Average Error: 6.7 → 0.3
Time: 11.2s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right) + \left(z - 1\right) \cdot \mathsf{fma}\left(-\sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1} \cdot \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right) + \left(z - 1\right) \cdot \mathsf{fma}\left(-\sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1} \cdot \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r70472 = x;
        double r70473 = 1.0;
        double r70474 = r70472 - r70473;
        double r70475 = y;
        double r70476 = log(r70475);
        double r70477 = r70474 * r70476;
        double r70478 = z;
        double r70479 = r70478 - r70473;
        double r70480 = r70473 - r70475;
        double r70481 = log(r70480);
        double r70482 = r70479 * r70481;
        double r70483 = r70477 + r70482;
        double r70484 = t;
        double r70485 = r70483 - r70484;
        return r70485;
}
↓
double f(double x, double y, double z, double t) {
        double r70486 = x;
        double r70487 = 1.0;
        double r70488 = r70486 - r70487;
        double r70489 = y;
        double r70490 = log(r70489);
        double r70491 = log(r70487);
        double r70492 = cbrt(r70491);
        double r70493 = r70492 * r70492;
        double r70494 = r70487 * r70489;
        double r70495 = 0.5;
        double r70496 = 2.0;
        double r70497 = pow(r70489, r70496);
        double r70498 = pow(r70487, r70496);
        double r70499 = r70497 / r70498;
        double r70500 = r70495 * r70499;
        double r70501 = r70494 + r70500;
        double r70502 = 1.0;
        double r70503 = r70501 * r70502;
        double r70504 = -r70503;
        double r70505 = fma(r70493, r70492, r70504);
        double r70506 = z;
        double r70507 = r70506 - r70487;
        double r70508 = r70505 * r70507;
        double r70509 = fma(r70488, r70490, r70508);
        double r70510 = sqrt(r70501);
        double r70511 = -r70510;
        double r70512 = sqrt(r70502);
        double r70513 = fma(r70487, r70489, r70500);
        double r70514 = r70512 * r70513;
        double r70515 = fma(r70511, r70510, r70514);
        double r70516 = r70507 * r70515;
        double r70517 = r70509 + r70516;
        double r70518 = t;
        double r70519 = r70517 - r70518;
        return r70519;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.7

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

  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. Using strategy rm

  4. Applied add-sqr-sqrt0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied prod-diff0.3

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

  7. Applied distribute-lft-in0.3

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

  8. Applied associate-+r+0.3

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

  9. Simplified0.3

    \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right)} + \left(z - 1\right) \cdot \mathsf{fma}\left(-\sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}} \cdot \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.3

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

  12. Applied sqrt-prod0.3

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

  13. Applied associate-*l*0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 +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))