Average Error: 2.0 → 1.0
Time: 24.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 4.53731180448953874 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right)}^{3}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;a \le 4.53731180448953874 \cdot 10^{-185}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right)}^{3}}}{y}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r132613 = x;
        double r132614 = y;
        double r132615 = z;
        double r132616 = log(r132615);
        double r132617 = r132614 * r132616;
        double r132618 = t;
        double r132619 = 1.0;
        double r132620 = r132618 - r132619;
        double r132621 = a;
        double r132622 = log(r132621);
        double r132623 = r132620 * r132622;
        double r132624 = r132617 + r132623;
        double r132625 = b;
        double r132626 = r132624 - r132625;
        double r132627 = exp(r132626);
        double r132628 = r132613 * r132627;
        double r132629 = r132628 / r132614;
        return r132629;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r132630 = a;
        double r132631 = 4.537311804489539e-185;
        bool r132632 = r132630 <= r132631;
        double r132633 = x;
        double r132634 = 1.0;
        double r132635 = r132634 / r132630;
        double r132636 = 1.0;
        double r132637 = pow(r132635, r132636);
        double r132638 = y;
        double r132639 = z;
        double r132640 = r132634 / r132639;
        double r132641 = log(r132640);
        double r132642 = log(r132635);
        double r132643 = t;
        double r132644 = b;
        double r132645 = fma(r132642, r132643, r132644);
        double r132646 = fma(r132638, r132641, r132645);
        double r132647 = exp(r132646);
        double r132648 = r132637 / r132647;
        double r132649 = r132633 * r132648;
        double r132650 = 3.0;
        double r132651 = pow(r132649, r132650);
        double r132652 = cbrt(r132651);
        double r132653 = r132652 / r132638;
        double r132654 = r132638 / r132648;
        double r132655 = r132634 / r132654;
        double r132656 = r132633 * r132655;
        double r132657 = r132632 ? r132653 : r132656;
        return r132657;
}

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 a < 4.537311804489539e-185

    1. Initial program 0.7

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.1

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube0.1

      \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\sqrt[3]{\left(e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{y}\]

    6. Applied add-cbrt-cube64.0

      \[\leadsto \frac{x \cdot \frac{\color{blue}{\sqrt[3]{\left({\left(\frac{1}{a}\right)}^{1} \cdot {\left(\frac{1}{a}\right)}^{1}\right) \cdot {\left(\frac{1}{a}\right)}^{1}}}}{\sqrt[3]{\left(e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

    7. Applied cbrt-undiv64.0

      \[\leadsto \frac{x \cdot \color{blue}{\sqrt[3]{\frac{\left({\left(\frac{1}{a}\right)}^{1} \cdot {\left(\frac{1}{a}\right)}^{1}\right) \cdot {\left(\frac{1}{a}\right)}^{1}}{\left(e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{y}\]

    8. Applied add-cbrt-cube64.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{\frac{\left({\left(\frac{1}{a}\right)}^{1} \cdot {\left(\frac{1}{a}\right)}^{1}\right) \cdot {\left(\frac{1}{a}\right)}^{1}}{\left(e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

    9. Applied cbrt-unprod64.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{\left({\left(\frac{1}{a}\right)}^{1} \cdot {\left(\frac{1}{a}\right)}^{1}\right) \cdot {\left(\frac{1}{a}\right)}^{1}}{\left(e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)} \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}\right) \cdot e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{y}\]

    10. Simplified2.2

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right)}^{3}}}}{y}\]

    if 4.537311804489539e-185 < a

    1. Initial program 2.2

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

    2. Taylor expanded around inf 2.2

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

    3. Simplified1.5

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
    4. Using strategy rm

    5. Applied associate-/l*0.6

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]
    6. Using strategy rm

    7. Applied div-inv0.7

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 4.53731180448953874 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right)}^{3}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))