Average Error: 6.8 → 0.4
Time: 31.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left({y}^{\frac{1}{6}}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right)\right)\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(\log 1 - \left(y \cdot 1 + \frac{y}{1} \cdot \left(\frac{1}{2} \cdot \frac{y}{1}\right)\right)\right) \cdot \left(z - 1\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left({y}^{\frac{1}{6}}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right)\right)\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(\log 1 - \left(y \cdot 1 + \frac{y}{1} \cdot \left(\frac{1}{2} \cdot \frac{y}{1}\right)\right)\right) \cdot \left(z - 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r3421145 = x;
        double r3421146 = 1.0;
        double r3421147 = r3421145 - r3421146;
        double r3421148 = y;
        double r3421149 = log(r3421148);
        double r3421150 = r3421147 * r3421149;
        double r3421151 = z;
        double r3421152 = r3421151 - r3421146;
        double r3421153 = r3421146 - r3421148;
        double r3421154 = log(r3421153);
        double r3421155 = r3421152 * r3421154;
        double r3421156 = r3421150 + r3421155;
        double r3421157 = t;
        double r3421158 = r3421156 - r3421157;
        return r3421158;
}
↓
double f(double x, double y, double z, double t) {
        double r3421159 = y;
        double r3421160 = 0.16666666666666666;
        double r3421161 = pow(r3421159, r3421160);
        double r3421162 = log(r3421161);
        double r3421163 = x;
        double r3421164 = 1.0;
        double r3421165 = r3421163 - r3421164;
        double r3421166 = r3421162 * r3421165;
        double r3421167 = sqrt(r3421159);
        double r3421168 = cbrt(r3421167);
        double r3421169 = log(r3421168);
        double r3421170 = r3421169 * r3421165;
        double r3421171 = r3421170 + r3421170;
        double r3421172 = r3421166 + r3421171;
        double r3421173 = log(r3421167);
        double r3421174 = r3421165 * r3421173;
        double r3421175 = log(r3421164);
        double r3421176 = r3421159 * r3421164;
        double r3421177 = r3421159 / r3421164;
        double r3421178 = 0.5;
        double r3421179 = r3421178 * r3421177;
        double r3421180 = r3421177 * r3421179;
        double r3421181 = r3421176 + r3421180;
        double r3421182 = r3421175 - r3421181;
        double r3421183 = z;
        double r3421184 = r3421183 - r3421164;
        double r3421185 = r3421182 * r3421184;
        double r3421186 = r3421174 + r3421185;
        double r3421187 = r3421172 + r3421186;
        double r3421188 = t;
        double r3421189 = r3421187 - r3421188;
        return r3421189;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 6.8

    \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  10. Applied add-cube-cbrt0.4

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

  11. Applied log-prod0.4

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

  12. Applied distribute-lft-in0.4

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

  13. Simplified0.4

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

  14. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt{y}}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + \left(x - 1\right) \cdot \log \color{blue}{\left({y}^{\frac{1}{6}}\right)}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(\left(\frac{1}{2} \cdot \frac{y}{1}\right) \cdot \frac{y}{1} + 1 \cdot y\right)\right)\right)\right) - t\]

  15. Final simplification0.4

    \[\leadsto \left(\left(\log \left({y}^{\frac{1}{6}}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{\sqrt{y}}\right) \cdot \left(x - 1\right)\right)\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(\log 1 - \left(y \cdot 1 + \frac{y}{1} \cdot \left(\frac{1}{2} \cdot \frac{y}{1}\right)\right)\right) \cdot \left(z - 1\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))