Average Error: 1.8 → 0.5
Time: 14.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \left({e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}{2}\right)} \cdot {e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}{2}\right)}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \left({e}^{\left(\frac{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)}{2}\right)} \cdot {e}^{\left(\frac{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)}{2}\right)}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r134539 = x;
        double r134540 = y;
        double r134541 = z;
        double r134542 = log(r134541);
        double r134543 = t;
        double r134544 = r134542 - r134543;
        double r134545 = r134540 * r134544;
        double r134546 = a;
        double r134547 = 1.0;
        double r134548 = r134547 - r134541;
        double r134549 = log(r134548);
        double r134550 = b;
        double r134551 = r134549 - r134550;
        double r134552 = r134546 * r134551;
        double r134553 = r134545 + r134552;
        double r134554 = exp(r134553);
        double r134555 = r134539 * r134554;
        return r134555;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r134556 = x;
        double r134557 = exp(1.0);
        double r134558 = y;
        double r134559 = z;
        double r134560 = log(r134559);
        double r134561 = t;
        double r134562 = r134560 - r134561;
        double r134563 = r134558 * r134562;
        double r134564 = a;
        double r134565 = 1.0;
        double r134566 = log(r134565);
        double r134567 = 0.5;
        double r134568 = 2.0;
        double r134569 = pow(r134559, r134568);
        double r134570 = pow(r134565, r134568);
        double r134571 = r134569 / r134570;
        double r134572 = r134567 * r134571;
        double r134573 = r134565 * r134559;
        double r134574 = r134572 + r134573;
        double r134575 = r134566 - r134574;
        double r134576 = b;
        double r134577 = r134575 - r134576;
        double r134578 = r134564 * r134577;
        double r134579 = r134563 + r134578;
        double r134580 = r134579 / r134568;
        double r134581 = pow(r134557, r134580);
        double r134582 = r134581 * r134581;
        double r134583 = r134556 * r134582;
        return r134583;
}

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 1.8

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

  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. Using strategy rm

  4. Applied *-un-lft-identity0.5

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

  5. Applied exp-prod0.5

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

  6. Simplified0.5

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

  8. Applied sqr-pow0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020081 
(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))))))