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

Average Error: 6.3 → 0.7

Time: 2.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -6.897924025846411 \cdot 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -7.078600279218089 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 3.756140649515336 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 1.4787498126002392 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -6.897924025846411e+168)
     t_0
     (if (<= (* x y) -7.078600279218089e-146)
       t_1
       (if (<= (* x y) 3.756140649515336e-242)
         t_0
         (if (<= (* x y) 1.4787498126002392e+157) 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 = x * (y / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -6.897924025846411e+168) {
		tmp = t_0;
	} else if ((x * y) <= -7.078600279218089e-146) {
		tmp = t_1;
	} else if ((x * y) <= 3.756140649515336e-242) {
		tmp = t_0;
	} else if ((x * y) <= 1.4787498126002392e+157) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.4
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -6.89792402584641086e168 or -7.0786002792180894e-146 < (*.f64 x y) < 3.756140649515336e-242 or 1.47874981260023918e157 < (*.f64 x y)

   1. Initial program 13.8

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

   2. Applied egg-rr 1.4

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

   3. Applied egg-rr 1.3

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

   4. Applied egg-rr 1.3

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

   if -6.89792402584641086e168 < (*.f64 x y) < -7.0786002792180894e-146 or 3.756140649515336e-242 < (*.f64 x y) < 1.47874981260023918e157

   1. Initial program 0.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -6.897924025846411 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -7.078600279218089 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.756140649515336 \cdot 10^{-242}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.4787498126002392 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))