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

Average Error: 6.2 → 0.8

Time: 4.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5.333425907820349 \cdot 10^{-234}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.554418957395864 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 3.9845616732991274 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))))
   (if (<= (* x y) (- INFINITY))
     t_0
     (if (<= (* x y) -5.333425907820349e-234)
       (/ (* x y) z)
       (if (<= (* x y) 5.554418957395864e-197)
         (/ x (/ z y))
         (if (<= (* x y) 3.9845616732991274e+100)
           (* (* x y) (/ 1.0 z))
           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 = x * (y / z);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((x * y) <= -5.333425907820349e-234) {
		tmp = (x * y) / z;
	} else if ((x * y) <= 5.554418957395864e-197) {
		tmp = x / (z / y);
	} else if ((x * y) <= 3.9845616732991274e+100) {
		tmp = (x * y) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	6.0
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -inf.0 or 3.9845616732991274e100 < (*.f64 x y)

   1. Initial program 22.9

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

   2. Applied *-un-lft-identity_binary64 22.9

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

   3. Applied times-frac_binary64 3.5

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

   4. Simplified 3.5

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

   if -inf.0 < (*.f64 x y) < -5.33342590782034878e-234

   1. Initial program 0.2

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

   2. Applied *-un-lft-identity_binary64 0.2

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

   3. Applied associate-/r*_binary64 0.2

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

   if -5.33342590782034878e-234 < (*.f64 x y) < 5.55441895739586378e-197

   1. Initial program 11.3

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

   2. Applied associate-/l*_binary64 0.6

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

   if 5.55441895739586378e-197 < (*.f64 x y) < 3.9845616732991274e100

   1. Initial program 0.2

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

   2. Applied div-inv_binary64 0.3

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -5.333425907820349 \cdot 10^{-234}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.554418957395864 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 3.9845616732991274 \cdot 10^{+100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))