Average Error: 1.9 → 0.5
Time: 15.8s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot \left(\sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}} \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\right)\right) \cdot \sqrt[3]{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot \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)\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)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r144432 = x;
        double r144433 = y;
        double r144434 = z;
        double r144435 = log(r144434);
        double r144436 = t;
        double r144437 = r144435 - r144436;
        double r144438 = r144433 * r144437;
        double r144439 = a;
        double r144440 = 1.0;
        double r144441 = r144440 - r144434;
        double r144442 = log(r144441);
        double r144443 = b;
        double r144444 = r144442 - r144443;
        double r144445 = r144439 * r144444;
        double r144446 = r144438 + r144445;
        double r144447 = exp(r144446);
        double r144448 = r144432 * r144447;
        return r144448;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r144449 = x;
        double r144450 = y;
        double r144451 = z;
        double r144452 = log(r144451);
        double r144453 = t;
        double r144454 = r144452 - r144453;
        double r144455 = r144450 * r144454;
        double r144456 = a;
        double r144457 = 1.0;
        double r144458 = log(r144457);
        double r144459 = 0.5;
        double r144460 = 2.0;
        double r144461 = pow(r144451, r144460);
        double r144462 = pow(r144457, r144460);
        double r144463 = r144461 / r144462;
        double r144464 = r144459 * r144463;
        double r144465 = r144457 * r144451;
        double r144466 = r144464 + r144465;
        double r144467 = r144458 - r144466;
        double r144468 = b;
        double r144469 = r144467 - r144468;
        double r144470 = r144456 * r144469;
        double r144471 = r144455 + r144470;
        double r144472 = exp(r144471);
        double r144473 = cbrt(r144472);
        double r144474 = r144473 * r144473;
        double r144475 = r144449 * r144474;
        double r144476 = r144475 * r144473;
        return r144476;
}

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. 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. Applied associate-*r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020034 
(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))))))