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

Average Error: 6.5 → 0.4

Time: 19.0s

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 r397514 = x;
        double r397515 = y;
        double r397516 = r397514 * r397515;
        double r397517 = z;
        double r397518 = r397516 / r397517;
        return r397518;
}

↓

double f(double x, double y, double z) {
        double r397519 = x;
        double r397520 = y;
        double r397521 = r397519 * r397520;
        double r397522 = -inf.0;
        bool r397523 = r397521 <= r397522;
        double r397524 = z;
        double r397525 = r397520 / r397524;
        double r397526 = r397519 * r397525;
        double r397527 = -8.262890138825605e-174;
        bool r397528 = r397521 <= r397527;
        double r397529 = 1.7752395367691266e-219;
        bool r397530 = r397521 <= r397529;
        double r397531 = !r397530;
        double r397532 = 4.6541035776417586e+206;
        bool r397533 = r397521 <= r397532;
        bool r397534 = r397531 && r397533;
        bool r397535 = r397528 || r397534;
        double r397536 = r397521 / r397524;
        double r397537 = r397519 / r397524;
        double r397538 = r397537 * r397520;
        double r397539 = r397535 ? r397536 : r397538;
        double r397540 = r397523 ? r397526 : r397539;
        return r397540;
}

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))