Average Error: 2.0 → 1.5
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 8.727404754869684 \cdot 10^{-255} \lor \neg \left(y \le 1.3353578076234787 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{e^{(y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right)}}\\ \end{array}\]

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 < 8.727404754869684e-255 or 1.3353578076234787e-30 < y

    1. Initial program 1.5

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

    3. Applied add-log-exp1.5

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

    4. Applied rem-exp-log1.5

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

    5. Simplified1.5

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

    if 8.727404754869684e-255 < y < 1.3353578076234787e-30

    1. Initial program 4.1

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

    3. Applied add-log-exp4.1

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

    4. Applied rem-exp-log4.1

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

    5. Simplified4.1

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

    6. Taylor expanded around inf 4.1

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

    7. Simplified2.7

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

    9. Applied associate-/l*1.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 8.727404754869684 \cdot 10^{-255} \lor \neg \left(y \le 1.3353578076234787 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{e^{(y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018235 +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))