Average Error: 2.0 → 0.5
Time: 32.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\left(\left(\left(\log 1 - z \cdot 1\right) - \frac{z \cdot z}{1} \cdot \frac{\frac{1}{2}}{1}\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\left(\left(\left(\log 1 - z \cdot 1\right) - \frac{z \cdot z}{1} \cdot \frac{\frac{1}{2}}{1}\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r87263 = x;
        double r87264 = y;
        double r87265 = z;
        double r87266 = log(r87265);
        double r87267 = t;
        double r87268 = r87266 - r87267;
        double r87269 = r87264 * r87268;
        double r87270 = a;
        double r87271 = 1.0;
        double r87272 = r87271 - r87265;
        double r87273 = log(r87272);
        double r87274 = b;
        double r87275 = r87273 - r87274;
        double r87276 = r87270 * r87275;
        double r87277 = r87269 + r87276;
        double r87278 = exp(r87277);
        double r87279 = r87263 * r87278;
        return r87279;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r87280 = x;
        double r87281 = 1.0;
        double r87282 = log(r87281);
        double r87283 = z;
        double r87284 = r87283 * r87281;
        double r87285 = r87282 - r87284;
        double r87286 = r87283 * r87283;
        double r87287 = r87286 / r87281;
        double r87288 = 0.5;
        double r87289 = r87288 / r87281;
        double r87290 = r87287 * r87289;
        double r87291 = r87285 - r87290;
        double r87292 = b;
        double r87293 = r87291 - r87292;
        double r87294 = a;
        double r87295 = r87293 * r87294;
        double r87296 = log(r87283);
        double r87297 = t;
        double r87298 = r87296 - r87297;
        double r87299 = y;
        double r87300 = r87298 * r87299;
        double r87301 = r87295 + r87300;
        double r87302 = exp(r87301);
        double r87303 = r87280 * r87302;
        return r87303;
}

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

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019195 
(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))))))