Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 1.2

Time: 33.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r163602 = x;
        double r163603 = y;
        double r163604 = z;
        double r163605 = log(r163604);
        double r163606 = r163603 * r163605;
        double r163607 = t;
        double r163608 = 1.0;
        double r163609 = r163607 - r163608;
        double r163610 = a;
        double r163611 = log(r163610);
        double r163612 = r163609 * r163611;
        double r163613 = r163606 + r163612;
        double r163614 = b;
        double r163615 = r163613 - r163614;
        double r163616 = exp(r163615);
        double r163617 = r163602 * r163616;
        double r163618 = r163617 / r163603;
        return r163618;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r163619 = 1.0;
        double r163620 = a;
        double r163621 = 1.0;
        double r163622 = pow(r163620, r163621);
        double r163623 = pow(r163622, r163621);
        double r163624 = z;
        double r163625 = r163619 / r163624;
        double r163626 = log(r163625);
        double r163627 = y;
        double r163628 = r163626 * r163627;
        double r163629 = r163619 / r163620;
        double r163630 = log(r163629);
        double r163631 = t;
        double r163632 = r163630 * r163631;
        double r163633 = b;
        double r163634 = r163632 + r163633;
        double r163635 = r163628 + r163634;
        double r163636 = exp(r163635);
        double r163637 = r163636 * r163627;
        double r163638 = x;
        double r163639 = r163637 / r163638;
        double r163640 = r163623 * r163639;
        double r163641 = r163619 / r163640;
        return r163641;
}

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

Derivation

1. Initial program 2.0

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

2. Taylor expanded around inf 2.0

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

3. Simplified 1.2

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

5. Applied add-cube-cbrt 1.3

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\left(\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]

6. Applied add-sqr-sqrt 1.3

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right)}^{1}}{\left(\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

7. Applied *-un-lft-identity 1.3

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{\color{blue}{1 \cdot 1}}{\sqrt{a} \cdot \sqrt{a}}\right)}^{1}}{\left(\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

8. Applied times-frac 1.3

   \[\leadsto \frac{x \cdot \frac{{\color{blue}{\left(\frac{1}{\sqrt{a}} \cdot \frac{1}{\sqrt{a}}\right)}}^{1}}{\left(\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

9. Applied unpow-prod-down 1.3

   \[\leadsto \frac{x \cdot \frac{\color{blue}{{\left(\frac{1}{\sqrt{a}}\right)}^{1} \cdot {\left(\frac{1}{\sqrt{a}}\right)}^{1}}}{\left(\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right) \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

10. Applied times-frac 1.3

    \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right)}}{y}\]

11. Applied associate-*r* 1.3

    \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]
12. Using strategy rm

13. Applied clear-num 1.3

    \[\leadsto \color{blue}{\frac{1}{\frac{y}{\left(x \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}\]

14. Simplified 6.2

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y}{x}}{\frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{\left(2 \cdot 1\right)}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\sqrt[3]{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}\]

15. Taylor expanded around inf 1.2

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

16. Final simplification 1.2

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

Reproduce

herbie shell --seed 2020056 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))