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

Average Error: 16.9 → 14.4

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.3394468926366395 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le -9.53108059989476 \cdot 10^{-269}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{\frac{y}{\sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r684643 = x;
        double r684644 = y;
        double r684645 = z;
        double r684646 = r684644 * r684645;
        double r684647 = t;
        double r684648 = r684646 / r684647;
        double r684649 = r684643 + r684648;
        double r684650 = a;
        double r684651 = 1.0;
        double r684652 = r684650 + r684651;
        double r684653 = b;
        double r684654 = r684644 * r684653;
        double r684655 = r684654 / r684647;
        double r684656 = r684652 + r684655;
        double r684657 = r684649 / r684656;
        return r684657;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r684658 = t;
        double r684659 = -2.3394468926366395e-06;
        bool r684660 = r684658 <= r684659;
        double r684661 = y;
        double r684662 = r684661 / r684658;
        double r684663 = z;
        double r684664 = x;
        double r684665 = fma(r684662, r684663, r684664);
        double r684666 = a;
        double r684667 = 1.0;
        double r684668 = r684666 + r684667;
        double r684669 = b;
        double r684670 = r684669 / r684658;
        double r684671 = r684661 * r684670;
        double r684672 = r684668 + r684671;
        double r684673 = r684665 / r684672;
        double r684674 = -9.53108059989476e-269;
        bool r684675 = r684658 <= r684674;
        double r684676 = r684661 * r684663;
        double r684677 = r684676 / r684658;
        double r684678 = r684664 + r684677;
        double r684679 = 1.0;
        double r684680 = r684661 * r684669;
        double r684681 = r684680 / r684658;
        double r684682 = r684668 + r684681;
        double r684683 = r684679 / r684682;
        double r684684 = r684678 * r684683;
        double r684685 = cbrt(r684658);
        double r684686 = r684661 / r684685;
        double r684687 = r684669 / r684685;
        double r684688 = r684686 * r684687;
        double r684689 = r684688 / r684685;
        double r684690 = r684668 + r684689;
        double r684691 = r684665 / r684690;
        double r684692 = r684675 ? r684684 : r684691;
        double r684693 = r684660 ? r684673 : r684692;
        return r684693;
}

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	16.9
Target	13.4
Herbie	14.4

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

Derivation

1. Split input into 3 regimes

2. if t < -2.3394468926366395e-06

   1. Initial program 11.1

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

   3. Applied *-un-lft-identity 11.1

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

   4. Applied associate-/r* 11.1

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

   5. Simplified 8.3

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

   7. Applied *-un-lft-identity 8.3

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

   8. Applied times-frac 3.9

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

   9. Simplified 3.9

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

   if -2.3394468926366395e-06 < t < -9.53108059989476e-269

   1. Initial program 21.8

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

   3. Applied div-inv 21.8

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

   if -9.53108059989476e-269 < t

   1. Initial program 17.9

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

   3. Applied *-un-lft-identity 17.9

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

   4. Applied associate-/r* 17.9

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

   5. Simplified 18.0

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

   7. Applied add-cube-cbrt 18.1

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

   8. Applied times-frac 16.6

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

   10. Applied associate-*r/ 16.5

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

   11. Simplified 16.9

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

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3394468926366395 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le -9.53108059989476 \cdot 10^{-269}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{\frac{y}{\sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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))))