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

Average Error: 5.8 → 0.3

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.6967839648114804 \cdot 10^{268}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -7.97061052598067788 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 6.0589118027638559 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.29204454885475858 \cdot 10^{212}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r718344 = x;
        double r718345 = y;
        double r718346 = r718344 * r718345;
        double r718347 = z;
        double r718348 = r718346 / r718347;
        return r718348;
}

↓

double f(double x, double y, double z) {
        double r718349 = x;
        double r718350 = y;
        double r718351 = r718349 * r718350;
        double r718352 = -2.6967839648114804e+268;
        bool r718353 = r718351 <= r718352;
        double r718354 = z;
        double r718355 = r718350 / r718354;
        double r718356 = r718349 * r718355;
        double r718357 = -7.970610525980678e-294;
        bool r718358 = r718351 <= r718357;
        double r718359 = r718351 / r718354;
        double r718360 = 6.058911802763856e-195;
        bool r718361 = r718351 <= r718360;
        double r718362 = 1.2920445488547586e+212;
        bool r718363 = r718351 <= r718362;
        double r718364 = r718354 / r718350;
        double r718365 = r718349 / r718364;
        double r718366 = r718363 ? r718359 : r718365;
        double r718367 = r718361 ? r718356 : r718366;
        double r718368 = r718358 ? r718359 : r718367;
        double r718369 = r718353 ? r718356 : r718368;
        return r718369;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	5.9
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -2.6967839648114804e+268 or -7.970610525980678e-294 < (* x y) < 6.058911802763856e-195

   1. Initial program 15.3

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

   3. Applied *-un-lft-identity 15.3

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

   4. Applied times-frac 0.5

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

   5. Simplified 0.5

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

   if -2.6967839648114804e+268 < (* x y) < -7.970610525980678e-294 or 6.058911802763856e-195 < (* x y) < 1.2920445488547586e+212

   1. Initial program 0.2

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

   if 1.2920445488547586e+212 < (* x y)

   1. Initial program 28.8

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

   3. Applied associate-/l* 1.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.6967839648114804 \cdot 10^{268}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -7.97061052598067788 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 6.0589118027638559 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.29204454885475858 \cdot 10^{212}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))