Average Error: 2.1 → 0.5
Time: 32.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\left(a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right) - b\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right) \cdot y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\left(a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right) - b\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right) \cdot y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r7118224 = x;
        double r7118225 = y;
        double r7118226 = z;
        double r7118227 = log(r7118226);
        double r7118228 = t;
        double r7118229 = r7118227 - r7118228;
        double r7118230 = r7118225 * r7118229;
        double r7118231 = a;
        double r7118232 = 1.0;
        double r7118233 = r7118232 - r7118226;
        double r7118234 = log(r7118233);
        double r7118235 = b;
        double r7118236 = r7118234 - r7118235;
        double r7118237 = r7118231 * r7118236;
        double r7118238 = r7118230 + r7118237;
        double r7118239 = exp(r7118238);
        double r7118240 = r7118224 * r7118239;
        return r7118240;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7118241 = x;
        double r7118242 = a;
        double r7118243 = 1.0;
        double r7118244 = log(r7118243);
        double r7118245 = z;
        double r7118246 = r7118245 * r7118243;
        double r7118247 = r7118244 - r7118246;
        double r7118248 = 0.5;
        double r7118249 = r7118245 / r7118243;
        double r7118250 = r7118249 * r7118249;
        double r7118251 = r7118248 * r7118250;
        double r7118252 = r7118247 - r7118251;
        double r7118253 = b;
        double r7118254 = r7118252 - r7118253;
        double r7118255 = r7118242 * r7118254;
        double r7118256 = cbrt(r7118245);
        double r7118257 = log(r7118256);
        double r7118258 = t;
        double r7118259 = r7118257 - r7118258;
        double r7118260 = y;
        double r7118261 = r7118259 * r7118260;
        double r7118262 = r7118255 + r7118261;
        double r7118263 = r7118256 * r7118256;
        double r7118264 = log(r7118263);
        double r7118265 = r7118264 * r7118260;
        double r7118266 = r7118262 + r7118265;
        double r7118267 = exp(r7118266);
        double r7118268 = r7118241 * r7118267;
        return r7118268;
}

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 2.1

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied log-prod0.5

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

  7. Applied associate--l+0.5

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

  8. Applied distribute-rgt-in0.5

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

  9. Applied associate-+l+0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 
(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))))))