Average Error: 1.8 → 0.9
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left({z}^{y} \cdot {a}^{t}\right) \cdot x}{{a}^{1.0} \cdot e^{b}}}{y} \le 9.29567238458204 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\frac{e^{b} \cdot y}{{z}^{y} \cdot {a}^{\left(t - 1.0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 2 regimes

  2. if (/ (/ (* (* (pow z y) (pow a t)) x) (* (pow a 1.0) (exp b))) y) < 9.29567238458204e+148

    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. Using strategy rm

    3. Applied sub-neg2.4

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

    4. Applied exp-sum2.4

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

    5. Applied simplify1.6

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

    7. Applied associate-/l*1.0

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

    8. Applied simplify1.0

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

    if 9.29567238458204e+148 < (/ (/ (* (* (pow z y) (pow a t)) x) (* (pow a 1.0) (exp b))) y)

    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

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

Runtime

Time bar (total: 1.8m)Debug logProfile

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