Average Error: 1.9 → 3.1
Time: 13.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.39009728408410565 \cdot 10^{94}:\\ \;\;\;\;x \cdot \log \left(e^{e^{y \cdot \left(\log z - t\right) + \log 1 \cdot a}}\right)\\ \mathbf{elif}\;y \le -1.24035582158853264:\\ \;\;\;\;x \cdot e^{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}\\ \mathbf{elif}\;y \le 46889490030544568:\\ \;\;\;\;x \cdot \sqrt[3]{{\left({z}^{y}\right)}^{3} \cdot {\left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(t \cdot y + \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(e^{e^{y \cdot \left(\log z - t\right) + \log 1 \cdot a}}\right)\\ \end{array}\]
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 -1.39009728408410565 \cdot 10^{94}:\\
\;\;\;\;x \cdot \log \left(e^{e^{y \cdot \left(\log z - t\right) + \log 1 \cdot a}}\right)\\

\mathbf{elif}\;y \le -1.24035582158853264:\\
\;\;\;\;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{elif}\;y \le 46889490030544568:\\
\;\;\;\;x \cdot \sqrt[3]{{\left({z}^{y}\right)}^{3} \cdot {\left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(t \cdot y + \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\right)}^{3}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r109119 = x;
        double r109120 = y;
        double r109121 = z;
        double r109122 = log(r109121);
        double r109123 = t;
        double r109124 = r109122 - r109123;
        double r109125 = r109120 * r109124;
        double r109126 = a;
        double r109127 = 1.0;
        double r109128 = r109127 - r109121;
        double r109129 = log(r109128);
        double r109130 = b;
        double r109131 = r109129 - r109130;
        double r109132 = r109126 * r109131;
        double r109133 = r109125 + r109132;
        double r109134 = exp(r109133);
        double r109135 = r109119 * r109134;
        return r109135;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r109136 = y;
        double r109137 = -1.3900972840841056e+94;
        bool r109138 = r109136 <= r109137;
        double r109139 = x;
        double r109140 = z;
        double r109141 = log(r109140);
        double r109142 = t;
        double r109143 = r109141 - r109142;
        double r109144 = r109136 * r109143;
        double r109145 = 1.0;
        double r109146 = log(r109145);
        double r109147 = a;
        double r109148 = r109146 * r109147;
        double r109149 = r109144 + r109148;
        double r109150 = exp(r109149);
        double r109151 = exp(r109150);
        double r109152 = log(r109151);
        double r109153 = r109139 * r109152;
        double r109154 = -1.2403558215885326;
        bool r109155 = r109136 <= r109154;
        double r109156 = b;
        double r109157 = r109147 * r109156;
        double r109158 = r109147 * r109140;
        double r109159 = r109145 * r109158;
        double r109160 = 0.5;
        double r109161 = 2.0;
        double r109162 = pow(r109140, r109161);
        double r109163 = r109147 * r109162;
        double r109164 = r109160 * r109163;
        double r109165 = r109159 + r109164;
        double r109166 = r109157 + r109165;
        double r109167 = -r109166;
        double r109168 = exp(r109167);
        double r109169 = r109139 * r109168;
        double r109170 = 4.688949003054457e+16;
        bool r109171 = r109136 <= r109170;
        double r109172 = pow(r109140, r109136);
        double r109173 = 3.0;
        double r109174 = pow(r109172, r109173);
        double r109175 = pow(r109145, r109147);
        double r109176 = 0.5;
        double r109177 = pow(r109145, r109161);
        double r109178 = r109163 / r109177;
        double r109179 = r109176 * r109178;
        double r109180 = r109142 * r109136;
        double r109181 = r109157 + r109159;
        double r109182 = r109180 + r109181;
        double r109183 = r109179 + r109182;
        double r109184 = exp(r109183);
        double r109185 = r109175 / r109184;
        double r109186 = pow(r109185, r109173);
        double r109187 = r109174 * r109186;
        double r109188 = cbrt(r109187);
        double r109189 = r109139 * r109188;
        double r109190 = r109171 ? r109189 : r109153;
        double r109191 = r109155 ? r109169 : r109190;
        double r109192 = r109138 ? r109153 : r109191;
        return r109192;
}

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 < -1.3900972840841056e+94 or 4.688949003054457e+16 < y

    1. Initial program 1.3

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

    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-log-exp1.1

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

    5. Taylor expanded around 0 5.2

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

    6. Simplified4.9

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

    if -1.3900972840841056e+94 < y < -1.2403558215885326

    1. Initial program 1.7

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

    2. Taylor expanded around 0 0.2

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

    3. Taylor expanded around inf 23.7

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

    if -1.2403558215885326 < y < 4.688949003054457e+16

    1. Initial program 2.3

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

    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. Using strategy rm

    4. Applied add-cbrt-cube0.2

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

    5. Simplified0.2

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

    6. Taylor expanded around inf 0.2

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

    7. Simplified0.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.39009728408410565 \cdot 10^{94}:\\ \;\;\;\;x \cdot \log \left(e^{e^{y \cdot \left(\log z - t\right) + \log 1 \cdot a}}\right)\\ \mathbf{elif}\;y \le -1.24035582158853264:\\ \;\;\;\;x \cdot e^{-\left(a \cdot b + \left(1 \cdot \left(a \cdot z\right) + 0.5 \cdot \left(a \cdot {z}^{2}\right)\right)\right)}\\ \mathbf{elif}\;y \le 46889490030544568:\\ \;\;\;\;x \cdot \sqrt[3]{{\left({z}^{y}\right)}^{3} \cdot {\left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(t \cdot y + \left(a \cdot b + 1 \cdot \left(a \cdot z\right)\right)\right)}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(e^{e^{y \cdot \left(\log z - t\right) + \log 1 \cdot a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))))