Average Error: 1.9 → 0.2
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \cdot \left(\left({z}^{y} \cdot {a}^{t}\right) \cdot x\right) \le -5.357333654843567 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\frac{\frac{\frac{y}{x}}{{z}^{y}}}{{a}^{\left(t - 1.0\right)}} \cdot e^{b}}\\ \mathbf{if}\;\frac{1}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \cdot \left(\left({z}^{y} \cdot {a}^{t}\right) \cdot x\right) \le 1.5694896574592005 \cdot 10^{+288}:\\ \;\;\;\;\frac{\left({z}^{y} \cdot {a}^{t}\right) \cdot x}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \log a \cdot \left(t - 1.0\right)\right) - b}\right)} \cdot x}{y}\\ \end{array}\]

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 3 regimes

  2. if (* (* x (* (* (pow z y) (pow a t)) 1)) (/ 1 (* y (* (pow a 1.0) (exp b))))) < -5.357333654843567e+307

    1. Initial program 6.3

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

    3. Applied sub-neg6.3

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

    4. Applied exp-sum6.3

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

    5. Applied simplify0.7

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

    7. Applied clear-num0.7

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

    8. Applied simplify0.8

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

    if -5.357333654843567e+307 < (* (* x (* (* (pow z y) (pow a t)) 1)) (/ 1 (* y (* (pow a 1.0) (exp b))))) < 1.5694896574592005e+288

    1. Initial program 2.5

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

    3. Applied sub-neg2.5

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

    4. Applied exp-sum2.5

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

    5. Applied simplify1.7

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

    7. Applied exp-neg1.7

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

    8. Applied pow-sub1.6

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

    9. Applied associate-*r/1.6

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

    10. Applied frac-times1.6

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

    11. Applied associate-*r/1.6

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

    12. Applied associate-/l/0.1

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

    if 1.5694896574592005e+288 < (* (* x (* (* (pow z y) (pow a t)) 1)) (/ 1 (* y (* (pow a 1.0) (exp b)))))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.4

      \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot \sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot \sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}{y}\]

    4. Applied exp-prod0.4

      \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot \sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)}}}{y}\]

    5. Applied simplify0.4

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \cdot \left(\left({z}^{y} \cdot {a}^{t}\right) \cdot x\right) \le -5.357333654843567 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\frac{\frac{\frac{y}{x}}{{z}^{y}}}{{a}^{\left(t - 1.0\right)}} \cdot e^{b}}\\ \mathbf{if}\;\frac{1}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y} \cdot \left(\left({z}^{y} \cdot {a}^{t}\right) \cdot x\right) \le 1.5694896574592005 \cdot 10^{+288}:\\ \;\;\;\;\frac{\left({z}^{y} \cdot {a}^{t}\right) \cdot x}{\left({a}^{1.0} \cdot e^{b}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \log a \cdot \left(t - 1.0\right)\right) - b}\right)} \cdot x}{y}\\ \end{array}}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018199 +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))