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

Average Error: 17.0 → 12.6

Time: 22.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.186697388398434257372885015533230598923 \cdot 10^{-48}:\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{\left(\frac{y}{\frac{t}{b}} + 1\right) + a}\\ \mathbf{elif}\;t \le 126368.2961535421491134911775588989257812:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(y \cdot b\right) \cdot \frac{1}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}{\left(a + 1\right) + \left(\frac{b}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r39120719 = x;
        double r39120720 = y;
        double r39120721 = z;
        double r39120722 = r39120720 * r39120721;
        double r39120723 = t;
        double r39120724 = r39120722 / r39120723;
        double r39120725 = r39120719 + r39120724;
        double r39120726 = a;
        double r39120727 = 1.0;
        double r39120728 = r39120726 + r39120727;
        double r39120729 = b;
        double r39120730 = r39120720 * r39120729;
        double r39120731 = r39120730 / r39120723;
        double r39120732 = r39120728 + r39120731;
        double r39120733 = r39120725 / r39120732;
        return r39120733;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r39120734 = t;
        double r39120735 = -4.186697388398434e-48;
        bool r39120736 = r39120734 <= r39120735;
        double r39120737 = y;
        double r39120738 = z;
        double r39120739 = r39120738 / r39120734;
        double r39120740 = r39120737 * r39120739;
        double r39120741 = x;
        double r39120742 = r39120740 + r39120741;
        double r39120743 = b;
        double r39120744 = r39120734 / r39120743;
        double r39120745 = r39120737 / r39120744;
        double r39120746 = 1.0;
        double r39120747 = r39120745 + r39120746;
        double r39120748 = a;
        double r39120749 = r39120747 + r39120748;
        double r39120750 = r39120742 / r39120749;
        double r39120751 = 126368.29615354215;
        bool r39120752 = r39120734 <= r39120751;
        double r39120753 = r39120737 * r39120738;
        double r39120754 = r39120753 / r39120734;
        double r39120755 = r39120741 + r39120754;
        double r39120756 = r39120737 * r39120743;
        double r39120757 = 1.0;
        double r39120758 = r39120757 / r39120734;
        double r39120759 = r39120756 * r39120758;
        double r39120760 = r39120748 + r39120746;
        double r39120761 = r39120759 + r39120760;
        double r39120762 = r39120755 / r39120761;
        double r39120763 = cbrt(r39120734);
        double r39120764 = r39120738 / r39120763;
        double r39120765 = cbrt(r39120737);
        double r39120766 = r39120765 / r39120763;
        double r39120767 = r39120764 * r39120766;
        double r39120768 = r39120765 * r39120765;
        double r39120769 = r39120768 / r39120763;
        double r39120770 = r39120767 * r39120769;
        double r39120771 = r39120741 + r39120770;
        double r39120772 = r39120743 / r39120763;
        double r39120773 = r39120772 * r39120766;
        double r39120774 = r39120773 * r39120769;
        double r39120775 = r39120760 + r39120774;
        double r39120776 = r39120771 / r39120775;
        double r39120777 = r39120752 ? r39120762 : r39120776;
        double r39120778 = r39120736 ? r39120750 : r39120777;
        return r39120778;
}

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	17.0
Target	13.5
Herbie	12.6

\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -4.186697388398434e-48

   1. Initial program 11.7

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

   3. Applied add-cube-cbrt 11.8

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

   4. Applied times-frac 8.7

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

   6. Applied add-cube-cbrt 8.7

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

   7. Applied times-frac 5.2

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

   9. Applied *-un-lft-identity 5.2

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

   10. Applied *-un-lft-identity 5.2

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

   11. Applied times-frac 5.2

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

   12. Simplified 5.2

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

   13. Simplified 4.9

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

   if -4.186697388398434e-48 < t < 126368.29615354215

   1. Initial program 22.5

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

   3. Applied div-inv 22.5

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

   if 126368.29615354215 < t

   1. Initial program 13.6

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

   3. Applied add-cube-cbrt 13.8

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

   4. Applied times-frac 9.5

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

   6. Applied add-cube-cbrt 9.5

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

   7. Applied times-frac 4.5

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

   9. Applied add-cube-cbrt 4.6

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

   10. Applied times-frac 4.6

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

   11. Applied associate-*l* 4.6

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

   13. Applied add-cube-cbrt 4.6

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

   14. Applied times-frac 4.6

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

   15. Applied associate-*l* 4.4

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.186697388398434257372885015533230598923 \cdot 10^{-48}:\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{\left(\frac{y}{\frac{t}{b}} + 1\right) + a}\\ \mathbf{elif}\;t \le 126368.2961535421491134911775588989257812:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(y \cdot b\right) \cdot \frac{1}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}{\left(a + 1\right) + \left(\frac{b}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))