Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 4.1 → 5.6

Time: 13.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r101685 = x;
        double r101686 = y;
        double r101687 = 2.0;
        double r101688 = z;
        double r101689 = t;
        double r101690 = a;
        double r101691 = r101689 + r101690;
        double r101692 = sqrt(r101691);
        double r101693 = r101688 * r101692;
        double r101694 = r101693 / r101689;
        double r101695 = b;
        double r101696 = c;
        double r101697 = r101695 - r101696;
        double r101698 = 5.0;
        double r101699 = 6.0;
        double r101700 = r101698 / r101699;
        double r101701 = r101690 + r101700;
        double r101702 = 3.0;
        double r101703 = r101689 * r101702;
        double r101704 = r101687 / r101703;
        double r101705 = r101701 - r101704;
        double r101706 = r101697 * r101705;
        double r101707 = r101694 - r101706;
        double r101708 = r101687 * r101707;
        double r101709 = exp(r101708);
        double r101710 = r101686 * r101709;
        double r101711 = r101685 + r101710;
        double r101712 = r101685 / r101711;
        return r101712;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r101713 = t;
        double r101714 = -2.3617424863871238e-92;
        bool r101715 = r101713 <= r101714;
        double r101716 = x;
        double r101717 = y;
        double r101718 = 2.0;
        double r101719 = c;
        double r101720 = a;
        double r101721 = 0.8333333333333334;
        double r101722 = r101720 + r101721;
        double r101723 = r101719 * r101722;
        double r101724 = b;
        double r101725 = r101720 * r101724;
        double r101726 = r101723 - r101725;
        double r101727 = r101718 * r101726;
        double r101728 = exp(r101727);
        double r101729 = r101717 * r101728;
        double r101730 = r101716 + r101729;
        double r101731 = r101716 / r101730;
        double r101732 = 9.255404473728014e-17;
        bool r101733 = r101713 <= r101732;
        double r101734 = z;
        double r101735 = r101713 + r101720;
        double r101736 = sqrt(r101735);
        double r101737 = r101734 * r101736;
        double r101738 = 5.0;
        double r101739 = 6.0;
        double r101740 = r101738 / r101739;
        double r101741 = r101720 - r101740;
        double r101742 = 3.0;
        double r101743 = r101713 * r101742;
        double r101744 = r101741 * r101743;
        double r101745 = r101737 * r101744;
        double r101746 = r101724 - r101719;
        double r101747 = r101720 * r101720;
        double r101748 = r101740 * r101740;
        double r101749 = r101747 - r101748;
        double r101750 = r101749 * r101743;
        double r101751 = r101741 * r101718;
        double r101752 = r101750 - r101751;
        double r101753 = r101746 * r101752;
        double r101754 = r101713 * r101753;
        double r101755 = r101745 - r101754;
        double r101756 = r101713 * r101744;
        double r101757 = r101755 / r101756;
        double r101758 = r101718 * r101757;
        double r101759 = exp(r101758);
        double r101760 = r101717 * r101759;
        double r101761 = r101716 + r101760;
        double r101762 = r101716 / r101761;
        double r101763 = r101737 / r101713;
        double r101764 = r101720 + r101740;
        double r101765 = r101718 / r101713;
        double r101766 = r101765 / r101742;
        double r101767 = exp(r101766);
        double r101768 = log(r101767);
        double r101769 = 3.0;
        double r101770 = pow(r101768, r101769);
        double r101771 = 0.3333333333333333;
        double r101772 = pow(r101770, r101771);
        double r101773 = r101764 - r101772;
        double r101774 = r101746 * r101773;
        double r101775 = r101763 - r101774;
        double r101776 = r101718 * r101775;
        double r101777 = exp(r101776);
        double r101778 = r101717 * r101777;
        double r101779 = r101716 + r101778;
        double r101780 = r101716 / r101779;
        double r101781 = r101733 ? r101762 : r101780;
        double r101782 = r101715 ? r101731 : r101781;
        return r101782;
}

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

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if t < -2.3617424863871238e-92

   1. Initial program 3.6

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Taylor expanded around inf 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]

   if -2.3617424863871238e-92 < t < 9.255404473728014e-17

   1. Initial program 5.4

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied flip-+ 8.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

   4. Applied frac-sub 8.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   5. Applied associate-*r/ 8.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

   6. Applied frac-sub 6.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]

   if 9.255404473728014e-17 < t

   1. Initial program 2.9

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot \color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}}\right)\right)}}\]

   4. Applied add-cbrt-cube 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}\right)\right)}}\]

   5. Applied cbrt-unprod 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

   6. Applied add-cbrt-cube 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}\right)\right)}}\]

   7. Applied cbrt-undiv 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

   8. Simplified 2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{\color{blue}{{\left(\frac{2}{t \cdot 3}\right)}^{3}}}\right)\right)}}\]
   9. Using strategy rm

   10. Applied pow1/3 2.9

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{{\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}}\right)\right)}}\]
   11. Using strategy rm

   12. Applied add-log-exp 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\color{blue}{\left(\log \left(e^{\frac{2}{t \cdot 3}}\right)\right)}}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]

   13. Simplified 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \color{blue}{\left(e^{\frac{\frac{2}{t}}{3}}\right)}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))