Average Error: 1.9 → 0.4
Time: 23.3s
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 r84353 = x;
        double r84354 = y;
        double r84355 = z;
        double r84356 = log(r84355);
        double r84357 = t;
        double r84358 = r84356 - r84357;
        double r84359 = r84354 * r84358;
        double r84360 = a;
        double r84361 = 1.0;
        double r84362 = r84361 - r84355;
        double r84363 = log(r84362);
        double r84364 = b;
        double r84365 = r84363 - r84364;
        double r84366 = r84360 * r84365;
        double r84367 = r84359 + r84366;
        double r84368 = exp(r84367);
        double r84369 = r84353 * r84368;
        return r84369;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r84370 = x;
        double r84371 = y;
        double r84372 = z;
        double r84373 = log(r84372);
        double r84374 = t;
        double r84375 = r84373 - r84374;
        double r84376 = r84371 * r84375;
        double r84377 = a;
        double r84378 = 1.0;
        double r84379 = log(r84378);
        double r84380 = 0.5;
        double r84381 = 2.0;
        double r84382 = pow(r84372, r84381);
        double r84383 = pow(r84378, r84381);
        double r84384 = r84382 / r84383;
        double r84385 = r84380 * r84384;
        double r84386 = r84378 * r84372;
        double r84387 = r84385 + r84386;
        double r84388 = r84379 - r84387;
        double r84389 = b;
        double r84390 = r84388 - r84389;
        double r84391 = r84377 * r84390;
        double r84392 = r84376 + r84391;
        double r84393 = exp(r84392);
        double r84394 = r84370 * r84393;
        return r84394;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(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))))))