Average Error: 1.9 → 0.5
Time: 11.7s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \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)}
double f(double x, double y, double z, double t, double a, double b) {
        double r100296 = x;
        double r100297 = y;
        double r100298 = z;
        double r100299 = log(r100298);
        double r100300 = t;
        double r100301 = r100299 - r100300;
        double r100302 = r100297 * r100301;
        double r100303 = a;
        double r100304 = 1.0;
        double r100305 = r100304 - r100298;
        double r100306 = log(r100305);
        double r100307 = b;
        double r100308 = r100306 - r100307;
        double r100309 = r100303 * r100308;
        double r100310 = r100302 + r100309;
        double r100311 = exp(r100310);
        double r100312 = r100296 * r100311;
        return r100312;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r100313 = x;
        double r100314 = y;
        double r100315 = z;
        double r100316 = log(r100315);
        double r100317 = t;
        double r100318 = r100316 - r100317;
        double r100319 = r100314 * r100318;
        double r100320 = a;
        double r100321 = 1.0;
        double r100322 = log(r100321);
        double r100323 = 0.5;
        double r100324 = 2.0;
        double r100325 = pow(r100315, r100324);
        double r100326 = pow(r100321, r100324);
        double r100327 = r100325 / r100326;
        double r100328 = r100323 * r100327;
        double r100329 = r100321 * r100315;
        double r100330 = r100328 + r100329;
        double r100331 = r100322 - r100330;
        double r100332 = b;
        double r100333 = r100331 - r100332;
        double r100334 = r100320 * r100333;
        double r100335 = r100319 + r100334;
        double r100336 = exp(r100335);
        double r100337 = r100313 * r100336;
        return r100337;
}

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 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 
(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))))))