Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Average Error: 16.9 → 13.5

Time: 11.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.697433997227182860846341323864432125566 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{x + \frac{y}{t} \cdot z}{\sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}}}}{\sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \le 1.625660671319587146448715347668017899923 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + y \cdot \frac{b}{t}}{x + \frac{y}{\frac{t}{z}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r455824 = x;
        double r455825 = y;
        double r455826 = z;
        double r455827 = r455825 * r455826;
        double r455828 = t;
        double r455829 = r455827 / r455828;
        double r455830 = r455824 + r455829;
        double r455831 = a;
        double r455832 = 1.0;
        double r455833 = r455831 + r455832;
        double r455834 = b;
        double r455835 = r455825 * r455834;
        double r455836 = r455835 / r455828;
        double r455837 = r455833 + r455836;
        double r455838 = r455830 / r455837;
        return r455838;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r455839 = t;
        double r455840 = -2.697433997227183e+75;
        bool r455841 = r455839 <= r455840;
        double r455842 = x;
        double r455843 = y;
        double r455844 = r455843 / r455839;
        double r455845 = z;
        double r455846 = r455844 * r455845;
        double r455847 = r455842 + r455846;
        double r455848 = a;
        double r455849 = 1.0;
        double r455850 = r455848 + r455849;
        double r455851 = b;
        double r455852 = r455851 / r455839;
        double r455853 = r455843 * r455852;
        double r455854 = r455850 + r455853;
        double r455855 = cbrt(r455854);
        double r455856 = r455855 * r455855;
        double r455857 = r455847 / r455856;
        double r455858 = r455857 / r455855;
        double r455859 = 1.6256606713195871e-15;
        bool r455860 = r455839 <= r455859;
        double r455861 = r455843 * r455845;
        double r455862 = r455861 / r455839;
        double r455863 = r455842 + r455862;
        double r455864 = 1.0;
        double r455865 = r455843 * r455851;
        double r455866 = r455839 / r455865;
        double r455867 = r455864 / r455866;
        double r455868 = r455850 + r455867;
        double r455869 = r455863 / r455868;
        double r455870 = r455839 / r455845;
        double r455871 = r455843 / r455870;
        double r455872 = r455842 + r455871;
        double r455873 = r455854 / r455872;
        double r455874 = r455864 / r455873;
        double r455875 = r455860 ? r455869 : r455874;
        double r455876 = r455841 ? r455858 : r455875;
        return r455876;
}

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

Target

Original	16.9
Target	13.1
Herbie	13.5

\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.697433997227183e+75

   1. Initial program 11.6

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.6

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

   4. Applied times-frac 8.3

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

   5. Simplified 8.3

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

   7. Applied associate-/l* 2.5

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

   9. Applied associate-/r/ 2.9

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

   11. Applied add-cube-cbrt 3.4

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

   12. Applied associate-/r* 3.4

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

   if -2.697433997227183e+75 < t < 1.6256606713195871e-15

   1. Initial program 21.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied clear-num 21.4

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

   if 1.6256606713195871e-15 < t

   1. Initial program 11.8

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.8

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

   4. Applied times-frac 9.2

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

   5. Simplified 9.2

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

   7. Applied associate-/l* 4.7

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

   9. Applied clear-num 5.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + y \cdot \frac{b}{t}}{x + \frac{y}{\frac{t}{z}}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.697433997227182860846341323864432125566 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{x + \frac{y}{t} \cdot z}{\sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}}}}{\sqrt[3]{\left(a + 1\right) + y \cdot \frac{b}{t}}}\\ \mathbf{elif}\;t \le 1.625660671319587146448715347668017899923 \cdot 10^{-15}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + y \cdot \frac{b}{t}}{x + \frac{y}{\frac{t}{z}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))