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

Average Error: 16.6 → 13.5

Time: 21.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.488080714023587319262296678365557292608 \cdot 10^{-72}:\\ \;\;\;\;\frac{x + \frac{y}{t} \cdot z}{\left(\frac{b}{t} \cdot y + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{1 + \left(a + \frac{b \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r493842 = x;
        double r493843 = y;
        double r493844 = z;
        double r493845 = r493843 * r493844;
        double r493846 = t;
        double r493847 = r493845 / r493846;
        double r493848 = r493842 + r493847;
        double r493849 = a;
        double r493850 = 1.0;
        double r493851 = r493849 + r493850;
        double r493852 = b;
        double r493853 = r493843 * r493852;
        double r493854 = r493853 / r493846;
        double r493855 = r493851 + r493854;
        double r493856 = r493848 / r493855;
        return r493856;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r493857 = t;
        double r493858 = -1.4880807140235873e-72;
        bool r493859 = r493857 <= r493858;
        double r493860 = x;
        double r493861 = y;
        double r493862 = r493861 / r493857;
        double r493863 = z;
        double r493864 = r493862 * r493863;
        double r493865 = r493860 + r493864;
        double r493866 = b;
        double r493867 = r493866 / r493857;
        double r493868 = r493867 * r493861;
        double r493869 = a;
        double r493870 = r493868 + r493869;
        double r493871 = 1.0;
        double r493872 = r493870 + r493871;
        double r493873 = r493865 / r493872;
        double r493874 = cbrt(r493861);
        double r493875 = r493863 * r493874;
        double r493876 = cbrt(r493857);
        double r493877 = r493875 / r493876;
        double r493878 = r493874 * r493874;
        double r493879 = r493876 * r493876;
        double r493880 = r493878 / r493879;
        double r493881 = r493877 * r493880;
        double r493882 = r493860 + r493881;
        double r493883 = r493866 * r493874;
        double r493884 = r493883 / r493876;
        double r493885 = r493884 * r493880;
        double r493886 = r493869 + r493885;
        double r493887 = r493871 + r493886;
        double r493888 = r493882 / r493887;
        double r493889 = r493859 ? r493873 : r493888;
        return r493889;
}

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	16.6
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 2 regimes

2. if t < -1.4880807140235873e-72

   1. Initial program 11.7

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

   2. Simplified 5.8

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

   4. Applied div-inv 5.8

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

   5. Applied associate-*l* 6.0

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

   6. Simplified 6.0

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

   if -1.4880807140235873e-72 < t

   1. Initial program 18.8

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

   2. Simplified 18.3

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

   4. Applied add-cube-cbrt 18.5

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

   5. Applied add-cube-cbrt 18.5

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

   6. Applied times-frac 18.5

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

   7. Applied associate-*l* 18.5

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

   8. Simplified 18.9

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

   10. Applied add-cube-cbrt 19.0

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

   11. Applied add-cube-cbrt 19.0

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

   12. Applied times-frac 19.0

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

   13. Applied associate-*l* 16.4

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

   14. Simplified 16.9

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

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.488080714023587319262296678365557292608 \cdot 10^{-72}:\\ \;\;\;\;\frac{x + \frac{y}{t} \cdot z}{\left(\frac{b}{t} \cdot y + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{1 + \left(a + \frac{b \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

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