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

Average Error: 6.4 → 6.4

Time: 6.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r659811 = x;
        double r659812 = y;
        double r659813 = r659811 * r659812;
        double r659814 = z;
        double r659815 = r659813 / r659814;
        return r659815;
}

↓

double f(double x, double y, double z) {
        double r659816 = x;
        double r659817 = y;
        double r659818 = z;
        double r659819 = r659817 / r659818;
        double r659820 = r659816 * r659819;
        return r659820;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.5
Herbie	6.4

\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ (* x y) z) < -7.76860356189378e+305

   1. Initial program 62.6

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

   3. Applied *-un-lft-identity 62.6

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

   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 -7.76860356189378e+305 < (/ (* x y) z) < -5.989061428235972e-77 or -0.0 < (/ (* x y) z) < 3.534109248982644e+298

   1. Initial program 2.2

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

   if -5.989061428235972e-77 < (/ (* x y) z) < -0.0

   1. Initial program 5.7

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

   3. Applied associate-/l* 6.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 6.4

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

   6. Applied add-cube-cbrt 7.0

      \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot y}}\]

   7. Applied times-frac 7.0

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}}}\]

   8. Applied associate-/r* 5.3

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}}}{\frac{\sqrt[3]{z}}{y}}}\]

   9. Simplified 5.3

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{\sqrt[3]{z}}{y}}\]

   if 3.534109248982644e+298 < (/ (* x y) z)

   1. Initial program 57.5

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

   3. Applied associate-/l* 1.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019303 
(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 1.70421306606504721e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))