Average Error: 1.8 → 1.6
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.0587678570776074 \cdot 10^{+30} \lor \neg \left(y \le 2.1276374615859492 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(e^{\sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left(\sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)\right)}{y}\\ \end{array}\]

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 y < -1.0587678570776074e+30 or 2.1276374615859492e-38 < y

    1. Initial program 0.2

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

    2. Taylor expanded around 0 0.2

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

    4. Applied associate-/l*0.2

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

    if -1.0587678570776074e+30 < y < 2.1276374615859492e-38

    1. Initial program 3.3

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

    2. Taylor expanded around 0 3.3

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

    3. Taylor expanded around inf 3.3

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

    4. Simplified3.0

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

    6. Applied add-cube-cbrt3.0

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0587678570776074 \cdot 10^{+30} \lor \neg \left(y \le 2.1276374615859492 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{\left(\log a \cdot \left(t - 1.0\right) + \log z \cdot y\right) - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(e^{\sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left(\sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \sqrt[3]{(t \cdot \left(\log a\right) + \left(-b\right))_*}\right)} \cdot \left({z}^{y} \cdot {a}^{\left(-1.0\right)}\right)\right)}{y}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.91.60.01.914.5%
herbie shell --seed 2018352 +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))