Average Error: 7.1 → 0.4
Time: 21.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x - 1\right) \cdot \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left({\left(\frac{1}{y}\right)}^{\left(\frac{-1}{3}\right)}\right)\right) + \left(\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(x - 1\right) \cdot \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left({\left(\frac{1}{y}\right)}^{\left(\frac{-1}{3}\right)}\right)\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 r52077 = x;
        double r52078 = 1.0;
        double r52079 = r52077 - r52078;
        double r52080 = y;
        double r52081 = log(r52080);
        double r52082 = r52079 * r52081;
        double r52083 = z;
        double r52084 = r52083 - r52078;
        double r52085 = r52078 - r52080;
        double r52086 = log(r52085);
        double r52087 = r52084 * r52086;
        double r52088 = r52082 + r52087;
        double r52089 = t;
        double r52090 = r52088 - r52089;
        return r52090;
}
↓
double f(double x, double y, double z, double t) {
        double r52091 = x;
        double r52092 = 1.0;
        double r52093 = r52091 - r52092;
        double r52094 = y;
        double r52095 = cbrt(r52094);
        double r52096 = r52095 * r52095;
        double r52097 = log(r52096);
        double r52098 = 1.0;
        double r52099 = r52098 / r52094;
        double r52100 = -1.0;
        double r52101 = 3.0;
        double r52102 = r52100 / r52101;
        double r52103 = pow(r52099, r52102);
        double r52104 = log(r52103);
        double r52105 = r52097 + r52104;
        double r52106 = r52093 * r52105;
        double r52107 = z;
        double r52108 = r52107 - r52092;
        double r52109 = log(r52092);
        double r52110 = r52092 * r52094;
        double r52111 = 2.0;
        double r52112 = r52098 / r52111;
        double r52113 = pow(r52094, r52111);
        double r52114 = pow(r52092, r52111);
        double r52115 = r52113 / r52114;
        double r52116 = r52112 * r52115;
        double r52117 = r52110 + r52116;
        double r52118 = r52109 - r52117;
        double r52119 = r52108 * r52118;
        double r52120 = t;
        double r52121 = r52119 - r52120;
        double r52122 = r52106 + r52121;
        return r52122;
}

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 7.1

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(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))