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

Average Error: 5.8 → 2.9

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.0212306242717315 \cdot 10^{115}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -2.08794536631012826 \cdot 10^{-85}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r714093 = x;
        double r714094 = y;
        double r714095 = r714093 * r714094;
        double r714096 = z;
        double r714097 = r714095 / r714096;
        return r714097;
}

↓

double f(double x, double y, double z) {
        double r714098 = x;
        double r714099 = y;
        double r714100 = r714098 * r714099;
        double r714101 = -2.0212306242717315e+115;
        bool r714102 = r714100 <= r714101;
        double r714103 = z;
        double r714104 = r714099 / r714103;
        double r714105 = r714098 * r714104;
        double r714106 = -2.0879453663101283e-85;
        bool r714107 = r714100 <= r714106;
        double r714108 = 1.0;
        double r714109 = r714108 / r714103;
        double r714110 = r714100 * r714109;
        double r714111 = 0.0;
        bool r714112 = r714100 <= r714111;
        double r714113 = r714098 / r714103;
        double r714114 = r714113 * r714099;
        double r714115 = r714112 ? r714114 : r714110;
        double r714116 = r714107 ? r714110 : r714115;
        double r714117 = r714102 ? r714105 : r714116;
        return r714117;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	5.9
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -2.0212306242717315e+115

   1. Initial program 13.5

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

   3. Applied *-un-lft-identity 13.5

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

   4. Applied times-frac 3.0

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

   5. Simplified 3.0

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

   if -2.0212306242717315e+115 < (* x y) < -2.0879453663101283e-85 or 0.0 < (* x y)

   1. Initial program 3.1

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

   3. Applied div-inv 3.1

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

   if -2.0879453663101283e-85 < (* x y) < 0.0

   1. Initial program 9.0

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

   3. Applied associate-/l* 1.9

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

   5. Applied associate-/r/ 2.2

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.0212306242717315 \cdot 10^{115}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -2.08794536631012826 \cdot 10^{-85}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))