Average Error: 2.0 → 0.5
Time: 2.4m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 2.0779705295437254 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \mathbf{if}\;\frac{e^{b}}{{a}^{\left(t - 1.0\right)}} \le 8.528437596246273 \cdot 10^{+301}:\\ \;\;\;\;\left(\frac{{z}^{y}}{e^{b} \cdot y} \cdot {a}^{\left(-1.0\right)}\right) \cdot \left(x \cdot {a}^{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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 (/ (exp b) (pow a (- t 1.0))) < 2.0779705295437254e-47 or 8.528437596246273e+301 < (/ (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 2.0779705295437254e-47 < (/ (exp b) (pow a (- t 1.0))) < 8.528437596246273e+301

    1. Initial program 8.4

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

    2. Taylor expanded around inf 8.4

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

    3. Applied simplify5.3

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

    5. Applied pow-neg5.3

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

    6. Applied associate-*l/5.3

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

    7. Applied frac-times5.3

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

    8. Applied associate-/r/0.5

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

    9. Applied simplify0.6

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(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))