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

Average Error: 10.5 → 2.9

Time: 6.9s

Precision: 64

Math

\[\frac{x - y \cdot z}{t - a \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.622785819064071 \cdot 10^{37} \lor \neg \left(z \le 5.957888797169072 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{t - a \cdot z} \cdot z\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((z <= -2.622785819064071e+37) || !(z <= 5.957888797169072e-85))) {
		VAR = ((double) (((double) (x / ((double) (t - ((double) (a * z)))))) - ((double) (y / ((double) (((double) (t / z)) - a))))));
	} else {
		VAR = ((double) (((double) (x / ((double) (t - ((double) (a * z)))))) - ((double) (((double) (y / ((double) (t - ((double) (a * z)))))) * z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.5
Target	1.8
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.622785819064071e+37 or 5.957888797169072e-85 < z

   1. Initial program 19.1

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

   3. Applied div-sub 19.1

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

   5. Applied associate-/l* 12.3

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

   6. Taylor expanded around 0 2.9

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

   if -2.622785819064071e+37 < z < 5.957888797169072e-85

   1. Initial program 0.4

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

   3. Applied div-sub 0.4

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

   5. Applied associate-/l* 3.1

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{\frac{t - a \cdot z}{z}}}\]
   6. Using strategy rm

   7. Applied associate-/r/ 2.8

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{t - a \cdot z} \cdot z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.622785819064071 \cdot 10^{37} \lor \neg \left(z \le 5.957888797169072 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{t - a \cdot z} \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))