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

Average Error: 6.2 → 0.4

Time: 2.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1.1825558902623664 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -9.603133343069813 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 7.199250155588428 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 1.0061014461818346 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (/ z x))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -1.1825558902623664e+183)
     t_0
     (if (<= (* x y) -9.603133343069813e-216)
       t_1
       (if (<= (* x y) 7.199250155588428e-278)
         t_0
         (if (<= (* x y) 1.0061014461818346e+298) t_1 t_0))))))

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 / (z / x);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1.1825558902623664e+183) {
		tmp = t_0;
	} else if ((x * y) <= -9.603133343069813e-216) {
		tmp = t_1;
	} else if ((x * y) <= 7.199250155588428e-278) {
		tmp = t_0;
	} else if ((x * y) <= 1.0061014461818346e+298) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	6.3
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 2 regimes

2. if (*.f64 x y) < -1.1825558902623664e183 or -9.6031333430698133e-216 < (*.f64 x y) < 7.1992501555884278e-278 or 1.0061014461818346e298 < (*.f64 x y)

   1. Initial program 18.8

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.7

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

   4. Taylor expanded in x around 0 0.6

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

   if -1.1825558902623664e183 < (*.f64 x y) < -9.6031333430698133e-216 or 7.1992501555884278e-278 < (*.f64 x y) < 1.0061014461818346e298

   1. Initial program 0.2

      \[\frac{x \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.1825558902623664 \cdot 10^{+183}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -9.603133343069813 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.199250155588428 \cdot 10^{-278}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq 1.0061014461818346 \cdot 10^{+298}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))