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

Average Error: 6.4 → 0.6

Time: 5.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq -1.1047486872844918 \cdot 10^{-289}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_0 \leq 6.439721645607079 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 (- INFINITY))
     (* x (/ y z))
     (if (<= t_0 -1.1047486872844918e-289)
       t_0
       (if (<= t_0 0.0)
         (* y (/ x z))
         (if (<= t_0 6.439721645607079e+300) t_0 (/ 1.0 (/ (/ z y) x))))))))

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 (t_0 <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if (t_0 <= -1.1047486872844918e-289) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = y * (x / z);
	} else if (t_0 <= 6.439721645607079e+300) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((z / y) / x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.2
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 4 regimes

2. if (/.f64 (*.f64 x y) z) < -inf.0

   1. Initial program 64.0

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

   2. Applied *-un-lft-identity_binary64 64.0

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

   3. Applied times-frac_binary64 0.3

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

   4. Simplified 0.3

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

   if -inf.0 < (/.f64 (*.f64 x y) z) < -1.1047486872844918e-289 or -0.0 < (/.f64 (*.f64 x y) z) < 6.43972164560707933e300

   1. Initial program 0.5

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

   if -1.1047486872844918e-289 < (/.f64 (*.f64 x y) z) < -0.0

   1. Initial program 11.8

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

   2. Applied associate-/l*_binary64 0.9

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

   3. Applied associate-/r/_binary64 0.7

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

   if 6.43972164560707933e300 < (/.f64 (*.f64 x y) z)

   1. Initial program 58.2

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

   2. Applied associate-/l*_binary64 1.8

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

   3. Applied clear-num_binary64 1.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -1.1047486872844918 \cdot 10^{-289}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 6.439721645607079 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(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))