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

Average Error: 6.5 → 1.6

Time: 12.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x \cdot y}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.6207759867400651 \cdot 10^{260}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -5.6900917550780378 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.3629336611497376 \cdot 10^{104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r887999 = x;
        double r888000 = y;
        double r888001 = r887999 * r888000;
        double r888002 = z;
        double r888003 = r888001 / r888002;
        return r888003;
}

↓

double f(double x, double y, double z) {
        double r888004 = x;
        double r888005 = y;
        double r888006 = r888004 * r888005;
        double r888007 = -1.6207759867400651e+260;
        bool r888008 = r888006 <= r888007;
        double r888009 = z;
        double r888010 = r888005 / r888009;
        double r888011 = r888004 * r888010;
        double r888012 = -5.690091755078038e-61;
        bool r888013 = r888006 <= r888012;
        double r888014 = r888006 / r888009;
        double r888015 = -0.0;
        bool r888016 = r888006 <= r888015;
        double r888017 = 4.3629336611497376e+104;
        bool r888018 = r888006 <= r888017;
        double r888019 = r888009 / r888005;
        double r888020 = r888004 / r888019;
        double r888021 = r888018 ? r888014 : r888020;
        double r888022 = r888016 ? r888011 : r888021;
        double r888023 = r888013 ? r888014 : r888022;
        double r888024 = r888008 ? r888011 : r888023;
        return r888024;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.4
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x y) < -1.6207759867400651e+260 or -5.690091755078038e-61 < (* x y) < -0.0

   1. Initial program 13.1

      \[\frac{x \cdot y}{z}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 13.1

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot z}}\]

   4. Applied times-frac 2.7

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\]

   5. Simplified 2.7

      \[\leadsto \color{blue}{x} \cdot \frac{y}{z}\]

   if -1.6207759867400651e+260 < (* x y) < -5.690091755078038e-61 or -0.0 < (* x y) < 4.3629336611497376e+104

   1. Initial program 0.4

      \[\frac{x \cdot y}{z}\]

   if 4.3629336611497376e+104 < (* x y)

   1. Initial program 14.3

      \[\frac{x \cdot y}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 3.9

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.6207759867400651 \cdot 10^{260}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -5.6900917550780378 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.3629336611497376 \cdot 10^{104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))