Average Error: 2.0 → 1.9
Time: 42.8s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -131967522831829665381196942903493345148900:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b\right)}}{y}\\ \mathbf{elif}\;x \le 2.21645226035916800897563050125442800086 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{x}{y}}{e^{b - \mathsf{fma}\left(t - 1, \log a, \log z \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}\right)\right)} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -131967522831829665381196942903493345148900:\\
\;\;\;\;\frac{x \cdot {e}^{\left(\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b\right)}}{y}\\

\mathbf{elif}\;x \le 2.21645226035916800897563050125442800086 \cdot 10^{86}:\\
\;\;\;\;\frac{\frac{x}{y}}{e^{b - \mathsf{fma}\left(t - 1, \log a, \log z \cdot y\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}\right)\right)} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r3187568 = x;
        double r3187569 = y;
        double r3187570 = z;
        double r3187571 = log(r3187570);
        double r3187572 = r3187569 * r3187571;
        double r3187573 = t;
        double r3187574 = 1.0;
        double r3187575 = r3187573 - r3187574;
        double r3187576 = a;
        double r3187577 = log(r3187576);
        double r3187578 = r3187575 * r3187577;
        double r3187579 = r3187572 + r3187578;
        double r3187580 = b;
        double r3187581 = r3187579 - r3187580;
        double r3187582 = exp(r3187581);
        double r3187583 = r3187568 * r3187582;
        double r3187584 = r3187583 / r3187569;
        return r3187584;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3187585 = x;
        double r3187586 = -1.3196752283182967e+41;
        bool r3187587 = r3187585 <= r3187586;
        double r3187588 = exp(1.0);
        double r3187589 = a;
        double r3187590 = log(r3187589);
        double r3187591 = t;
        double r3187592 = 1.0;
        double r3187593 = r3187591 - r3187592;
        double r3187594 = r3187590 * r3187593;
        double r3187595 = z;
        double r3187596 = log(r3187595);
        double r3187597 = y;
        double r3187598 = r3187596 * r3187597;
        double r3187599 = r3187594 + r3187598;
        double r3187600 = b;
        double r3187601 = r3187599 - r3187600;
        double r3187602 = pow(r3187588, r3187601);
        double r3187603 = r3187585 * r3187602;
        double r3187604 = r3187603 / r3187597;
        double r3187605 = 2.216452260359168e+86;
        bool r3187606 = r3187585 <= r3187605;
        double r3187607 = r3187585 / r3187597;
        double r3187608 = fma(r3187593, r3187590, r3187598);
        double r3187609 = r3187600 - r3187608;
        double r3187610 = exp(r3187609);
        double r3187611 = r3187607 / r3187610;
        double r3187612 = exp(r3187601);
        double r3187613 = r3187585 * r3187612;
        double r3187614 = r3187613 / r3187597;
        double r3187615 = cbrt(r3187614);
        double r3187616 = r3187615 * r3187615;
        double r3187617 = r3187615 * r3187616;
        double r3187618 = cbrt(r3187617);
        double r3187619 = r3187615 * r3187618;
        double r3187620 = log1p(r3187614);
        double r3187621 = expm1(r3187620);
        double r3187622 = cbrt(r3187621);
        double r3187623 = r3187622 * r3187615;
        double r3187624 = r3187623 * r3187615;
        double r3187625 = cbrt(r3187624);
        double r3187626 = r3187619 * r3187625;
        double r3187627 = r3187606 ? r3187611 : r3187626;
        double r3187628 = r3187587 ? r3187604 : r3187627;
        return r3187628;
}

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 3 regimes

  2. if x < -1.3196752283182967e+41

    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. Using strategy rm

    3. Applied *-un-lft-identity0.7

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

    4. Applied exp-prod0.7

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

    5. Simplified0.7

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

    if -1.3196752283182967e+41 < x < 2.216452260359168e+86

    1. Initial program 2.8

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

    2. Simplified1.7

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

    if 2.216452260359168e+86 < x

    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. Using strategy rm

    3. Applied add-cube-cbrt0.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.7

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.7

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}}\]
    8. Using strategy rm

    9. Applied expm1-log1p-u4.1

      \[\leadsto \left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \cdot \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\right)\right)}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -131967522831829665381196942903493345148900:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b\right)}}{y}\\ \mathbf{elif}\;x \le 2.21645226035916800897563050125442800086 \cdot 10^{86}:\\ \;\;\;\;\frac{\frac{x}{y}}{e^{b - \mathsf{fma}\left(t - 1, \log a, \log z \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \left(\sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}\right)\right)} \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}\right) \cdot \sqrt[3]{\frac{x \cdot e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b}}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))