Average Error: 1.8 → 0.4
Time: 1.4m
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]
↓
\[e^{a \cdot \left(\left(\left(\log 1.0 - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - z \cdot 1.0\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}
↓
e^{a \cdot \left(\left(\left(\log 1.0 - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - z \cdot 1.0\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 r7129496 = x;
        double r7129497 = y;
        double r7129498 = z;
        double r7129499 = log(r7129498);
        double r7129500 = t;
        double r7129501 = r7129499 - r7129500;
        double r7129502 = r7129497 * r7129501;
        double r7129503 = a;
        double r7129504 = 1.0;
        double r7129505 = r7129504 - r7129498;
        double r7129506 = log(r7129505);
        double r7129507 = b;
        double r7129508 = r7129506 - r7129507;
        double r7129509 = r7129503 * r7129508;
        double r7129510 = r7129502 + r7129509;
        double r7129511 = exp(r7129510);
        double r7129512 = r7129496 * r7129511;
        return r7129512;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7129513 = a;
        double r7129514 = 1.0;
        double r7129515 = log(r7129514);
        double r7129516 = z;
        double r7129517 = r7129516 / r7129514;
        double r7129518 = r7129517 * r7129517;
        double r7129519 = 0.5;
        double r7129520 = r7129518 * r7129519;
        double r7129521 = r7129515 - r7129520;
        double r7129522 = r7129516 * r7129514;
        double r7129523 = r7129521 - r7129522;
        double r7129524 = b;
        double r7129525 = r7129523 - r7129524;
        double r7129526 = r7129513 * r7129525;
        double r7129527 = log(r7129516);
        double r7129528 = t;
        double r7129529 = r7129527 - r7129528;
        double r7129530 = y;
        double r7129531 = r7129529 * r7129530;
        double r7129532 = r7129526 + r7129531;
        double r7129533 = exp(r7129532);
        double r7129534 = x;
        double r7129535 = r7129533 * r7129534;
        return r7129535;
}

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 1.8

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.4

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1.0 - \left(1.0 \cdot z + \frac{1}{2} \cdot \frac{{z}^{2}}{{1.0}^{2}}\right)\right)} - b\right)}\]

  3. Simplified0.4

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\left(\log 1.0 - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - z \cdot 1.0\right)} - b\right)}\]

  4. Final simplification0.4

    \[\leadsto e^{a \cdot \left(\left(\left(\log 1.0 - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - z \cdot 1.0\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x\]

Reproduce

herbie shell --seed 2019158 
(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))))))