Average Error: 1.9 → 1.0
Time: 55.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \left(x \cdot \frac{\sqrt{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}
↓
\frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \left(x \cdot \frac{\sqrt{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r2722374 = x;
        double r2722375 = y;
        double r2722376 = z;
        double r2722377 = log(r2722376);
        double r2722378 = r2722375 * r2722377;
        double r2722379 = t;
        double r2722380 = 1.0;
        double r2722381 = r2722379 - r2722380;
        double r2722382 = a;
        double r2722383 = log(r2722382);
        double r2722384 = r2722381 * r2722383;
        double r2722385 = r2722378 + r2722384;
        double r2722386 = b;
        double r2722387 = r2722385 - r2722386;
        double r2722388 = exp(r2722387);
        double r2722389 = r2722374 * r2722388;
        double r2722390 = r2722389 / r2722375;
        return r2722390;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2722391 = t;
        double r2722392 = 1.0;
        double r2722393 = r2722391 - r2722392;
        double r2722394 = a;
        double r2722395 = log(r2722394);
        double r2722396 = y;
        double r2722397 = z;
        double r2722398 = log(r2722397);
        double r2722399 = r2722396 * r2722398;
        double r2722400 = fma(r2722393, r2722395, r2722399);
        double r2722401 = b;
        double r2722402 = r2722400 - r2722401;
        double r2722403 = exp(r2722402);
        double r2722404 = sqrt(r2722403);
        double r2722405 = cbrt(r2722396);
        double r2722406 = r2722404 / r2722405;
        double r2722407 = x;
        double r2722408 = exp(1.0);
        double r2722409 = pow(r2722408, r2722402);
        double r2722410 = cbrt(r2722409);
        double r2722411 = r2722410 * r2722410;
        double r2722412 = cbrt(r2722403);
        double r2722413 = r2722411 * r2722412;
        double r2722414 = sqrt(r2722413);
        double r2722415 = r2722405 * r2722405;
        double r2722416 = r2722414 / r2722415;
        double r2722417 = r2722407 * r2722416;
        double r2722418 = r2722406 * r2722417;
        return r2722418;
}

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

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

  2. Simplified2.0

    \[\leadsto \color{blue}{x \cdot \frac{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}{y}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt2.0

    \[\leadsto x \cdot \frac{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

  5. Applied add-sqr-sqrt2.0

    \[\leadsto x \cdot \frac{\color{blue}{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}} \cdot \sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

  6. Applied times-frac2.0

    \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\right)}\]

  7. Applied associate-*r*1.0

    \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity1.0

    \[\leadsto \left(x \cdot \frac{\sqrt{e^{\color{blue}{1 \cdot \left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]

  10. Applied exp-prod1.0

    \[\leadsto \left(x \cdot \frac{\sqrt{\color{blue}{{\left(e^{1}\right)}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]

  11. Simplified1.0

    \[\leadsto \left(x \cdot \frac{\sqrt{{\color{blue}{e}}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt1.0

    \[\leadsto \left(x \cdot \frac{\sqrt{\color{blue}{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}\right) \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]
  14. Using strategy rm

  15. Applied e-exp-11.0

    \[\leadsto \left(x \cdot \frac{\sqrt{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}\right) \cdot \sqrt[3]{{\color{blue}{\left(e^{1}\right)}}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]

  16. Applied pow-exp1.0

    \[\leadsto \left(x \cdot \frac{\sqrt{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}\right) \cdot \sqrt[3]{\color{blue}{e^{1 \cdot \left(\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, \log z \cdot y\right) - b}}}{\sqrt[3]{y}}\]

  17. Final simplification1.0

    \[\leadsto \frac{\sqrt{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}{\sqrt[3]{y}} \cdot \left(x \cdot \frac{\sqrt{\left(\sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}} \cdot \sqrt[3]{{e}^{\left(\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(t - 1.0, \log a, y \cdot \log z\right) - b}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))