Average Error: 1.8 → 0.3
Time: 20.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left({e}^{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-b, 1, b\right), a, \mathsf{fma}\left(y, \log z - t, a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z} + \sqrt{\log 1}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -1 \cdot b\right)\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-b, 1, b\right), a, \mathsf{fma}\left(y, \log z - t, a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z} + \sqrt{\log 1}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -1 \cdot b\right)\right)\right)}{2}\right)}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left({e}^{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-b, 1, b\right), a, \mathsf{fma}\left(y, \log z - t, a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z} + \sqrt{\log 1}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -1 \cdot b\right)\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-b, 1, b\right), a, \mathsf{fma}\left(y, \log z - t, a \cdot \mathsf{fma}\left(\sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z} + \sqrt{\log 1}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -1 \cdot b\right)\right)\right)}{2}\right)}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r158321 = x;
        double r158322 = y;
        double r158323 = z;
        double r158324 = log(r158323);
        double r158325 = t;
        double r158326 = r158324 - r158325;
        double r158327 = r158322 * r158326;
        double r158328 = a;
        double r158329 = 1.0;
        double r158330 = r158329 - r158323;
        double r158331 = log(r158330);
        double r158332 = b;
        double r158333 = r158331 - r158332;
        double r158334 = r158328 * r158333;
        double r158335 = r158327 + r158334;
        double r158336 = exp(r158335);
        double r158337 = r158321 * r158336;
        return r158337;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r158338 = x;
        double r158339 = exp(1.0);
        double r158340 = b;
        double r158341 = -r158340;
        double r158342 = 1.0;
        double r158343 = fma(r158341, r158342, r158340);
        double r158344 = a;
        double r158345 = y;
        double r158346 = z;
        double r158347 = log(r158346);
        double r158348 = t;
        double r158349 = r158347 - r158348;
        double r158350 = 0.5;
        double r158351 = 2.0;
        double r158352 = pow(r158346, r158351);
        double r158353 = 1.0;
        double r158354 = pow(r158353, r158351);
        double r158355 = r158352 / r158354;
        double r158356 = r158350 * r158355;
        double r158357 = r158353 * r158346;
        double r158358 = r158356 + r158357;
        double r158359 = sqrt(r158358);
        double r158360 = log(r158353);
        double r158361 = sqrt(r158360);
        double r158362 = r158359 + r158361;
        double r158363 = r158361 - r158359;
        double r158364 = -1.0;
        double r158365 = r158364 * r158340;
        double r158366 = fma(r158362, r158363, r158365);
        double r158367 = r158344 * r158366;
        double r158368 = fma(r158345, r158349, r158367);
        double r158369 = fma(r158343, r158344, r158368);
        double r158370 = r158369 / r158351;
        double r158371 = pow(r158339, r158370);
        double r158372 = r158371 * r158371;
        double r158373 = r158338 * r158372;
        return r158373;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 1.8

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.5

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

  4. Applied *-un-lft-identity0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied difference-of-squares0.5

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

  8. Applied prod-diff0.5

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

  9. Applied distribute-lft-in0.5

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

  10. Applied associate-+r+0.5

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

  11. Simplified0.3

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

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

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

  14. Applied exp-prod0.3

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

  15. Simplified0.3

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

  17. Applied sqr-pow0.3

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

  18. Simplified0.3

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))