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

Average Error: 6.3 → 0.9

Time: 1.5s

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 r2182546 = x;
        double r2182547 = y;
        double r2182548 = r2182546 * r2182547;
        double r2182549 = z;
        double r2182550 = r2182548 / r2182549;
        return r2182550;
}

↓

double f(double x, double y, double z) {
        double r2182551 = x;
        double r2182552 = y;
        double r2182553 = r2182551 * r2182552;
        double r2182554 = z;
        double r2182555 = r2182553 / r2182554;
        double r2182556 = -8.66942912235429e+277;
        bool r2182557 = r2182555 <= r2182556;
        double r2182558 = r2182554 / r2182552;
        double r2182559 = r2182551 / r2182558;
        double r2182560 = -5.97338617635297e-306;
        bool r2182561 = r2182555 <= r2182560;
        double r2182562 = 0.0;
        bool r2182563 = r2182555 <= r2182562;
        double r2182564 = r2182551 / r2182554;
        double r2182565 = r2182564 * r2182552;
        double r2182566 = 2.471842223307634e+273;
        bool r2182567 = r2182555 <= r2182566;
        double r2182568 = r2182567 ? r2182555 : r2182559;
        double r2182569 = r2182563 ? r2182565 : r2182568;
        double r2182570 = r2182561 ? r2182555 : r2182569;
        double r2182571 = r2182557 ? r2182559 : r2182570;
        return r2182571;
}

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 +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))