Average Error: 1.8 → 0.4
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 1.474099479005752 \cdot 10^{-82}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 1.3356108855427758 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 (/ (exp b) (pow a (- t 1.0))) < 1.474099479005752e-82 or 1.3356108855427758e+303 < (/ (exp b) (pow a (- t 1.0)))

    1. Initial program 0.3

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

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

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

    4. Applied exp-prod0.3

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

    5. Applied simplify0.3

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

    if 1.474099479005752e-82 < (/ (exp b) (pow a (- t 1.0))) < 1.3356108855427758e+303

    1. Initial program 7.8

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

    2. Taylor expanded around inf 7.8

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

    3. Applied simplify5.6

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

    5. Applied associate-*l/5.6

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

    6. Applied associate-/l/0.7

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))