Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 1.9 → 0.1

Time: 13.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1015.931981034841896871512290090322494507:\\ \;\;\;\;\left(\left(x \cdot \frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;x \le 717450678909797:\\ \;\;\;\;\frac{{\left(\frac{1}{{a}^{1}}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{1}{y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r81594 = x;
        double r81595 = y;
        double r81596 = z;
        double r81597 = log(r81596);
        double r81598 = r81595 * r81597;
        double r81599 = t;
        double r81600 = 1.0;
        double r81601 = r81599 - r81600;
        double r81602 = a;
        double r81603 = log(r81602);
        double r81604 = r81601 * r81603;
        double r81605 = r81598 + r81604;
        double r81606 = b;
        double r81607 = r81605 - r81606;
        double r81608 = exp(r81607);
        double r81609 = r81594 * r81608;
        double r81610 = r81609 / r81595;
        return r81610;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r81611 = x;
        double r81612 = -1015.9319810348419;
        bool r81613 = r81611 <= r81612;
        double r81614 = 1.0;
        double r81615 = a;
        double r81616 = r81614 / r81615;
        double r81617 = 1.0;
        double r81618 = pow(r81616, r81617);
        double r81619 = sqrt(r81618);
        double r81620 = y;
        double r81621 = z;
        double r81622 = r81614 / r81621;
        double r81623 = log(r81622);
        double r81624 = log(r81616);
        double r81625 = t;
        double r81626 = b;
        double r81627 = fma(r81624, r81625, r81626);
        double r81628 = fma(r81620, r81623, r81627);
        double r81629 = exp(r81628);
        double r81630 = cbrt(r81629);
        double r81631 = r81630 * r81630;
        double r81632 = r81619 / r81631;
        double r81633 = r81611 * r81632;
        double r81634 = r81619 / r81630;
        double r81635 = r81633 * r81634;
        double r81636 = r81614 / r81620;
        double r81637 = r81635 * r81636;
        double r81638 = 717450678909797.0;
        bool r81639 = r81611 <= r81638;
        double r81640 = pow(r81615, r81617);
        double r81641 = r81614 / r81640;
        double r81642 = pow(r81641, r81617);
        double r81643 = r81642 / r81629;
        double r81644 = r81611 / r81620;
        double r81645 = r81643 * r81644;
        double r81646 = r81618 / r81629;
        double r81647 = r81611 * r81646;
        double r81648 = sqrt(r81647);
        double r81649 = r81648 * r81648;
        double r81650 = r81649 * r81636;
        double r81651 = r81639 ? r81645 : r81650;
        double r81652 = r81613 ? r81637 : r81651;
        return r81652;
}

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. Split input into 3 regimes

2. if x < -1015.9319810348419

   1. Initial program 0.7

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

   2. Taylor expanded around inf 0.7

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

   3. Simplified 0.1

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

   5. Applied div-inv 0.1

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

   7. Applied add-cube-cbrt 0.1

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

   8. Applied add-sqr-sqrt 0.1

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

   9. Applied times-frac 0.1

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

   10. Applied associate-*r* 0.1

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

   if -1015.9319810348419 < x < 717450678909797.0

   1. Initial program 3.0

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

   2. Taylor expanded around inf 3.0

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

   3. Simplified 2.2

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

   5. Applied div-inv 2.2

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

   7. Applied add-cube-cbrt 2.3

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

   8. Applied add-sqr-sqrt 2.3

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

   9. Applied times-frac 2.3

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

   10. Applied associate-*r* 2.3

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

   11. Taylor expanded around inf 0.1

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

   12. Simplified 0.1

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

   if 717450678909797.0 < x

   1. Initial program 0.7

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

   2. Taylor expanded around inf 0.7

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

   3. Simplified 0.1

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

   5. Applied div-inv 0.1

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

   7. Applied add-sqr-sqrt 0.1

      \[\leadsto \color{blue}{\left(\sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right)} \cdot \frac{1}{y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1015.931981034841896871512290090322494507:\\ \;\;\;\;\left(\left(x \cdot \frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;x \le 717450678909797:\\ \;\;\;\;\frac{{\left(\frac{1}{{a}^{1}}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \sqrt{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(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))