Average Error: 1.9 → 0.5
Time: 30.8s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[e^{a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
e^{a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r7081442 = x;
        double r7081443 = y;
        double r7081444 = z;
        double r7081445 = log(r7081444);
        double r7081446 = t;
        double r7081447 = r7081445 - r7081446;
        double r7081448 = r7081443 * r7081447;
        double r7081449 = a;
        double r7081450 = 1.0;
        double r7081451 = r7081450 - r7081444;
        double r7081452 = log(r7081451);
        double r7081453 = b;
        double r7081454 = r7081452 - r7081453;
        double r7081455 = r7081449 * r7081454;
        double r7081456 = r7081448 + r7081455;
        double r7081457 = exp(r7081456);
        double r7081458 = r7081442 * r7081457;
        return r7081458;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7081459 = a;
        double r7081460 = 1.0;
        double r7081461 = log(r7081460);
        double r7081462 = z;
        double r7081463 = r7081462 * r7081460;
        double r7081464 = r7081461 - r7081463;
        double r7081465 = 0.5;
        double r7081466 = r7081460 / r7081462;
        double r7081467 = r7081466 * r7081466;
        double r7081468 = r7081465 / r7081467;
        double r7081469 = r7081464 - r7081468;
        double r7081470 = b;
        double r7081471 = r7081469 - r7081470;
        double r7081472 = r7081459 * r7081471;
        double r7081473 = log(r7081462);
        double r7081474 = t;
        double r7081475 = r7081473 - r7081474;
        double r7081476 = y;
        double r7081477 = r7081475 * r7081476;
        double r7081478 = r7081472 + r7081477;
        double r7081479 = exp(r7081478);
        double r7081480 = x;
        double r7081481 = r7081479 * r7081480;
        return r7081481;
}

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

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))