Average Error: 2.0 → 0.4
Time: 24.1s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot 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)}
double f(double x, double y, double z, double t, double a, double b) {
        double r121413 = x;
        double r121414 = y;
        double r121415 = z;
        double r121416 = log(r121415);
        double r121417 = t;
        double r121418 = r121416 - r121417;
        double r121419 = r121414 * r121418;
        double r121420 = a;
        double r121421 = 1.0;
        double r121422 = r121421 - r121415;
        double r121423 = log(r121422);
        double r121424 = b;
        double r121425 = r121423 - r121424;
        double r121426 = r121420 * r121425;
        double r121427 = r121419 + r121426;
        double r121428 = exp(r121427);
        double r121429 = r121413 * r121428;
        return r121429;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r121430 = x;
        double r121431 = y;
        double r121432 = z;
        double r121433 = log(r121432);
        double r121434 = t;
        double r121435 = r121433 - r121434;
        double r121436 = r121431 * r121435;
        double r121437 = a;
        double r121438 = 1.0;
        double r121439 = log(r121438);
        double r121440 = 0.5;
        double r121441 = 2.0;
        double r121442 = pow(r121432, r121441);
        double r121443 = pow(r121438, r121441);
        double r121444 = r121442 / r121443;
        double r121445 = r121440 * r121444;
        double r121446 = r121438 * r121432;
        double r121447 = r121445 + r121446;
        double r121448 = r121439 - r121447;
        double r121449 = b;
        double r121450 = r121448 - r121449;
        double r121451 = r121437 * r121450;
        double r121452 = r121436 + r121451;
        double r121453 = exp(r121452);
        double r121454 = r121430 * r121453;
        return r121454;
}

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 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 
(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))))))