Average Error: 1.9 → 0.4
Time: 24.3s
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 r77499 = x;
        double r77500 = y;
        double r77501 = z;
        double r77502 = log(r77501);
        double r77503 = t;
        double r77504 = r77502 - r77503;
        double r77505 = r77500 * r77504;
        double r77506 = a;
        double r77507 = 1.0;
        double r77508 = r77507 - r77501;
        double r77509 = log(r77508);
        double r77510 = b;
        double r77511 = r77509 - r77510;
        double r77512 = r77506 * r77511;
        double r77513 = r77505 + r77512;
        double r77514 = exp(r77513);
        double r77515 = r77499 * r77514;
        return r77515;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r77516 = x;
        double r77517 = y;
        double r77518 = z;
        double r77519 = log(r77518);
        double r77520 = t;
        double r77521 = r77519 - r77520;
        double r77522 = r77517 * r77521;
        double r77523 = a;
        double r77524 = 1.0;
        double r77525 = log(r77524);
        double r77526 = 0.5;
        double r77527 = 2.0;
        double r77528 = pow(r77518, r77527);
        double r77529 = pow(r77524, r77527);
        double r77530 = r77528 / r77529;
        double r77531 = r77526 * r77530;
        double r77532 = r77524 * r77518;
        double r77533 = r77531 + r77532;
        double r77534 = r77525 - r77533;
        double r77535 = b;
        double r77536 = r77534 - r77535;
        double r77537 = r77523 * r77536;
        double r77538 = r77522 + r77537;
        double r77539 = exp(r77538);
        double r77540 = r77516 * r77539;
        return r77540;
}

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 0.4

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

  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 2019323 
(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))))))