Average Error: 1.9 → 0.5
Time: 13.9s
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 r122308 = x;
        double r122309 = y;
        double r122310 = z;
        double r122311 = log(r122310);
        double r122312 = t;
        double r122313 = r122311 - r122312;
        double r122314 = r122309 * r122313;
        double r122315 = a;
        double r122316 = 1.0;
        double r122317 = r122316 - r122310;
        double r122318 = log(r122317);
        double r122319 = b;
        double r122320 = r122318 - r122319;
        double r122321 = r122315 * r122320;
        double r122322 = r122314 + r122321;
        double r122323 = exp(r122322);
        double r122324 = r122308 * r122323;
        return r122324;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r122325 = x;
        double r122326 = y;
        double r122327 = z;
        double r122328 = log(r122327);
        double r122329 = t;
        double r122330 = r122328 - r122329;
        double r122331 = r122326 * r122330;
        double r122332 = a;
        double r122333 = 1.0;
        double r122334 = log(r122333);
        double r122335 = 0.5;
        double r122336 = 2.0;
        double r122337 = pow(r122327, r122336);
        double r122338 = pow(r122333, r122336);
        double r122339 = r122337 / r122338;
        double r122340 = r122335 * r122339;
        double r122341 = r122333 * r122327;
        double r122342 = r122340 + r122341;
        double r122343 = r122334 - r122342;
        double r122344 = b;
        double r122345 = r122343 - r122344;
        double r122346 = r122332 * r122345;
        double r122347 = r122331 + r122346;
        double r122348 = exp(r122347);
        double r122349 = 3.0;
        double r122350 = pow(r122348, r122349);
        double r122351 = cbrt(r122350);
        double r122352 = r122325 * r122351;
        return r122352;
}

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-cbrt-cube0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019347 
(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))))))