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

Average Error: 6.3 → 0.9

Time: 1.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -8.6694291223542898 \cdot 10^{277}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -5.97338617635296963 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 2.4718422233076342 \cdot 10^{273}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r717698 = x;
        double r717699 = y;
        double r717700 = r717698 * r717699;
        double r717701 = z;
        double r717702 = r717700 / r717701;
        return r717702;
}

↓

double f(double x, double y, double z) {
        double r717703 = x;
        double r717704 = y;
        double r717705 = r717703 * r717704;
        double r717706 = z;
        double r717707 = r717705 / r717706;
        double r717708 = -8.66942912235429e+277;
        bool r717709 = r717707 <= r717708;
        double r717710 = r717706 / r717704;
        double r717711 = r717703 / r717710;
        double r717712 = -5.97338617635297e-306;
        bool r717713 = r717707 <= r717712;
        double r717714 = 0.0;
        bool r717715 = r717707 <= r717714;
        double r717716 = r717703 / r717706;
        double r717717 = r717716 * r717704;
        double r717718 = 2.471842223307634e+273;
        bool r717719 = r717707 <= r717718;
        double r717720 = r717719 ? r717707 : r717711;
        double r717721 = r717715 ? r717717 : r717720;
        double r717722 = r717713 ? r717707 : r717721;
        double r717723 = r717709 ? r717711 : r717722;
        return r717723;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.4
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -8.66942912235429e+277 or 2.471842223307634e+273 < (/ (* x y) z)

   1. Initial program 43.0

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

   3. Applied associate-/l* 6.1

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

   if -8.66942912235429e+277 < (/ (* x y) z) < -5.97338617635297e-306 or 0.0 < (/ (* x y) z) < 2.471842223307634e+273

   1. Initial program 0.5

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

   3. Applied associate-/l* 8.0

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

   4. Taylor expanded around 0 0.5

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

   if -5.97338617635297e-306 < (/ (* x y) z) < 0.0

   1. Initial program 11.0

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

   3. Applied associate-/l* 0.3

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

   5. Applied associate-/r/ 0.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -8.6694291223542898 \cdot 10^{277}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -5.97338617635296963 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 2.4718422233076342 \cdot 10^{273}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))