Average Error: 2.1 → 0.5
Time: 33.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]
↓
\[\left(\sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}} \cdot \sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}}\right) \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}
↓
\left(\sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}} \cdot \sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}}\right) \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r7672292 = x;
        double r7672293 = y;
        double r7672294 = z;
        double r7672295 = log(r7672294);
        double r7672296 = t;
        double r7672297 = r7672295 - r7672296;
        double r7672298 = r7672293 * r7672297;
        double r7672299 = a;
        double r7672300 = 1.0;
        double r7672301 = r7672300 - r7672294;
        double r7672302 = log(r7672301);
        double r7672303 = b;
        double r7672304 = r7672302 - r7672303;
        double r7672305 = r7672299 * r7672304;
        double r7672306 = r7672298 + r7672305;
        double r7672307 = exp(r7672306);
        double r7672308 = r7672292 * r7672307;
        return r7672308;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7672309 = a;
        double r7672310 = 1.0;
        double r7672311 = log(r7672310);
        double r7672312 = z;
        double r7672313 = r7672312 * r7672310;
        double r7672314 = r7672311 - r7672313;
        double r7672315 = 0.5;
        double r7672316 = r7672312 / r7672310;
        double r7672317 = r7672316 * r7672316;
        double r7672318 = r7672315 * r7672317;
        double r7672319 = r7672314 - r7672318;
        double r7672320 = b;
        double r7672321 = r7672319 - r7672320;
        double r7672322 = r7672309 * r7672321;
        double r7672323 = log(r7672312);
        double r7672324 = t;
        double r7672325 = r7672323 - r7672324;
        double r7672326 = y;
        double r7672327 = r7672325 * r7672326;
        double r7672328 = r7672322 + r7672327;
        double r7672329 = exp(r7672328);
        double r7672330 = sqrt(r7672329);
        double r7672331 = r7672330 * r7672330;
        double r7672332 = x;
        double r7672333 = r7672331 * r7672332;
        return r7672333;
}

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

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1.0 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1.0 - \left(1.0 \cdot z + \frac{1}{2} \cdot \frac{{z}^{2}}{{1.0}^{2}}\right)\right)} - b\right)}\]

  3. Simplified0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\left(\log 1.0 - 1.0 \cdot z\right) - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right)} - b\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.5

    \[\leadsto x \cdot \color{blue}{\left(\sqrt{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\left(\log 1.0 - 1.0 \cdot z\right) - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - b\right)}} \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\left(\log 1.0 - 1.0 \cdot z\right) - \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right) \cdot \frac{1}{2}\right) - b\right)}}\right)}\]

  6. Final simplification0.5

    \[\leadsto \left(\sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}} \cdot \sqrt{e^{a \cdot \left(\left(\left(\log 1.0 - z \cdot 1.0\right) - \frac{1}{2} \cdot \left(\frac{z}{1.0} \cdot \frac{z}{1.0}\right)\right) - b\right) + \left(\log z - t\right) \cdot y}}\right) \cdot x\]

Reproduce

herbie shell --seed 2019162 
(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))))))