Average Error: 2.0 → 0.8
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\sqrt{{z}^{y} \cdot {a}^{t}} \cdot \sqrt{\frac{{z}^{y}}{e^{b}} \cdot {a}^{t}}\right)}{y \cdot \left(\sqrt{e^{b} \cdot {a}^{1.0}} \cdot \sqrt{{a}^{1.0}}\right)} \le 5.2119627451565103 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot \left(\sqrt{{z}^{y} \cdot {a}^{t}} \cdot \sqrt{\frac{{z}^{y}}{e^{b}} \cdot {a}^{t}}\right)}{y \cdot \left(\sqrt{e^{b} \cdot {a}^{1.0}} \cdot \sqrt{{a}^{1.0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {\left({e}^{\left(\sqrt[3]{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*}\right)}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)}}{y}\\ \end{array}\]

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 (/ (* x (* (sqrt (* (pow z y) (pow a t))) (sqrt (* (/ (pow z y) (exp b)) (pow a t))))) (* y (* (sqrt (* (exp b) (pow a 1.0))) (sqrt (pow a 1.0))))) < 5.2119627451565103e+272

    1. Initial program 2.7

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

    3. Applied add-sqr-sqrt2.7

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

    4. Applied simplify2.6

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

    5. Applied simplify1.9

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

    7. Applied pow-sub1.9

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

    8. Applied associate-*r/1.9

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

    9. Applied sqrt-div1.9

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

    10. Applied pow-sub1.9

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

    11. Applied frac-times1.8

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

    12. Applied sqrt-div1.9

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

    13. Applied frac-times1.9

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

    14. Applied associate-*r/1.9

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

    15. Applied associate-/l/1.0

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

    if 5.2119627451565103e+272 < (/ (* x (* (sqrt (* (pow z y) (pow a t))) (sqrt (* (/ (pow z y) (exp b)) (pow a t))))) (* y (* (sqrt (* (exp b) (pow a 1.0))) (sqrt (pow a 1.0)))))

    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

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

    7. Applied add-cube-cbrt0.3

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

    8. Applied pow-unpow0.3

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

    9. Applied simplify0.3

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

Runtime

Time bar (total: 2.4m)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))