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

Average Error: 6.3 → 0.6

Time: 4.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -7.422106636486913 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 1.1372816739969421 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 8.217236327076306 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= (* x y) (- INFINITY))
     t_0
     (let* ((t_1 (/ (* x y) z)))
       (if (<= (* x y) -7.422106636486913e-286)
         t_1
         (if (<= (* x y) 1.1372816739969421e-180)
           t_0
           (if (<= (* x y) 8.217236327076306e+122) t_1 (* x (/ y z)))))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_0;
	} else {
		double t_1 = (x * y) / z;
		double tmp_1;
		if ((x * y) <= -7.422106636486913e-286) {
			tmp_1 = t_1;
		} else if ((x * y) <= 1.1372816739969421e-180) {
			tmp_1 = t_0;
		} else if ((x * y) <= 8.217236327076306e+122) {
			tmp_1 = t_1;
		} else {
			tmp_1 = x * (y / z);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.3
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) < -inf.0 or -7.42210663648691259e-286 < (*.f64 x y) < 1.13728167399694212e-180

   1. Initial program 16.2

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

   2. Applied associate-/l*_binary64 0.5

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

   3. Applied div-inv_binary64 0.5

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

   4. Applied associate-/r*_binary64 0.6

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

   5. Applied associate-/r/_binary64 0.6

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

   if -inf.0 < (*.f64 x y) < -7.42210663648691259e-286 or 1.13728167399694212e-180 < (*.f64 x y) < 8.21723632707630642e122

   1. Initial program 0.2

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

   if 8.21723632707630642e122 < (*.f64 x y)

   1. Initial program 17.3

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

   2. Applied *-un-lft-identity_binary64 17.3

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

   3. Applied times-frac_binary64 3.3

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

   4. Simplified 3.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -7.422106636486913 \cdot 10^{-286}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.1372816739969421 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 8.217236327076306 \cdot 10^{+122}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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))