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

Average Error: 6.1 → 0.6

Time: 3.8s

Precision: binary64

\[[x, y]=\mathsf{sort}([x, y])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.901655898454945 \cdot 10^{+221}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -8.727398383638099 \cdot 10^{-254}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2.316744091879154 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 1.8629534670010378 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -3.901655898454945e+221)
   (/ 1.0 (* (/ 1.0 x) (/ z y)))
   (if (<= (* x y) -8.727398383638099e-254)
     (/ (* x y) z)
     (if (<= (* x y) 2.316744091879154e-131)
       (/ x (/ z y))
       (if (<= (* x y) 1.8629534670010378e+225)
         (/ (* x y) z)
         (/ 1.0 (* (/ 1.0 x) (/ z y))))))))

C

double code(double x, double y, double z) {
	return (x * y) / z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -3.901655898454945e+221) {
		tmp = 1.0 / ((1.0 / x) * (z / y));
	} else if ((x * y) <= -8.727398383638099e-254) {
		tmp = (x * y) / z;
	} else if ((x * y) <= 2.316744091879154e-131) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1.8629534670010378e+225) {
		tmp = (x * y) / z;
	} else {
		tmp = 1.0 / ((1.0 / x) * (z / y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.90165589845494465e221 or 1.8629534670010378e225 < (*.f64 x y)

   1. Initial program 31.5

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

   3. Applied clear-num_binary64_17809 31.6

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity_binary64_17810 31.6

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

   6. Applied times-frac_binary64_17816 1.3

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

   if -3.90165589845494465e221 < (*.f64 x y) < -8.72739838363809855e-254 or 2.31674409187915405e-131 < (*.f64 x y) < 1.8629534670010378e225

   1. Initial program 0.2

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

   if -8.72739838363809855e-254 < (*.f64 x y) < 2.31674409187915405e-131

   1. Initial program 9.4

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

   3. Applied associate-/l*_binary64_17755 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.901655898454945 \cdot 10^{+221}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -8.727398383638099 \cdot 10^{-254}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2.316744091879154 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 1.8629534670010378 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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))