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

Average Error: 6.5 → 0.4

Time: 19.4s

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 r549280 = x;
        double r549281 = y;
        double r549282 = r549280 * r549281;
        double r549283 = z;
        double r549284 = r549282 / r549283;
        return r549284;
}

↓

double f(double x, double y, double z) {
        double r549285 = x;
        double r549286 = y;
        double r549287 = r549285 * r549286;
        double r549288 = -inf.0;
        bool r549289 = r549287 <= r549288;
        double r549290 = z;
        double r549291 = r549286 / r549290;
        double r549292 = r549285 * r549291;
        double r549293 = -8.262890138825605e-174;
        bool r549294 = r549287 <= r549293;
        double r549295 = 1.7752395367691266e-219;
        bool r549296 = r549287 <= r549295;
        double r549297 = !r549296;
        double r549298 = 4.6541035776417586e+206;
        bool r549299 = r549287 <= r549298;
        bool r549300 = r549297 && r549299;
        bool r549301 = r549294 || r549300;
        double r549302 = r549287 / r549290;
        double r549303 = r549285 / r549290;
        double r549304 = r549303 * r549286;
        double r549305 = r549301 ? r549302 : r549304;
        double r549306 = r549289 ? r549292 : r549305;
        return r549306;
}

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 +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))