Average Error: 2.1 → 0.5
Time: 17.5s
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 r115474 = x;
        double r115475 = y;
        double r115476 = z;
        double r115477 = log(r115476);
        double r115478 = t;
        double r115479 = r115477 - r115478;
        double r115480 = r115475 * r115479;
        double r115481 = a;
        double r115482 = 1.0;
        double r115483 = r115482 - r115476;
        double r115484 = log(r115483);
        double r115485 = b;
        double r115486 = r115484 - r115485;
        double r115487 = r115481 * r115486;
        double r115488 = r115480 + r115487;
        double r115489 = exp(r115488);
        double r115490 = r115474 * r115489;
        return r115490;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r115491 = x;
        double r115492 = 2.0;
        double r115493 = exp(r115492);
        double r115494 = y;
        double r115495 = z;
        double r115496 = log(r115495);
        double r115497 = t;
        double r115498 = r115496 - r115497;
        double r115499 = r115494 * r115498;
        double r115500 = a;
        double r115501 = 1.0;
        double r115502 = log(r115501);
        double r115503 = 0.5;
        double r115504 = pow(r115495, r115492);
        double r115505 = pow(r115501, r115492);
        double r115506 = r115504 / r115505;
        double r115507 = r115503 * r115506;
        double r115508 = r115501 * r115495;
        double r115509 = r115507 + r115508;
        double r115510 = r115502 - r115509;
        double r115511 = b;
        double r115512 = r115510 - r115511;
        double r115513 = r115500 * r115512;
        double r115514 = r115499 + r115513;
        double r115515 = pow(r115493, r115514);
        double r115516 = pow(r115515, r115503);
        double r115517 = r115491 * r115516;
        return r115517;
}

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))))))