Average Error: 1.9 → 0.5
Time: 19.9s
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 r488 = x;
        double r489 = y;
        double r490 = z;
        double r491 = log(r490);
        double r492 = t;
        double r493 = r491 - r492;
        double r494 = r489 * r493;
        double r495 = a;
        double r496 = 1.0;
        double r497 = r496 - r490;
        double r498 = log(r497);
        double r499 = b;
        double r500 = r498 - r499;
        double r501 = r495 * r500;
        double r502 = r494 + r501;
        double r503 = exp(r502);
        double r504 = r488 * r503;
        return r504;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r505 = x;
        double r506 = y;
        double r507 = z;
        double r508 = log(r507);
        double r509 = t;
        double r510 = r508 - r509;
        double r511 = r506 * r510;
        double r512 = a;
        double r513 = 1.0;
        double r514 = log(r513);
        double r515 = 0.5;
        double r516 = 2.0;
        double r517 = pow(r507, r516);
        double r518 = pow(r513, r516);
        double r519 = r517 / r518;
        double r520 = r515 * r519;
        double r521 = r513 * r507;
        double r522 = r520 + r521;
        double r523 = r514 - r522;
        double r524 = b;
        double r525 = r523 - r524;
        double r526 = r512 * r525;
        double r527 = r511 + r526;
        double r528 = exp(r527);
        double r529 = r505 * r528;
        return r529;
}

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))))))