Average Error: 2.1 → 8.0
Time: 25.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.199638483849183435540034190073792906329 \cdot 10^{-152}:\\ \;\;\;\;x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\frac{\left(y \cdot \left(\log z \cdot \log z - t \cdot t\right)\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right) + \left(\log z + t\right) \cdot \left(a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)\right)}{\left(\log z + t\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right)}}}\right)\\ \mathbf{elif}\;y \le 5.487655812586681992965441010366711704904 \cdot 10^{-8}:\\ \;\;\;\;x \cdot e^{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z \cdot y + a \cdot \log 1\right) - t \cdot y}\\ \end{array}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.199638483849183435540034190073792906329 \cdot 10^{-152}:\\
\;\;\;\;x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\frac{\left(y \cdot \left(\log z \cdot \log z - t \cdot t\right)\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right) + \left(\log z + t\right) \cdot \left(a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)\right)}{\left(\log z + t\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right)}}}\right)\\

\mathbf{elif}\;y \le 5.487655812586681992965441010366711704904 \cdot 10^{-8}:\\
\;\;\;\;x \cdot e^{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot e^{\left(\log z \cdot y + a \cdot \log 1\right) - t \cdot y}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r173684 = x;
        double r173685 = y;
        double r173686 = z;
        double r173687 = log(r173686);
        double r173688 = t;
        double r173689 = r173687 - r173688;
        double r173690 = r173685 * r173689;
        double r173691 = a;
        double r173692 = 1.0;
        double r173693 = r173692 - r173686;
        double r173694 = log(r173693);
        double r173695 = b;
        double r173696 = r173694 - r173695;
        double r173697 = r173691 * r173696;
        double r173698 = r173690 + r173697;
        double r173699 = exp(r173698);
        double r173700 = r173684 * r173699;
        return r173700;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r173701 = y;
        double r173702 = -5.1996384838491834e-152;
        bool r173703 = r173701 <= r173702;
        double r173704 = x;
        double r173705 = z;
        double r173706 = log(r173705);
        double r173707 = t;
        double r173708 = r173706 - r173707;
        double r173709 = r173701 * r173708;
        double r173710 = a;
        double r173711 = 1.0;
        double r173712 = log(r173711);
        double r173713 = 0.5;
        double r173714 = 2.0;
        double r173715 = pow(r173705, r173714);
        double r173716 = pow(r173711, r173714);
        double r173717 = r173715 / r173716;
        double r173718 = r173713 * r173717;
        double r173719 = r173711 * r173705;
        double r173720 = r173718 + r173719;
        double r173721 = r173712 - r173720;
        double r173722 = b;
        double r173723 = r173721 - r173722;
        double r173724 = r173710 * r173723;
        double r173725 = r173709 + r173724;
        double r173726 = exp(r173725);
        double r173727 = cbrt(r173726);
        double r173728 = r173727 * r173727;
        double r173729 = r173706 * r173706;
        double r173730 = r173707 * r173707;
        double r173731 = r173729 - r173730;
        double r173732 = r173701 * r173731;
        double r173733 = r173721 + r173722;
        double r173734 = r173732 * r173733;
        double r173735 = r173706 + r173707;
        double r173736 = r173721 * r173721;
        double r173737 = r173722 * r173722;
        double r173738 = r173736 - r173737;
        double r173739 = r173710 * r173738;
        double r173740 = r173735 * r173739;
        double r173741 = r173734 + r173740;
        double r173742 = r173735 * r173733;
        double r173743 = r173741 / r173742;
        double r173744 = exp(r173743);
        double r173745 = cbrt(r173744);
        double r173746 = r173728 * r173745;
        double r173747 = r173704 * r173746;
        double r173748 = 5.487655812586682e-08;
        bool r173749 = r173701 <= r173748;
        double r173750 = r173710 * r173722;
        double r173751 = r173710 * r173705;
        double r173752 = r173711 * r173751;
        double r173753 = 0.5;
        double r173754 = r173710 * r173715;
        double r173755 = r173753 * r173754;
        double r173756 = r173752 + r173755;
        double r173757 = r173750 + r173756;
        double r173758 = -r173757;
        double r173759 = exp(r173758);
        double r173760 = r173704 * r173759;
        double r173761 = r173706 * r173701;
        double r173762 = r173710 * r173712;
        double r173763 = r173761 + r173762;
        double r173764 = r173707 * r173701;
        double r173765 = r173763 - r173764;
        double r173766 = exp(r173765);
        double r173767 = r173704 * r173766;
        double r173768 = r173749 ? r173760 : r173767;
        double r173769 = r173703 ? r173747 : r173768;
        return r173769;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if y < -5.1996384838491834e-152

    1. Initial program 2.8

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

    2. Taylor expanded around 0 1.1

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.2

      \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)}\]
    5. Using strategy rm

    6. Applied flip--4.7

      \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \color{blue}{\frac{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b}}}}\right)\]

    7. Applied associate-*r/5.0

      \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + \color{blue}{\frac{a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b}}}}\right)\]

    8. Applied flip--6.0

      \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \color{blue}{\frac{\log z \cdot \log z - t \cdot t}{\log z + t}} + \frac{a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b}}}\right)\]

    9. Applied associate-*r/6.4

      \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\color{blue}{\frac{y \cdot \left(\log z \cdot \log z - t \cdot t\right)}{\log z + t}} + \frac{a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b}}}\right)\]

    10. Applied frac-add17.5

      \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\color{blue}{\frac{\left(y \cdot \left(\log z \cdot \log z - t \cdot t\right)\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right) + \left(\log z + t\right) \cdot \left(a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)\right)}{\left(\log z + t\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right)}}}}\right)\]

    if -5.1996384838491834e-152 < y < 5.487655812586682e-08

    1. Initial program 2.4

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

    2. Taylor expanded around 0 0.1

      \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]

    3. Taylor expanded around inf 5.8

      \[\leadsto x \cdot e^{\color{blue}{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}}\]

    if 5.487655812586682e-08 < y

    1. Initial program 1.0

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

    2. Taylor expanded around 0 2.4

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.199638483849183435540034190073792906329 \cdot 10^{-152}:\\ \;\;\;\;x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\frac{\left(y \cdot \left(\log z \cdot \log z - t \cdot t\right)\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right) + \left(\log z + t\right) \cdot \left(a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) \cdot \left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b \cdot b\right)\right)}{\left(\log z + t\right) \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) + b\right)}}}\right)\\ \mathbf{elif}\;y \le 5.487655812586681992965441010366711704904 \cdot 10^{-8}:\\ \;\;\;\;x \cdot e^{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\log z \cdot y + a \cdot \log 1\right) - t \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))