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

Average Error: 6.0 → 0.4

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.166521247594354925726439343625696917212 \cdot 10^{205}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.343713460892298755823259752013121029537 \cdot 10^{-191} \lor \neg \left(x \cdot y \le 4.152478566064296352109998791440075706057 \cdot 10^{-204}\right) \land x \cdot y \le 9.404611212472955651642851083785275703569 \cdot 10^{253}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r1180175 = x;
        double r1180176 = y;
        double r1180177 = r1180175 * r1180176;
        double r1180178 = z;
        double r1180179 = r1180177 / r1180178;
        return r1180179;
}

↓

double f(double x, double y, double z) {
        double r1180180 = x;
        double r1180181 = y;
        double r1180182 = r1180180 * r1180181;
        double r1180183 = -2.166521247594355e+205;
        bool r1180184 = r1180182 <= r1180183;
        double r1180185 = z;
        double r1180186 = r1180185 / r1180181;
        double r1180187 = r1180180 / r1180186;
        double r1180188 = -1.3437134608922988e-191;
        bool r1180189 = r1180182 <= r1180188;
        double r1180190 = 4.152478566064296e-204;
        bool r1180191 = r1180182 <= r1180190;
        double r1180192 = !r1180191;
        double r1180193 = 9.404611212472956e+253;
        bool r1180194 = r1180182 <= r1180193;
        bool r1180195 = r1180192 && r1180194;
        bool r1180196 = r1180189 || r1180195;
        double r1180197 = r1180182 / r1180185;
        double r1180198 = r1180181 / r1180185;
        double r1180199 = r1180198 * r1180180;
        double r1180200 = r1180196 ? r1180197 : r1180199;
        double r1180201 = r1180184 ? r1180187 : r1180200;
        return r1180201;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	6.2
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) < -2.166521247594355e+205

   1. Initial program 27.2

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

   3. Applied associate-/l* 1.5

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

   if -2.166521247594355e+205 < (* x y) < -1.3437134608922988e-191 or 4.152478566064296e-204 < (* x y) < 9.404611212472956e+253

   1. Initial program 0.2

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

   if -1.3437134608922988e-191 < (* x y) < 4.152478566064296e-204 or 9.404611212472956e+253 < (* x y)

   1. Initial program 13.8

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

   2. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.166521247594354925726439343625696917212 \cdot 10^{205}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.343713460892298755823259752013121029537 \cdot 10^{-191} \lor \neg \left(x \cdot y \le 4.152478566064296352109998791440075706057 \cdot 10^{-204}\right) \land x \cdot y \le 9.404611212472955651642851083785275703569 \cdot 10^{253}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))