Average Error: 2.0 → 1.0
Time: 2.2m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{{e}^{\left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*\right)}} \le -0.0044516954053401055:\\ \;\;\;\;\frac{x}{\frac{y}{{e}^{\left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*\right)}}}\\ \mathbf{if}\;\frac{y}{{e}^{\left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*\right)}} \le 1.3486367112678076 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{{a}^{\left(t - 1.0\right)} \cdot \left(\left(-x\right) \cdot {z}^{y}\right)}{e^{b}}}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{{e}^{\left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left((y \cdot \left(\log z\right) + \left(-b\right))_*\right))_*\right)}}}\\ \end{array}\]

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 (/ y (pow E (fma (log a) (- t 1.0) (fma y (log z) (- b))))) < -0.0044516954053401055 or 1.3486367112678076e+72 < (/ y (pow E (fma (log a) (- t 1.0) (fma y (log z) (- b)))))

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied exp-prod0.8

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

    5. Applied simplify0.8

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

    7. Applied associate-/l*0.3

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

    8. Applied simplify0.3

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

    if -0.0044516954053401055 < (/ y (pow E (fma (log a) (- t 1.0) (fma y (log z) (- b))))) < 1.3486367112678076e+72

    1. Initial program 10.3

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

    3. Applied sub-neg10.3

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

    4. Applied exp-sum10.3

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

    5. Applied simplify6.2

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

    7. Applied frac-2neg6.2

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

    8. Applied simplify6.3

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018201 +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))