

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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))