Average Error: 7.3 → 0.9
Time: 11.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\sqrt[3]{\log y \cdot \left(x - 1\right)} \cdot \sqrt[3]{\log y \cdot \left(x - 1\right)}\right) \cdot \sqrt[3]{\log y \cdot \left(x - 1\right)} + \left(\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\sqrt[3]{\log y \cdot \left(x - 1\right)} \cdot \sqrt[3]{\log y \cdot \left(x - 1\right)}\right) \cdot \sqrt[3]{\log y \cdot \left(x - 1\right)} + \left(\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r53186 = x;
        double r53187 = 1.0;
        double r53188 = r53186 - r53187;
        double r53189 = y;
        double r53190 = log(r53189);
        double r53191 = r53188 * r53190;
        double r53192 = z;
        double r53193 = r53192 - r53187;
        double r53194 = r53187 - r53189;
        double r53195 = log(r53194);
        double r53196 = r53193 * r53195;
        double r53197 = r53191 + r53196;
        double r53198 = t;
        double r53199 = r53197 - r53198;
        return r53199;
}
↓
double f(double x, double y, double z, double t) {
        double r53200 = y;
        double r53201 = log(r53200);
        double r53202 = x;
        double r53203 = 1.0;
        double r53204 = r53202 - r53203;
        double r53205 = r53201 * r53204;
        double r53206 = cbrt(r53205);
        double r53207 = r53206 * r53206;
        double r53208 = r53207 * r53206;
        double r53209 = z;
        double r53210 = r53209 - r53203;
        double r53211 = log(r53203);
        double r53212 = r53203 * r53200;
        double r53213 = 0.5;
        double r53214 = 2.0;
        double r53215 = pow(r53200, r53214);
        double r53216 = pow(r53203, r53214);
        double r53217 = r53215 / r53216;
        double r53218 = r53213 * r53217;
        double r53219 = r53212 + r53218;
        double r53220 = r53211 - r53219;
        double r53221 = r53210 * r53220;
        double r53222 = t;
        double r53223 = r53221 - r53222;
        double r53224 = r53208 + r53223;
        return r53224;
}

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 7.3

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

  2. Simplified7.3

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

  3. Taylor expanded around 0 0.3

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

  5. Applied fma-udef0.3

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

  7. Applied add-cube-cbrt0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))