Average Error: 1.9 → 2.2
Time: 2.2m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(t - 1.0\right) \cdot \log a \le -591.6896912817447:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(\sqrt[3]{\left(t - 1.0\right) \cdot \log a} \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) - b}}{y}\\ \mathbf{if}\;\left(t - 1.0\right) \cdot \log a \le -167.9720135823427:\\ \;\;\;\;\frac{\frac{\frac{x}{e^{b}}}{{z}^{\left(-y\right)}}}{\frac{y}{\frac{{a}^{t}}{{a}^{1.0}}}}\\ \mathbf{if}\;\left(t - 1.0\right) \cdot \log a \le +\infty:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(\sqrt[3]{\left(t - 1.0\right) \cdot \log a} \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(\sqrt[3]{\left(t - 1.0\right) \cdot \log a} \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) \cdot \sqrt[3]{\left(t - 1.0\right) \cdot \log a}\right) - b}}{y}\\ \end{array}\]

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 (* (- t 1.0) (log a)) < -591.6896912817447 or -167.9720135823427 < (* (- t 1.0) (log a))

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt1.3

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

    if -591.6896912817447 < (* (- t 1.0) (log a)) < -167.9720135823427

    1. Initial program 5.9

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

    2. Taylor expanded around inf 5.9

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

    3. Applied simplify6.7

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

Runtime

Time bar (total: 2.2m)Debug log

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +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))