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

Average Error: 6.4 → 2.0

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -2.66985847088286362 \cdot 10^{271}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -2.2813575988703053 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 4.12286438253875463 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 2.2046490361409421 \cdot 10^{173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r641817 = x;
        double r641818 = y;
        double r641819 = r641817 * r641818;
        double r641820 = z;
        double r641821 = r641819 / r641820;
        return r641821;
}

↓

double f(double x, double y, double z) {
        double r641822 = x;
        double r641823 = y;
        double r641824 = r641822 * r641823;
        double r641825 = z;
        double r641826 = r641824 / r641825;
        double r641827 = -2.6698584708828636e+271;
        bool r641828 = r641826 <= r641827;
        double r641829 = r641825 / r641823;
        double r641830 = r641822 / r641829;
        double r641831 = -2.2813575988703053e-294;
        bool r641832 = r641826 <= r641831;
        double r641833 = 4.1228643825387546e-275;
        bool r641834 = r641826 <= r641833;
        double r641835 = r641822 / r641825;
        double r641836 = r641835 * r641823;
        double r641837 = 2.204649036140942e+173;
        bool r641838 = r641826 <= r641837;
        double r641839 = r641838 ? r641826 : r641830;
        double r641840 = r641834 ? r641836 : r641839;
        double r641841 = r641832 ? r641826 : r641840;
        double r641842 = r641828 ? r641830 : r641841;
        return r641842;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.4
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -2.6698584708828636e+271 or 2.204649036140942e+173 < (/ (* x y) z)

   1. Initial program 26.4

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

   3. Applied associate-/l* 10.5

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

   if -2.6698584708828636e+271 < (/ (* x y) z) < -2.2813575988703053e-294 or 4.1228643825387546e-275 < (/ (* x y) z) < 2.204649036140942e+173

   1. Initial program 0.5

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

   if -2.2813575988703053e-294 < (/ (* x y) z) < 4.1228643825387546e-275

   1. Initial program 10.8

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

   3. Applied associate-/l* 1.5

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

   5. Applied associate-/r/ 1.6

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -2.66985847088286362 \cdot 10^{271}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -2.2813575988703053 \cdot 10^{-294}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 4.12286438253875463 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 2.2046490361409421 \cdot 10^{173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))