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

Average Error: 16.2 → 13.3

Time: 9.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r249827 = x;
        double r249828 = y;
        double r249829 = z;
        double r249830 = r249828 * r249829;
        double r249831 = t;
        double r249832 = r249830 / r249831;
        double r249833 = r249827 + r249832;
        double r249834 = a;
        double r249835 = 1.0;
        double r249836 = r249834 + r249835;
        double r249837 = b;
        double r249838 = r249828 * r249837;
        double r249839 = r249838 / r249831;
        double r249840 = r249836 + r249839;
        double r249841 = r249833 / r249840;
        return r249841;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r249842 = y;
        double r249843 = -6.3581812405522e-131;
        bool r249844 = r249842 <= r249843;
        double r249845 = 7.593646484466209e-92;
        bool r249846 = r249842 <= r249845;
        double r249847 = !r249846;
        bool r249848 = r249844 || r249847;
        double r249849 = x;
        double r249850 = z;
        double r249851 = t;
        double r249852 = r249850 / r249851;
        double r249853 = r249842 * r249852;
        double r249854 = r249849 + r249853;
        double r249855 = a;
        double r249856 = 1.0;
        double r249857 = r249855 + r249856;
        double r249858 = b;
        double r249859 = r249858 / r249851;
        double r249860 = r249842 * r249859;
        double r249861 = r249857 + r249860;
        double r249862 = r249854 / r249861;
        double r249863 = r249842 * r249850;
        double r249864 = r249863 / r249851;
        double r249865 = r249849 + r249864;
        double r249866 = r249842 * r249858;
        double r249867 = cbrt(r249851);
        double r249868 = r249867 * r249867;
        double r249869 = r249866 / r249868;
        double r249870 = r249869 / r249867;
        double r249871 = r249857 + r249870;
        double r249872 = r249865 / r249871;
        double r249873 = r249848 ? r249862 : r249872;
        return r249873;
}

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	16.2
Target	13.2
Herbie	13.3

\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -6.3581812405522e-131 or 7.593646484466209e-92 < y

   1. Initial program 23.2

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

   3. Applied *-un-lft-identity 23.2

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

   4. Applied times-frac 21.7

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

   5. Simplified 21.7

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

   7. Applied *-un-lft-identity 21.7

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

   8. Applied times-frac 18.8

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

   9. Simplified 18.8

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

   if -6.3581812405522e-131 < y < 7.593646484466209e-92

   1. Initial program 2.1

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

   3. Applied add-cube-cbrt 2.2

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

   4. Applied associate-/r* 2.2

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

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.3581812405522004 \cdot 10^{-131} \lor \neg \left(y \le 7.59364648446620877 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))