Average Error: 2.0 → 1.7
Time: 6.2m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le -2610577381.2603726:\\ \;\;\;\;\frac{x \cdot \sqrt[3]{{\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}}}{y}\\ \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le 541.69668280295:\\ \;\;\;\;\frac{\frac{\frac{{z}^{y}}{{a}^{1.0}}}{\frac{e^{b}}{{a}^{t}}}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sqrt[3]{{\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}}}{y}\\ \end{array}\]

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 (fma (- t 1.0) (log a) (- b)) < -2610577381.2603726 or 541.69668280295 < (fma (- t 1.0) (log a) (- b))

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.6

      \[\leadsto \frac{x \cdot \color{blue}{\sqrt[3]{\left(e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b} \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right) \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}{y}\]

    4. Applied simplify0.6

      \[\leadsto \frac{x \cdot \sqrt[3]{\color{blue}{{\left(e^{2 + 1}\right)}^{\left((y \cdot \left(\log z\right) + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*\right)}}}}{y}\]

    if -2610577381.2603726 < (fma (- t 1.0) (log a) (- b)) < 541.69668280295

    1. Initial program 7.1

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

    2. Taylor expanded around inf 7.1

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

    3. Applied simplify4.6

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

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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))