Average Error: 1.8 → 0.6
Time: 12.8s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \sqrt[3]{{\left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\right)}^{3}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \sqrt[3]{{\left(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)}^{3}}
double f(double x, double y, double z, double t, double a, double b) {
        double r151255 = x;
        double r151256 = y;
        double r151257 = z;
        double r151258 = log(r151257);
        double r151259 = t;
        double r151260 = r151258 - r151259;
        double r151261 = r151256 * r151260;
        double r151262 = a;
        double r151263 = 1.0;
        double r151264 = r151263 - r151257;
        double r151265 = log(r151264);
        double r151266 = b;
        double r151267 = r151265 - r151266;
        double r151268 = r151262 * r151267;
        double r151269 = r151261 + r151268;
        double r151270 = exp(r151269);
        double r151271 = r151255 * r151270;
        return r151271;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r151272 = x;
        double r151273 = y;
        double r151274 = z;
        double r151275 = log(r151274);
        double r151276 = t;
        double r151277 = r151275 - r151276;
        double r151278 = r151273 * r151277;
        double r151279 = a;
        double r151280 = 1.0;
        double r151281 = log(r151280);
        double r151282 = 0.5;
        double r151283 = 2.0;
        double r151284 = pow(r151274, r151283);
        double r151285 = pow(r151280, r151283);
        double r151286 = r151284 / r151285;
        double r151287 = r151282 * r151286;
        double r151288 = r151280 * r151274;
        double r151289 = r151287 + r151288;
        double r151290 = r151281 - r151289;
        double r151291 = b;
        double r151292 = r151290 - r151291;
        double r151293 = r151279 * r151292;
        double r151294 = r151278 + r151293;
        double r151295 = exp(r151294);
        double r151296 = 3.0;
        double r151297 = pow(r151295, r151296);
        double r151298 = cbrt(r151297);
        double r151299 = r151272 * r151298;
        return r151299;
}

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 1.8

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

  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. Using strategy rm

  4. Applied add-cbrt-cube0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020027 
(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))))))