Average Error: 6.6 → 0.3
Time: 11.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \log y + \mathsf{fma}\left(\sqrt[3]{\log 1} \cdot \sqrt[3]{\log 1}, \sqrt[3]{\log 1}, -\left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right) \cdot 1\right) \cdot \left(z - 1\right)\right) + \left(z - 1\right) \cdot \mathsf{fma}\left(-\sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}, \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}} \cdot \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \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
double f(double x, double y, double z, double t) {
        double r72338 = x;
        double r72339 = 1.0;
        double r72340 = r72338 - r72339;
        double r72341 = y;
        double r72342 = log(r72341);
        double r72343 = r72340 * r72342;
        double r72344 = z;
        double r72345 = r72344 - r72339;
        double r72346 = r72339 - r72341;
        double r72347 = log(r72346);
        double r72348 = r72345 * r72347;
        double r72349 = r72343 + r72348;
        double r72350 = t;
        double r72351 = r72349 - r72350;
        return r72351;
}
↓
double f(double x, double y, double z, double t) {
        double r72352 = x;
        double r72353 = 1.0;
        double r72354 = r72352 - r72353;
        double r72355 = y;
        double r72356 = log(r72355);
        double r72357 = r72354 * r72356;
        double r72358 = log(r72353);
        double r72359 = cbrt(r72358);
        double r72360 = r72359 * r72359;
        double r72361 = r72353 * r72355;
        double r72362 = 0.5;
        double r72363 = 2.0;
        double r72364 = pow(r72355, r72363);
        double r72365 = pow(r72353, r72363);
        double r72366 = r72364 / r72365;
        double r72367 = r72362 * r72366;
        double r72368 = r72361 + r72367;
        double r72369 = 1.0;
        double r72370 = r72368 * r72369;
        double r72371 = -r72370;
        double r72372 = fma(r72360, r72359, r72371);
        double r72373 = z;
        double r72374 = r72373 - r72353;
        double r72375 = r72372 * r72374;
        double r72376 = r72357 + r72375;
        double r72377 = sqrt(r72368);
        double r72378 = -r72377;
        double r72379 = r72377 * r72377;
        double r72380 = fma(r72378, r72377, r72379);
        double r72381 = r72374 * r72380;
        double r72382 = r72376 + r72381;
        double r72383 = t;
        double r72384 = r72382 - r72383;
        return r72384;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.6

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

  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-sqrt0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied prod-diff0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+r+0.4

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

  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 fma-udef0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020039 +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))