Average Error: 2.0 → 0.5
Time: 32.9s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left({e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}{2}\right)} \cdot {e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}{2}\right)}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left({e}^{\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 {e}^{\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)
double f(double x, double y, double z, double t, double a, double b) {
        double r87324 = x;
        double r87325 = y;
        double r87326 = z;
        double r87327 = log(r87326);
        double r87328 = t;
        double r87329 = r87327 - r87328;
        double r87330 = r87325 * r87329;
        double r87331 = a;
        double r87332 = 1.0;
        double r87333 = r87332 - r87326;
        double r87334 = log(r87333);
        double r87335 = b;
        double r87336 = r87334 - r87335;
        double r87337 = r87331 * r87336;
        double r87338 = r87330 + r87337;
        double r87339 = exp(r87338);
        double r87340 = r87324 * r87339;
        return r87340;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r87341 = x;
        double r87342 = exp(1.0);
        double r87343 = y;
        double r87344 = z;
        double r87345 = log(r87344);
        double r87346 = t;
        double r87347 = r87345 - r87346;
        double r87348 = r87343 * r87347;
        double r87349 = a;
        double r87350 = 1.0;
        double r87351 = log(r87350);
        double r87352 = 0.5;
        double r87353 = 2.0;
        double r87354 = pow(r87344, r87353);
        double r87355 = pow(r87350, r87353);
        double r87356 = r87354 / r87355;
        double r87357 = r87352 * r87356;
        double r87358 = r87350 * r87344;
        double r87359 = r87357 + r87358;
        double r87360 = r87351 - r87359;
        double r87361 = b;
        double r87362 = r87360 - r87361;
        double r87363 = r87349 * r87362;
        double r87364 = r87348 + r87363;
        double r87365 = r87364 / r87353;
        double r87366 = pow(r87342, r87365);
        double r87367 = r87366 * r87366;
        double r87368 = r87341 * r87367;
        return r87368;
}

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 2.0

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

  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. Using strategy rm

  8. Applied sqr-pow0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019235 
(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))))))