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

Average Error: 6.3 → 0.4

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.183698632864184 \cdot 10^{+230}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -9.618042628314796 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.7638753128812204 \cdot 10^{-234}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.5348271514093593 \cdot 10^{+241}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -4.183698632864184e+230)
   (* x (/ y z))
   (if (<= (* x y) -9.618042628314796e-127)
     (/ (* x y) z)
     (if (<= (* x y) 1.7638753128812204e-234)
       (* x (/ y z))
       (if (<= (* x y) 1.5348271514093593e+241)
         (/ (* x y) z)
         (/ x (/ z y)))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((((double) (x * y)) <= -4.183698632864184e+230)) {
		tmp = ((double) (x * (y / z)));
	} else {
		double tmp_1;
		if ((((double) (x * y)) <= -9.618042628314796e-127)) {
			tmp_1 = (((double) (x * y)) / z);
		} else {
			double tmp_2;
			if ((((double) (x * y)) <= 1.7638753128812204e-234)) {
				tmp_2 = ((double) (x * (y / z)));
			} else {
				double tmp_3;
				if ((((double) (x * y)) <= 1.5348271514093593e+241)) {
					tmp_3 = (((double) (x * y)) / z);
				} else {
					tmp_3 = (x / (z / y));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	5.8
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.18369863286418431e230 or -9.6180426283147962e-127 < (*.f64 x y) < 1.7638753128812204e-234

   1. Initial program 12.4

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

   3. Applied *-un-lft-identity_binary64 12.4

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

   4. Applied times-frac_binary64 0.7

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

   5. Simplified 0.7

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

   if -4.18369863286418431e230 < (*.f64 x y) < -9.6180426283147962e-127 or 1.7638753128812204e-234 < (*.f64 x y) < 1.53482715140935926e241

   1. Initial program 0.3

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

   if 1.53482715140935926e241 < (*.f64 x y)

   1. Initial program 34.1

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

   3. Applied associate-/l*_binary64 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.183698632864184 \cdot 10^{+230}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -9.618042628314796 \cdot 10^{-127}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.7638753128812204 \cdot 10^{-234}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.5348271514093593 \cdot 10^{+241}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))