Average Error: 1.9 → 0.3
Time: 14.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot {\left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right)}^{\left(y \cdot \log 1 + \mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}\right) \cdot {\left(\sqrt[3]{e}\right)}^{\left(y \cdot \log 1 + \mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot {\left(\sqrt[3]{e} \cdot \sqrt[3]{e}\right)}^{\left(y \cdot \log 1 + \mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}\right) \cdot {\left(\sqrt[3]{e}\right)}^{\left(y \cdot \log 1 + \mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r113287 = x;
        double r113288 = y;
        double r113289 = z;
        double r113290 = log(r113289);
        double r113291 = t;
        double r113292 = r113290 - r113291;
        double r113293 = r113288 * r113292;
        double r113294 = a;
        double r113295 = 1.0;
        double r113296 = r113295 - r113289;
        double r113297 = log(r113296);
        double r113298 = b;
        double r113299 = r113297 - r113298;
        double r113300 = r113294 * r113299;
        double r113301 = r113293 + r113300;
        double r113302 = exp(r113301);
        double r113303 = r113287 * r113302;
        return r113303;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r113304 = x;
        double r113305 = exp(1.0);
        double r113306 = cbrt(r113305);
        double r113307 = r113306 * r113306;
        double r113308 = y;
        double r113309 = 1.0;
        double r113310 = log(r113309);
        double r113311 = r113308 * r113310;
        double r113312 = z;
        double r113313 = log(r113312);
        double r113314 = t;
        double r113315 = r113313 - r113314;
        double r113316 = 1.0;
        double r113317 = log(r113316);
        double r113318 = 0.5;
        double r113319 = 2.0;
        double r113320 = pow(r113312, r113319);
        double r113321 = pow(r113316, r113319);
        double r113322 = r113320 / r113321;
        double r113323 = r113316 * r113312;
        double r113324 = fma(r113318, r113322, r113323);
        double r113325 = b;
        double r113326 = r113324 + r113325;
        double r113327 = r113317 - r113326;
        double r113328 = a;
        double r113329 = r113327 * r113328;
        double r113330 = fma(r113308, r113315, r113329);
        double r113331 = r113311 + r113330;
        double r113332 = pow(r113307, r113331);
        double r113333 = r113304 * r113332;
        double r113334 = pow(r113306, r113331);
        double r113335 = r113333 * r113334;
        return r113335;
}

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 1.9

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

  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

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

  5. Applied log-prod0.5

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

  6. Applied associate--l+0.5

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

  7. Applied distribute-lft-in0.5

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

  8. Applied associate-+l+0.5

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

  9. Simplified0.3

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

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

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

  12. Applied exp-prod0.3

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

  13. Simplified0.3

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

  15. Applied add-cube-cbrt0.3

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

  16. Applied unpow-prod-down0.3

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

  17. Applied associate-*r*0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020033 +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))))))