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

Average Error: 16.2 → 15.0

Time: 15.6s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.8379120371637104 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{z}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} + x\right) \cdot \frac{1}{\frac{b}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} + \left(a + 1.0\right)}\\ \mathbf{elif}\;t \le -1.0344055968847788 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(a - 1.0\right) \cdot t\right) \cdot \frac{x + \frac{y \cdot z}{t}}{\left(b \cdot y\right) \cdot \left(a - 1.0\right) + t \cdot \left(\left(a + 1.0\right) \cdot \left(a - 1.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}} \cdot \left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}}\right)}{\frac{b}{\sqrt[3]{\sqrt[3]{t}}} \cdot \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} + \left(a + 1.0\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r13325686 = x;
        double r13325687 = y;
        double r13325688 = z;
        double r13325689 = r13325687 * r13325688;
        double r13325690 = t;
        double r13325691 = r13325689 / r13325690;
        double r13325692 = r13325686 + r13325691;
        double r13325693 = a;
        double r13325694 = 1.0;
        double r13325695 = r13325693 + r13325694;
        double r13325696 = b;
        double r13325697 = r13325687 * r13325696;
        double r13325698 = r13325697 / r13325690;
        double r13325699 = r13325695 + r13325698;
        double r13325700 = r13325692 / r13325699;
        return r13325700;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r13325701 = t;
        double r13325702 = -3.8379120371637104e-143;
        bool r13325703 = r13325701 <= r13325702;
        double r13325704 = z;
        double r13325705 = cbrt(r13325701);
        double r13325706 = r13325704 / r13325705;
        double r13325707 = y;
        double r13325708 = r13325705 * r13325705;
        double r13325709 = r13325707 / r13325708;
        double r13325710 = r13325706 * r13325709;
        double r13325711 = x;
        double r13325712 = r13325710 + r13325711;
        double r13325713 = 1.0;
        double r13325714 = b;
        double r13325715 = r13325714 / r13325705;
        double r13325716 = r13325715 * r13325709;
        double r13325717 = a;
        double r13325718 = 1.0;
        double r13325719 = r13325717 + r13325718;
        double r13325720 = r13325716 + r13325719;
        double r13325721 = r13325713 / r13325720;
        double r13325722 = r13325712 * r13325721;
        double r13325723 = -1.0344055968847788e-306;
        bool r13325724 = r13325701 <= r13325723;
        double r13325725 = r13325717 - r13325718;
        double r13325726 = r13325725 * r13325701;
        double r13325727 = r13325707 * r13325704;
        double r13325728 = r13325727 / r13325701;
        double r13325729 = r13325711 + r13325728;
        double r13325730 = r13325714 * r13325707;
        double r13325731 = r13325730 * r13325725;
        double r13325732 = r13325719 * r13325725;
        double r13325733 = r13325701 * r13325732;
        double r13325734 = r13325731 + r13325733;
        double r13325735 = r13325729 / r13325734;
        double r13325736 = r13325726 * r13325735;
        double r13325737 = cbrt(r13325704);
        double r13325738 = sqrt(r13325701);
        double r13325739 = cbrt(r13325738);
        double r13325740 = r13325737 / r13325739;
        double r13325741 = r13325737 * r13325737;
        double r13325742 = r13325741 / r13325739;
        double r13325743 = r13325709 * r13325742;
        double r13325744 = r13325740 * r13325743;
        double r13325745 = r13325711 + r13325744;
        double r13325746 = cbrt(r13325705);
        double r13325747 = r13325714 / r13325746;
        double r13325748 = cbrt(r13325708);
        double r13325749 = r13325709 / r13325748;
        double r13325750 = r13325747 * r13325749;
        double r13325751 = r13325750 + r13325719;
        double r13325752 = r13325745 / r13325751;
        double r13325753 = r13325724 ? r13325736 : r13325752;
        double r13325754 = r13325703 ? r13325722 : r13325753;
        return r13325754;
}

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	13.1
Herbie	15.0

\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -3.8379120371637104e-143

   1. Initial program 12.1

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 12.3

      \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

   4. Applied times-frac 10.6

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 10.6

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   7. Applied times-frac 8.0

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
   8. Using strategy rm

   9. Applied div-inv 8.1

      \[\leadsto \color{blue}{\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]

   if -3.8379120371637104e-143 < t < -1.0344055968847788e-306

   1. Initial program 27.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied flip-+ 34.8

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{\frac{a \cdot a - 1.0 \cdot 1.0}{a - 1.0}} + \frac{y \cdot b}{t}}\]

   4. Applied frac-add 34.9

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{\frac{\left(a \cdot a - 1.0 \cdot 1.0\right) \cdot t + \left(a - 1.0\right) \cdot \left(y \cdot b\right)}{\left(a - 1.0\right) \cdot t}}}\]

   5. Applied associate-/r/ 36.9

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{\left(a \cdot a - 1.0 \cdot 1.0\right) \cdot t + \left(a - 1.0\right) \cdot \left(y \cdot b\right)} \cdot \left(\left(a - 1.0\right) \cdot t\right)}\]

   6. Simplified 36.9

      \[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{\left(\left(a + 1.0\right) \cdot \left(a - 1.0\right)\right) \cdot t + \left(y \cdot b\right) \cdot \left(a - 1.0\right)}} \cdot \left(\left(a - 1.0\right) \cdot t\right)\]

   if -1.0344055968847788e-306 < t

   1. Initial program 16.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 16.5

      \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

   4. Applied times-frac 16.3

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 16.3

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

   7. Applied times-frac 14.8

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 14.8

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}}\]

   10. Applied cbrt-prod 14.8

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\color{blue}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}}\]

   11. Applied *-un-lft-identity 14.8

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{1 \cdot b}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\]

   12. Applied times-frac 14.9

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}\right)}}\]

   13. Applied associate-*r* 14.8

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \color{blue}{\left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}}\]

   14. Simplified 14.8

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \color{blue}{\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]
   15. Using strategy rm

   16. Applied add-sqr-sqrt 14.9

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}}}{\left(a + 1.0\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]

   17. Applied cbrt-prod 15.0

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\color{blue}{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}}}{\left(a + 1.0\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]

   18. Applied add-cube-cbrt 15.0

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt[3]{\sqrt{t}} \cdot \sqrt[3]{\sqrt{t}}}}{\left(a + 1.0\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]

   19. Applied times-frac 15.0

       \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}}\right)}}{\left(a + 1.0\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]

   20. Applied associate-*r* 14.3

       \[\leadsto \frac{x + \color{blue}{\left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}}}}{\left(a + 1.0\right) + \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{b}{\sqrt[3]{\sqrt[3]{t}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.8379120371637104 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{z}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} + x\right) \cdot \frac{1}{\frac{b}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} + \left(a + 1.0\right)}\\ \mathbf{elif}\;t \le -1.0344055968847788 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(a - 1.0\right) \cdot t\right) \cdot \frac{x + \frac{y \cdot z}{t}}{\left(b \cdot y\right) \cdot \left(a - 1.0\right) + t \cdot \left(\left(a + 1.0\right) \cdot \left(a - 1.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}} \cdot \left(\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{t}}}\right)}{\frac{b}{\sqrt[3]{\sqrt[3]{t}}} \cdot \frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} + \left(a + 1.0\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))