Average Error: 2.1 → 0.5
Time: 13.6s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot {\left({\left(e^{2}\right)}^{\left(y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}\right)}^{\frac{1}{2}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot {\left({\left(e^{2}\right)}^{\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)}\right)}^{\frac{1}{2}}
double f(double x, double y, double z, double t, double a, double b) {
        double r133710 = x;
        double r133711 = y;
        double r133712 = z;
        double r133713 = log(r133712);
        double r133714 = t;
        double r133715 = r133713 - r133714;
        double r133716 = r133711 * r133715;
        double r133717 = a;
        double r133718 = 1.0;
        double r133719 = r133718 - r133712;
        double r133720 = log(r133719);
        double r133721 = b;
        double r133722 = r133720 - r133721;
        double r133723 = r133717 * r133722;
        double r133724 = r133716 + r133723;
        double r133725 = exp(r133724);
        double r133726 = r133710 * r133725;
        return r133726;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r133727 = x;
        double r133728 = 2.0;
        double r133729 = exp(r133728);
        double r133730 = y;
        double r133731 = z;
        double r133732 = log(r133731);
        double r133733 = t;
        double r133734 = r133732 - r133733;
        double r133735 = r133730 * r133734;
        double r133736 = a;
        double r133737 = 1.0;
        double r133738 = log(r133737);
        double r133739 = 0.5;
        double r133740 = pow(r133731, r133728);
        double r133741 = pow(r133737, r133728);
        double r133742 = r133740 / r133741;
        double r133743 = r133739 * r133742;
        double r133744 = r133737 * r133731;
        double r133745 = r133743 + r133744;
        double r133746 = r133738 - r133745;
        double r133747 = b;
        double r133748 = r133746 - r133747;
        double r133749 = r133736 * r133748;
        double r133750 = r133735 + r133749;
        double r133751 = pow(r133729, r133750);
        double r133752 = pow(r133751, r133739);
        double r133753 = r133727 * r133752;
        return r133753;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.1

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

  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 add-sqr-sqrt0.5

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

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

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

  7. Applied exp-prod0.5

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

  8. Applied sqrt-pow10.5

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

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

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

  10. Applied exp-prod0.5

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

  11. Applied sqrt-pow10.5

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

  12. Applied pow-prod-down0.5

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

  13. Simplified0.5

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

  15. Applied div-inv0.5

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

  16. Applied pow-unpow0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 
(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))))))