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

Average Error: 5.8 → 0.3

Time: 4.0s

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 r897495 = x;
        double r897496 = y;
        double r897497 = r897495 * r897496;
        double r897498 = z;
        double r897499 = r897497 / r897498;
        return r897499;
}

↓

double f(double x, double y, double z) {
        double r897500 = x;
        double r897501 = y;
        double r897502 = r897500 * r897501;
        double r897503 = -2.6967839648114804e+268;
        bool r897504 = r897502 <= r897503;
        double r897505 = z;
        double r897506 = r897501 / r897505;
        double r897507 = r897500 * r897506;
        double r897508 = -7.970610525980678e-294;
        bool r897509 = r897502 <= r897508;
        double r897510 = r897502 / r897505;
        double r897511 = 6.058911802763856e-195;
        bool r897512 = r897502 <= r897511;
        double r897513 = 1.2920445488547586e+212;
        bool r897514 = r897502 <= r897513;
        double r897515 = r897505 / r897501;
        double r897516 = r897500 / r897515;
        double r897517 = r897514 ? r897510 : r897516;
        double r897518 = r897512 ? r897507 : r897517;
        double r897519 = r897509 ? r897510 : r897518;
        double r897520 = r897504 ? r897507 : r897519;
        return r897520;
}

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 
(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))