Average Error: 1.8 → 0.8
Time: 32.9s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[{\left(\sqrt{\frac{1}{a}}\right)}^{1} \cdot \frac{x}{\frac{y}{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
{\left(\sqrt{\frac{1}{a}}\right)}^{1} \cdot \frac{x}{\frac{y}{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r154510 = x;
        double r154511 = y;
        double r154512 = z;
        double r154513 = log(r154512);
        double r154514 = r154511 * r154513;
        double r154515 = t;
        double r154516 = 1.0;
        double r154517 = r154515 - r154516;
        double r154518 = a;
        double r154519 = log(r154518);
        double r154520 = r154517 * r154519;
        double r154521 = r154514 + r154520;
        double r154522 = b;
        double r154523 = r154521 - r154522;
        double r154524 = exp(r154523);
        double r154525 = r154510 * r154524;
        double r154526 = r154525 / r154511;
        return r154526;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r154527 = 1.0;
        double r154528 = a;
        double r154529 = r154527 / r154528;
        double r154530 = sqrt(r154529);
        double r154531 = 1.0;
        double r154532 = pow(r154530, r154531);
        double r154533 = x;
        double r154534 = y;
        double r154535 = z;
        double r154536 = r154527 / r154535;
        double r154537 = log(r154536);
        double r154538 = log(r154529);
        double r154539 = t;
        double r154540 = b;
        double r154541 = fma(r154538, r154539, r154540);
        double r154542 = fma(r154534, r154537, r154541);
        double r154543 = exp(r154542);
        double r154544 = r154532 / r154543;
        double r154545 = r154534 / r154544;
        double r154546 = r154533 / r154545;
        double r154547 = r154532 * r154546;
        return r154547;
}

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

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

  2. Taylor expanded around inf 1.9

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

  3. Simplified1.2

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

  5. Applied associate-/l*1.2

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

  7. Applied *-un-lft-identity1.2

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

  8. Applied add-sqr-sqrt1.2

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

  9. Applied unpow-prod-down1.2

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

  10. Applied times-frac1.2

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

  11. Applied *-un-lft-identity1.2

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

  12. Applied times-frac1.2

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

  13. Applied *-un-lft-identity1.2

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

  14. Applied times-frac0.8

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

  15. Simplified0.8

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

  16. Final simplification0.8

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))