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

Average Error: 16.0 → 12.5

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.2812855953559667 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{t} \cdot z + x}{\left(1.0 + b \cdot \frac{y}{t}\right) + a}\\ \mathbf{elif}\;t \le 6.1634254372012 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}} \cdot \left(\frac{z \cdot y}{t} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} \cdot z + x}{\left(1.0 + b \cdot \frac{y}{t}\right) + a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r36382832 = x;
        double r36382833 = y;
        double r36382834 = z;
        double r36382835 = r36382833 * r36382834;
        double r36382836 = t;
        double r36382837 = r36382835 / r36382836;
        double r36382838 = r36382832 + r36382837;
        double r36382839 = a;
        double r36382840 = 1.0;
        double r36382841 = r36382839 + r36382840;
        double r36382842 = b;
        double r36382843 = r36382833 * r36382842;
        double r36382844 = r36382843 / r36382836;
        double r36382845 = r36382841 + r36382844;
        double r36382846 = r36382838 / r36382845;
        return r36382846;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r36382847 = t;
        double r36382848 = -2.2812855953559667e-25;
        bool r36382849 = r36382847 <= r36382848;
        double r36382850 = y;
        double r36382851 = r36382850 / r36382847;
        double r36382852 = z;
        double r36382853 = r36382851 * r36382852;
        double r36382854 = x;
        double r36382855 = r36382853 + r36382854;
        double r36382856 = 1.0;
        double r36382857 = b;
        double r36382858 = r36382857 * r36382851;
        double r36382859 = r36382856 + r36382858;
        double r36382860 = a;
        double r36382861 = r36382859 + r36382860;
        double r36382862 = r36382855 / r36382861;
        double r36382863 = 6.1634254372012e-68;
        bool r36382864 = r36382847 <= r36382863;
        double r36382865 = 1.0;
        double r36382866 = r36382860 + r36382856;
        double r36382867 = r36382850 * r36382857;
        double r36382868 = r36382867 / r36382847;
        double r36382869 = r36382866 + r36382868;
        double r36382870 = r36382865 / r36382869;
        double r36382871 = r36382852 * r36382850;
        double r36382872 = r36382871 / r36382847;
        double r36382873 = r36382872 + r36382854;
        double r36382874 = r36382870 * r36382873;
        double r36382875 = r36382864 ? r36382874 : r36382862;
        double r36382876 = r36382849 ? r36382862 : r36382875;
        return r36382876;
}

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	16.0
Target	13.2
Herbie	12.5

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

Derivation

1. Split input into 2 regimes

2. if t < -2.2812855953559667e-25 or 6.1634254372012e-68 < t

   1. Initial program 10.9

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

   3. Applied associate-/l* 8.3

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

   5. Applied *-un-lft-identity 8.3

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

   6. Applied *-un-lft-identity 8.3

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

   7. Applied times-frac 8.3

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

   8. Simplified 8.3

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

   9. Simplified 5.0

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

   if -2.2812855953559667e-25 < t < 6.1634254372012e-68

   1. Initial program 23.5

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

   3. Applied div-inv 23.5

      \[\leadsto \color{blue}{\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2812855953559667 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{t} \cdot z + x}{\left(1.0 + b \cdot \frac{y}{t}\right) + a}\\ \mathbf{elif}\;t \le 6.1634254372012 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\left(a + 1.0\right) + \frac{y \cdot b}{t}} \cdot \left(\frac{z \cdot y}{t} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} \cdot z + x}{\left(1.0 + b \cdot \frac{y}{t}\right) + a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))))