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

Average Error: 6.4 → 2.0

Time: 2.1s

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 r730962 = x;
        double r730963 = y;
        double r730964 = r730962 * r730963;
        double r730965 = z;
        double r730966 = r730964 / r730965;
        return r730966;
}

↓

double f(double x, double y, double z) {
        double r730967 = x;
        double r730968 = y;
        double r730969 = r730967 * r730968;
        double r730970 = z;
        double r730971 = r730969 / r730970;
        double r730972 = -2.6698584708828636e+271;
        bool r730973 = r730971 <= r730972;
        double r730974 = r730970 / r730968;
        double r730975 = r730967 / r730974;
        double r730976 = -2.2813575988703053e-294;
        bool r730977 = r730971 <= r730976;
        double r730978 = 4.1228643825387546e-275;
        bool r730979 = r730971 <= r730978;
        double r730980 = r730967 / r730970;
        double r730981 = r730980 * r730968;
        double r730982 = 2.204649036140942e+173;
        bool r730983 = r730971 <= r730982;
        double r730984 = r730983 ? r730971 : r730975;
        double r730985 = r730979 ? r730981 : r730984;
        double r730986 = r730977 ? r730971 : r730985;
        double r730987 = r730973 ? r730975 : r730986;
        return r730987;
}

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 +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))