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

Average Error: 16.6 → 13.0

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0217902711402013 \cdot 10^{-15} \lor \neg \left(t \le 8.50855217814580292 \cdot 10^{98}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r713730 = x;
        double r713731 = y;
        double r713732 = z;
        double r713733 = r713731 * r713732;
        double r713734 = t;
        double r713735 = r713733 / r713734;
        double r713736 = r713730 + r713735;
        double r713737 = a;
        double r713738 = 1.0;
        double r713739 = r713737 + r713738;
        double r713740 = b;
        double r713741 = r713731 * r713740;
        double r713742 = r713741 / r713734;
        double r713743 = r713739 + r713742;
        double r713744 = r713736 / r713743;
        return r713744;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r713745 = t;
        double r713746 = -1.0217902711402013e-15;
        bool r713747 = r713745 <= r713746;
        double r713748 = 8.508552178145803e+98;
        bool r713749 = r713745 <= r713748;
        double r713750 = !r713749;
        bool r713751 = r713747 || r713750;
        double r713752 = x;
        double r713753 = y;
        double r713754 = z;
        double r713755 = r713754 / r713745;
        double r713756 = r713753 * r713755;
        double r713757 = r713752 + r713756;
        double r713758 = a;
        double r713759 = 1.0;
        double r713760 = r713758 + r713759;
        double r713761 = cbrt(r713745);
        double r713762 = r713761 * r713761;
        double r713763 = r713753 / r713762;
        double r713764 = b;
        double r713765 = r713764 / r713761;
        double r713766 = r713763 * r713765;
        double r713767 = r713760 + r713766;
        double r713768 = r713757 / r713767;
        double r713769 = 1.0;
        double r713770 = r713753 * r713754;
        double r713771 = r713745 / r713770;
        double r713772 = r713769 / r713771;
        double r713773 = r713752 + r713772;
        double r713774 = r713753 * r713764;
        double r713775 = r713745 / r713774;
        double r713776 = r713769 / r713775;
        double r713777 = r713760 + r713776;
        double r713778 = r713773 / r713777;
        double r713779 = r713751 ? r713768 : r713778;
        return r713779;
}

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	13.5
Herbie	13.0

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

Derivation

1. Split input into 2 regimes

2. if t < -1.0217902711402013e-15 or 8.508552178145803e+98 < t

   1. Initial program 12.0

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

   3. Applied add-cube-cbrt 12.1

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

   4. Applied times-frac 8.8

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

   6. Applied *-un-lft-identity 8.8

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

   7. Applied times-frac 4.0

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

   8. Simplified 4.0

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

   if -1.0217902711402013e-15 < t < 8.508552178145803e+98

   1. Initial program 20.4

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

   3. Applied clear-num 20.5

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

   5. Applied clear-num 20.5

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0217902711402013 \cdot 10^{-15} \lor \neg \left(t \le 8.50855217814580292 \cdot 10^{98}\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y \cdot z}}}{\left(a + 1\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))))