Average Error: 2.0 → 0.5
Time: 16.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left(\left(\sqrt[3]{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)}} \cdot \sqrt[3]{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)}}\right) \cdot \sqrt[3]{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)}}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r143424 = x;
        double r143425 = y;
        double r143426 = z;
        double r143427 = log(r143426);
        double r143428 = t;
        double r143429 = r143427 - r143428;
        double r143430 = r143425 * r143429;
        double r143431 = a;
        double r143432 = 1.0;
        double r143433 = r143432 - r143426;
        double r143434 = log(r143433);
        double r143435 = b;
        double r143436 = r143434 - r143435;
        double r143437 = r143431 * r143436;
        double r143438 = r143430 + r143437;
        double r143439 = exp(r143438);
        double r143440 = r143424 * r143439;
        return r143440;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r143441 = x;
        double r143442 = y;
        double r143443 = z;
        double r143444 = log(r143443);
        double r143445 = t;
        double r143446 = r143444 - r143445;
        double r143447 = r143442 * r143446;
        double r143448 = a;
        double r143449 = 1.0;
        double r143450 = log(r143449);
        double r143451 = 0.5;
        double r143452 = 2.0;
        double r143453 = pow(r143443, r143452);
        double r143454 = pow(r143449, r143452);
        double r143455 = r143453 / r143454;
        double r143456 = r143451 * r143455;
        double r143457 = r143449 * r143443;
        double r143458 = r143456 + r143457;
        double r143459 = r143450 - r143458;
        double r143460 = b;
        double r143461 = r143459 - r143460;
        double r143462 = r143448 * r143461;
        double r143463 = r143447 + r143462;
        double r143464 = exp(r143463);
        double r143465 = cbrt(r143464);
        double r143466 = r143465 * r143465;
        double r143467 = r143466 * r143465;
        double r143468 = r143441 * r143467;
        return r143468;
}

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 2.0

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.5

    \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)}\]

  5. Final simplification0.5

    \[\leadsto x \cdot \left(\left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)\]

Reproduce

herbie shell --seed 2020047 
(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))))))