Average Error: 2.0 → 1.1
Time: 41.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)\right)}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{{e}^{\left(\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)\right)}}}} \cdot \frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(\log a, t - 1, \log z \cdot y - b\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r3257471 = x;
        double r3257472 = y;
        double r3257473 = z;
        double r3257474 = log(r3257473);
        double r3257475 = r3257472 * r3257474;
        double r3257476 = t;
        double r3257477 = 1.0;
        double r3257478 = r3257476 - r3257477;
        double r3257479 = a;
        double r3257480 = log(r3257479);
        double r3257481 = r3257478 * r3257480;
        double r3257482 = r3257475 + r3257481;
        double r3257483 = b;
        double r3257484 = r3257482 - r3257483;
        double r3257485 = exp(r3257484);
        double r3257486 = r3257471 * r3257485;
        double r3257487 = r3257486 / r3257472;
        return r3257487;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r3257488 = x;
        double r3257489 = y;
        double r3257490 = cbrt(r3257489);
        double r3257491 = r3257490 * r3257490;
        double r3257492 = cbrt(r3257491);
        double r3257493 = cbrt(r3257490);
        double r3257494 = r3257492 * r3257493;
        double r3257495 = exp(1.0);
        double r3257496 = a;
        double r3257497 = log(r3257496);
        double r3257498 = t;
        double r3257499 = 1.0;
        double r3257500 = r3257498 - r3257499;
        double r3257501 = z;
        double r3257502 = log(r3257501);
        double r3257503 = r3257502 * r3257489;
        double r3257504 = b;
        double r3257505 = r3257503 - r3257504;
        double r3257506 = fma(r3257497, r3257500, r3257505);
        double r3257507 = pow(r3257495, r3257506);
        double r3257508 = cbrt(r3257507);
        double r3257509 = r3257494 / r3257508;
        double r3257510 = r3257488 / r3257509;
        double r3257511 = 1.0;
        double r3257512 = exp(r3257506);
        double r3257513 = cbrt(r3257512);
        double r3257514 = r3257513 * r3257513;
        double r3257515 = r3257491 / r3257514;
        double r3257516 = r3257511 / r3257515;
        double r3257517 = r3257510 * r3257516;
        return r3257517;
}

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 2.0

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

  3. Applied associate-/l*1.9

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

  4. Simplified1.9

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

  6. Applied add-cube-cbrt1.9

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied times-frac1.9

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

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

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

  10. Applied times-frac1.1

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

  12. Applied *-un-lft-identity1.1

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

  13. Applied exp-prod1.1

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

  14. Simplified1.1

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

  16. Applied add-cube-cbrt1.1

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

  17. Applied cbrt-prod1.1

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

  18. Final simplification1.1

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

Reproduce

herbie shell --seed 2019171 +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))