Average Error: 1.8 → 0.7
Time: 2.8m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{{z}^{y} \cdot {a}^{t}}{{a}^{1.0}}}{\left(\sqrt[3]{y \cdot \frac{e^{b}}{x}} \cdot \sqrt[3]{y \cdot \frac{e^{b}}{x}}\right) \cdot \sqrt[3]{y \cdot \frac{e^{b}}{x}}} \le -2.6091528987011057 \cdot 10^{+240}:\\ \;\;\;\;\frac{\frac{x}{{a}^{1.0}}}{y} \cdot \left(t \cdot \log a + y \cdot \log z\right) + \frac{\frac{x}{{a}^{1.0}}}{y}\\ \mathbf{if}\;\frac{\frac{{z}^{y} \cdot {a}^{t}}{{a}^{1.0}}}{\left(\sqrt[3]{y \cdot \frac{e^{b}}{x}} \cdot \sqrt[3]{y \cdot \frac{e^{b}}{x}}\right) \cdot \sqrt[3]{y \cdot \frac{e^{b}}{x}}} \le +\infty:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot {a}^{t}}{{a}^{1.0}}}{y \cdot \frac{e^{b}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 3 regimes

  2. if (/ (/ (* (pow z y) (pow a t)) (pow a 1.0)) (* (* (cbrt (* y (/ (exp b) x))) (cbrt (* y (/ (exp b) x)))) (cbrt (* y (/ (exp b) x))))) < -2.6091528987011057e+240

    1. Initial program 6.0

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

    2. Taylor expanded around 0 8.8

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

    3. Applied simplify3.7

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

    if -2.6091528987011057e+240 < (/ (/ (* (pow z y) (pow a t)) (pow a 1.0)) (* (* (cbrt (* y (/ (exp b) x))) (cbrt (* y (/ (exp b) x)))) (cbrt (* y (/ (exp b) x))))) < +inf.0

    1. Initial program 2.4

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

    2. Taylor expanded around inf 2.4

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

    3. Applied simplify0.9

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

    if +inf.0 < (/ (/ (* (pow z y) (pow a t)) (pow a 1.0)) (* (* (cbrt (* y (/ (exp b) x))) (cbrt (* y (/ (exp b) x)))) (cbrt (* y (/ (exp b) x)))))

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied exp-prod0.0

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

    5. Applied simplify0.0

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(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))