Average Error: 1.9 → 0.5
Time: 10.8s
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 r145461 = x;
        double r145462 = y;
        double r145463 = z;
        double r145464 = log(r145463);
        double r145465 = t;
        double r145466 = r145464 - r145465;
        double r145467 = r145462 * r145466;
        double r145468 = a;
        double r145469 = 1.0;
        double r145470 = r145469 - r145463;
        double r145471 = log(r145470);
        double r145472 = b;
        double r145473 = r145471 - r145472;
        double r145474 = r145468 * r145473;
        double r145475 = r145467 + r145474;
        double r145476 = exp(r145475);
        double r145477 = r145461 * r145476;
        return r145477;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r145478 = x;
        double r145479 = y;
        double r145480 = z;
        double r145481 = log(r145480);
        double r145482 = t;
        double r145483 = r145481 - r145482;
        double r145484 = r145479 * r145483;
        double r145485 = a;
        double r145486 = 1.0;
        double r145487 = log(r145486);
        double r145488 = 0.5;
        double r145489 = 2.0;
        double r145490 = pow(r145480, r145489);
        double r145491 = pow(r145486, r145489);
        double r145492 = r145490 / r145491;
        double r145493 = r145488 * r145492;
        double r145494 = r145486 * r145480;
        double r145495 = r145493 + r145494;
        double r145496 = r145487 - r145495;
        double r145497 = b;
        double r145498 = r145496 - r145497;
        double r145499 = r145485 * r145498;
        double r145500 = r145484 + r145499;
        double r145501 = exp(r145500);
        double r145502 = r145478 * r145501;
        return r145502;
}

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 1.9

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

  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. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 
(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))))))