Average Error: 2.0 → 0.8
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{a}^{t} \cdot {a}^{\left(-1.0\right)}}{y} \cdot \frac{{z}^{y} \cdot x}{e^{b}} \le 1.1707620736027092 \cdot 10^{+306}:\\ \;\;\;\;\frac{{a}^{t} \cdot {a}^{\left(-1.0\right)}}{y} \cdot \frac{{z}^{y} \cdot x}{e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)}}{y}\\ \end{array}\]

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 a (- 1.0))) y) (/ (* (pow z y) x) (exp b))) < 1.1707620736027092e+306

    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 exp-diff2.7

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

    4. Applied simplify1.9

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

    5. Taylor expanded around inf 2.7

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

    6. Applied simplify1.2

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

    8. Applied sub-neg1.2

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

    9. Applied unpow-prod-up1.1

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

    if 1.1707620736027092e+306 < (* (/ (* (pow a t) (pow a (- 1.0))) y) (/ (* (pow z y) x) (exp b)))

    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 add-cube-cbrt0.3

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

    4. Applied exp-prod0.3

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

    5. Applied simplify0.3

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018178 +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))