Average Error: 7.2 → 0.5
Time: 55.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(\log 1 - 1 \cdot y\right) - \frac{y}{1} \cdot \left(\frac{1}{2} \cdot \frac{y}{1}\right)\right) \cdot \left(z - 1\right) + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(\log 1 - 1 \cdot y\right) - \frac{y}{1} \cdot \left(\frac{1}{2} \cdot \frac{y}{1}\right)\right) \cdot \left(z - 1\right) + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r3933394 = x;
        double r3933395 = 1.0;
        double r3933396 = r3933394 - r3933395;
        double r3933397 = y;
        double r3933398 = log(r3933397);
        double r3933399 = r3933396 * r3933398;
        double r3933400 = z;
        double r3933401 = r3933400 - r3933395;
        double r3933402 = r3933395 - r3933397;
        double r3933403 = log(r3933402);
        double r3933404 = r3933401 * r3933403;
        double r3933405 = r3933399 + r3933404;
        double r3933406 = t;
        double r3933407 = r3933405 - r3933406;
        return r3933407;
}
↓
double f(double x, double y, double z, double t) {
        double r3933408 = 1.0;
        double r3933409 = log(r3933408);
        double r3933410 = y;
        double r3933411 = r3933408 * r3933410;
        double r3933412 = r3933409 - r3933411;
        double r3933413 = r3933410 / r3933408;
        double r3933414 = 0.5;
        double r3933415 = r3933414 * r3933413;
        double r3933416 = r3933413 * r3933415;
        double r3933417 = r3933412 - r3933416;
        double r3933418 = z;
        double r3933419 = r3933418 - r3933408;
        double r3933420 = r3933417 * r3933419;
        double r3933421 = cbrt(r3933410);
        double r3933422 = log(r3933421);
        double r3933423 = x;
        double r3933424 = r3933423 - r3933408;
        double r3933425 = r3933422 * r3933424;
        double r3933426 = r3933425 + r3933425;
        double r3933427 = r3933426 + r3933425;
        double r3933428 = r3933420 + r3933427;
        double r3933429 = t;
        double r3933430 = r3933428 - r3933429;
        return r3933430;
}

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 7.2

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

  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

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.5

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

  7. Applied distribute-lft-in0.5

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 
(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))