Average Error: 2.0 → 0.8
Time: 2.3m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le -1.7690701216996935 \cdot 10^{+28}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)} \cdot x}{y}\\ \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le 194.43798985741967:\\ \;\;\;\;\frac{\left({a}^{t} \cdot {z}^{y}\right) \cdot x}{y \cdot \left(e^{b} \cdot {a}^{1.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)} \cdot x}{y}\\ \end{array}\]

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)) < -1.7690701216996935e+28 or 194.43798985741967 < (fma (- t 1.0) (log a) (- b))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.5

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

    4. Applied exp-prod0.5

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

    5. Applied simplify0.5

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

    if -1.7690701216996935e+28 < (fma (- t 1.0) (log a) (- b)) < 194.43798985741967

    1. Initial program 7.3

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

    3. Applied sub-neg7.3

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

    4. Applied exp-sum7.6

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

    5. Applied simplify5.6

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

    7. Applied exp-neg5.6

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

    8. Applied pow-sub5.4

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

    9. Applied associate-*r/5.4

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

    10. Applied frac-times5.4

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

    11. Applied associate-*r/5.4

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

    12. Applied associate-/l/1.9

      \[\leadsto \color{blue}{\frac{x \cdot \left(\left({z}^{y} \cdot {a}^{t}\right) \cdot 1\right)}{y \cdot \left({a}^{1.0} \cdot e^{b}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le -1.7690701216996935 \cdot 10^{+28}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)} \cdot x}{y}\\ \mathbf{if}\;(\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_* \le 194.43798985741967:\\ \;\;\;\;\frac{\left({a}^{t} \cdot {z}^{y}\right) \cdot x}{y \cdot \left(e^{b} \cdot {a}^{1.0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*} \cdot \sqrt[3]{(\left(\log z\right) \cdot y + \left((\left(t - 1.0\right) \cdot \left(\log a\right) + \left(-b\right))_*\right))_*}}\right)}^{\left(\sqrt[3]{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}\right)} \cdot x}{y}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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))