Average Error: 2.2 → 0.5
Time: 14.0s
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 r118491 = x;
        double r118492 = y;
        double r118493 = z;
        double r118494 = log(r118493);
        double r118495 = t;
        double r118496 = r118494 - r118495;
        double r118497 = r118492 * r118496;
        double r118498 = a;
        double r118499 = 1.0;
        double r118500 = r118499 - r118493;
        double r118501 = log(r118500);
        double r118502 = b;
        double r118503 = r118501 - r118502;
        double r118504 = r118498 * r118503;
        double r118505 = r118497 + r118504;
        double r118506 = exp(r118505);
        double r118507 = r118491 * r118506;
        return r118507;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r118508 = x;
        double r118509 = y;
        double r118510 = z;
        double r118511 = log(r118510);
        double r118512 = t;
        double r118513 = r118511 - r118512;
        double r118514 = r118509 * r118513;
        double r118515 = a;
        double r118516 = 1.0;
        double r118517 = log(r118516);
        double r118518 = 0.5;
        double r118519 = 2.0;
        double r118520 = pow(r118510, r118519);
        double r118521 = pow(r118516, r118519);
        double r118522 = r118520 / r118521;
        double r118523 = r118518 * r118522;
        double r118524 = r118516 * r118510;
        double r118525 = r118523 + r118524;
        double r118526 = r118517 - r118525;
        double r118527 = b;
        double r118528 = r118526 - r118527;
        double r118529 = r118515 * r118528;
        double r118530 = r118514 + r118529;
        double r118531 = exp(r118530);
        double r118532 = r118508 * r118531;
        return r118532;
}

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 2.2

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

  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 2020045 
(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))))))