Average Error: 1.9 → 0.5
Time: 30.8s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(\sqrt{e^{a \cdot \left(\left(\log 1 - z \cdot 1\right) - \left(b + \frac{\frac{1}{2}}{1} \cdot \frac{z \cdot z}{1}\right)\right) + y \cdot \left(\log z - t\right)}} \cdot x\right) \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{{z}^{2}}{{1}^{2}} \cdot \frac{1}{2} + z \cdot 1\right)\right) - b\right)}}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(\sqrt{e^{a \cdot \left(\left(\log 1 - z \cdot 1\right) - \left(b + \frac{\frac{1}{2}}{1} \cdot \frac{z \cdot z}{1}\right)\right) + y \cdot \left(\log z - t\right)}} \cdot x\right) \cdot \sqrt{e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{{z}^{2}}{{1}^{2}} \cdot \frac{1}{2} + z \cdot 1\right)\right) - b\right)}}
double f(double x, double y, double z, double t, double a, double b) {
        double r114713 = x;
        double r114714 = y;
        double r114715 = z;
        double r114716 = log(r114715);
        double r114717 = t;
        double r114718 = r114716 - r114717;
        double r114719 = r114714 * r114718;
        double r114720 = a;
        double r114721 = 1.0;
        double r114722 = r114721 - r114715;
        double r114723 = log(r114722);
        double r114724 = b;
        double r114725 = r114723 - r114724;
        double r114726 = r114720 * r114725;
        double r114727 = r114719 + r114726;
        double r114728 = exp(r114727);
        double r114729 = r114713 * r114728;
        return r114729;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r114730 = a;
        double r114731 = 1.0;
        double r114732 = log(r114731);
        double r114733 = z;
        double r114734 = r114733 * r114731;
        double r114735 = r114732 - r114734;
        double r114736 = b;
        double r114737 = 0.5;
        double r114738 = r114737 / r114731;
        double r114739 = r114733 * r114733;
        double r114740 = r114739 / r114731;
        double r114741 = r114738 * r114740;
        double r114742 = r114736 + r114741;
        double r114743 = r114735 - r114742;
        double r114744 = r114730 * r114743;
        double r114745 = y;
        double r114746 = log(r114733);
        double r114747 = t;
        double r114748 = r114746 - r114747;
        double r114749 = r114745 * r114748;
        double r114750 = r114744 + r114749;
        double r114751 = exp(r114750);
        double r114752 = sqrt(r114751);
        double r114753 = x;
        double r114754 = r114752 * r114753;
        double r114755 = 2.0;
        double r114756 = pow(r114733, r114755);
        double r114757 = pow(r114731, r114755);
        double r114758 = r114756 / r114757;
        double r114759 = r114758 * r114737;
        double r114760 = r114759 + r114734;
        double r114761 = r114732 - r114760;
        double r114762 = r114761 - r114736;
        double r114763 = r114730 * r114762;
        double r114764 = r114749 + r114763;
        double r114765 = exp(r114764);
        double r114766 = sqrt(r114765);
        double r114767 = r114754 * r114766;
        return r114767;
}

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

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

  3. Simplified0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied associate-*r*0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019174 
(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))))))