Average Error: 1.8 → 0.5
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 7.964242519851838 \cdot 10^{-39} \lor \neg \left(\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 1.1538827626012094 \cdot 10^{+291}\right):\\ \;\;\;\;\frac{{e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{t} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)}{y \cdot e^{b}} \cdot x\\ \end{array}\]

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 (/ (exp b) (pow a (- t 1.0))) < 7.964242519851838e-39 or 1.1538827626012094e+291 < (/ (exp b) (pow a (- t 1.0)))

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied exp-prod0.5

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

    5. Applied simplify0.5

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

    if 7.964242519851838e-39 < (/ (exp b) (pow a (- t 1.0))) < 1.1538827626012094e+291

    1. Initial program 8.2

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

    2. Taylor expanded around inf 8.2

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

    3. Applied simplify6.3

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

    5. Applied associate-*l/6.3

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

    6. Applied associate-/r/0.4

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

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 7.964242519851838 \cdot 10^{-39} \lor \neg \left(\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 1.1538827626012094 \cdot 10^{+291}\right):\\ \;\;\;\;\frac{{e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{t} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)}{y \cdot e^{b}} \cdot x\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))