Average Error: 2.2 → 0.5
Time: 12.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot {e}^{\left(y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot {e}^{\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)}
double f(double x, double y, double z, double t, double a, double b) {
        double r129381 = x;
        double r129382 = y;
        double r129383 = z;
        double r129384 = log(r129383);
        double r129385 = t;
        double r129386 = r129384 - r129385;
        double r129387 = r129382 * r129386;
        double r129388 = a;
        double r129389 = 1.0;
        double r129390 = r129389 - r129383;
        double r129391 = log(r129390);
        double r129392 = b;
        double r129393 = r129391 - r129392;
        double r129394 = r129388 * r129393;
        double r129395 = r129387 + r129394;
        double r129396 = exp(r129395);
        double r129397 = r129381 * r129396;
        return r129397;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r129398 = x;
        double r129399 = exp(1.0);
        double r129400 = y;
        double r129401 = z;
        double r129402 = log(r129401);
        double r129403 = t;
        double r129404 = r129402 - r129403;
        double r129405 = r129400 * r129404;
        double r129406 = a;
        double r129407 = 1.0;
        double r129408 = log(r129407);
        double r129409 = 0.5;
        double r129410 = 2.0;
        double r129411 = pow(r129401, r129410);
        double r129412 = pow(r129407, r129410);
        double r129413 = r129411 / r129412;
        double r129414 = r129409 * r129413;
        double r129415 = r129407 * r129401;
        double r129416 = r129414 + r129415;
        double r129417 = r129408 - r129416;
        double r129418 = b;
        double r129419 = r129417 - r129418;
        double r129420 = r129406 * r129419;
        double r129421 = r129405 + r129420;
        double r129422 = pow(r129399, r129421);
        double r129423 = r129398 * r129422;
        return r129423;
}

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 2.2

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

  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 *-un-lft-identity0.5

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

  5. Applied exp-prod0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020049 
(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))))))