Average Error: 2.0 → 0.5
Time: 1.7m
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[{\left(e^{\sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)} \cdot \sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)}}\right)}^{\left(\sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)}\right)} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
{\left(e^{\sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)} \cdot \sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)}}\right)}^{\left(\sqrt[3]{\left(\log z - t\right) \cdot y + a \cdot \left(\left(\left(\log 1 - \left(\frac{z}{1} \cdot \frac{z}{1}\right) \cdot \frac{1}{2}\right) - z \cdot 1\right) - b\right)}\right)} \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r6229385 = x;
        double r6229386 = y;
        double r6229387 = z;
        double r6229388 = log(r6229387);
        double r6229389 = t;
        double r6229390 = r6229388 - r6229389;
        double r6229391 = r6229386 * r6229390;
        double r6229392 = a;
        double r6229393 = 1.0;
        double r6229394 = r6229393 - r6229387;
        double r6229395 = log(r6229394);
        double r6229396 = b;
        double r6229397 = r6229395 - r6229396;
        double r6229398 = r6229392 * r6229397;
        double r6229399 = r6229391 + r6229398;
        double r6229400 = exp(r6229399);
        double r6229401 = r6229385 * r6229400;
        return r6229401;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r6229402 = z;
        double r6229403 = log(r6229402);
        double r6229404 = t;
        double r6229405 = r6229403 - r6229404;
        double r6229406 = y;
        double r6229407 = r6229405 * r6229406;
        double r6229408 = a;
        double r6229409 = 1.0;
        double r6229410 = log(r6229409);
        double r6229411 = r6229402 / r6229409;
        double r6229412 = r6229411 * r6229411;
        double r6229413 = 0.5;
        double r6229414 = r6229412 * r6229413;
        double r6229415 = r6229410 - r6229414;
        double r6229416 = r6229402 * r6229409;
        double r6229417 = r6229415 - r6229416;
        double r6229418 = b;
        double r6229419 = r6229417 - r6229418;
        double r6229420 = r6229408 * r6229419;
        double r6229421 = r6229407 + r6229420;
        double r6229422 = cbrt(r6229421);
        double r6229423 = r6229422 * r6229422;
        double r6229424 = exp(r6229423);
        double r6229425 = pow(r6229424, r6229422);
        double r6229426 = x;
        double r6229427 = r6229425 * r6229426;
        return r6229427;
}

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 0.5

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

  3. Simplified0.5

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied exp-prod0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))