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

Average Error: 6.5 → 0.4

Time: 16.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -8.262890138825605465511184669818653865175 \cdot 10^{-174} \lor \neg \left(x \cdot y \le 1.775239536769126579734755404861004766773 \cdot 10^{-219}\right) \land x \cdot y \le 4.654103577641758604718829078963007975589 \cdot 10^{206}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r460175 = x;
        double r460176 = y;
        double r460177 = r460175 * r460176;
        double r460178 = z;
        double r460179 = r460177 / r460178;
        return r460179;
}

↓

double f(double x, double y, double z) {
        double r460180 = x;
        double r460181 = y;
        double r460182 = r460180 * r460181;
        double r460183 = -inf.0;
        bool r460184 = r460182 <= r460183;
        double r460185 = z;
        double r460186 = r460181 / r460185;
        double r460187 = r460180 * r460186;
        double r460188 = -8.262890138825605e-174;
        bool r460189 = r460182 <= r460188;
        double r460190 = 1.7752395367691266e-219;
        bool r460191 = r460182 <= r460190;
        double r460192 = !r460191;
        double r460193 = 4.6541035776417586e+206;
        bool r460194 = r460182 <= r460193;
        bool r460195 = r460192 && r460194;
        bool r460196 = r460189 || r460195;
        double r460197 = r460182 / r460185;
        double r460198 = r460180 / r460185;
        double r460199 = r460198 * r460181;
        double r460200 = r460196 ? r460197 : r460199;
        double r460201 = r460184 ? r460187 : r460200;
        return r460201;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.5
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x y) < -inf.0

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied times-frac 0.3

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

   5. Simplified 0.3

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

   if -inf.0 < (* x y) < -8.262890138825605e-174 or 1.7752395367691266e-219 < (* x y) < 4.6541035776417586e+206

   1. Initial program 0.2

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

   if -8.262890138825605e-174 < (* x y) < 1.7752395367691266e-219 or 4.6541035776417586e+206 < (* x y)

   1. Initial program 13.4

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

   3. Applied associate-/l* 0.8

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 0.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -8.262890138825605465511184669818653865175 \cdot 10^{-174} \lor \neg \left(x \cdot y \le 1.775239536769126579734755404861004766773 \cdot 10^{-219}\right) \land x \cdot y \le 4.654103577641758604718829078963007975589 \cdot 10^{206}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))