package examples
package smartExamples
import smartfloats._
import scala.collection.mutable.LinkedList
/**
* These are examples from the Technical Report.
*/
object PaperExamples {
/**
* Computes the cube root of 10 by Halley's method.
*/
def cubeRoot = {
import AffineFloat._
println("~~~ Cube root ~~~")
val a: AffineFloat = 10
var xn = AffineFloat(1.6)
for(i <- 1 until 5) {
xn = xn * ((xn*xn*xn + 2.0*a)/(2.0*xn*xn*xn + a))
}
println("final:" + xn.toStringWithAbsErrors)
println("intervals: " + xn.interval)
}
/**
* Computes the area of a triangle with Kahan's method.
*/
def triangleKahan(aa: SmartFloat, bb: SmartFloat, cc: SmartFloat): SmartFloat = {
import SmartFloat._
var a = aa
var b = bb
var c = cc
if(b < a) {
val t = a
if(c < b) {
a = c; c = t
}
else {
if(c < a) {
a = b; b = c; c = t
}
else {
a = b; b = t
}
}
}
else if(c < b) {
val t = c; c = b;
if(c < a) {
b = a; a = t
}
else {
b = t
}
}
sqrt((a+(b+c)) * (c-(a-b)) * (c+(a-b)) * (a+(b-c))) / 4.0
}
def triangleKahan(aa: AffineFloat, bb: AffineFloat, cc: AffineFloat) = {
import AffineFloat._
var a = aa
var b = bb
var c = cc
if(b < a) {
val t = a
if(c < b) {
a = c; c = t
}
else {
if(c < a) {
a = b; b = c; c = t
}
else {
a = b; b = t
}
}
}
else if(c < b) {
val t = c; c = b;
if(c < a) {
b = a; a = t
}
else {
b = t
}
}
sqrt((a+(b+c)) * (c-(a-b)) * (c+(a-b)) * (a+(b-c))) / 4.0
}
def triangleKahan(aa: IntervalFloat, bb: IntervalFloat, cc: IntervalFloat) = {
import IntervalFloat._
var a = aa
var b = bb
var c = cc
if(b < a) {
val t = a
if(c < b) {
a = c; c = t
}
else {
if(c < a) {
a = b; b = c; c = t
}
else {
a = b; b = t
}
}
}
else if(c < b) {
val t = c; c = b;
if(c < a) {
b = a; a = t
}
else {
b = t
}
}
sqrt((a+(b+c)) * (c-(a-b)) * (c+(a-b)) * (a+(b-c))) / 4.0
}
def triangleExampleSection = {
println("SmartFloat: textbook")
println(triangleTextbook(9.0, SmartFloat(4.8, 0.09), SmartFloat(4.8, 0.09)))
println(triangleTextbook(9.0, SmartFloat(4.7, 0.09), SmartFloat(4.7, 0.09)))
println("\nIntervalFloat: textbook")
println(triangleTextbook(9.0, IntervalFloat(4.8, 0.09), IntervalFloat(4.8, 0.09)))
println(triangleTextbook(9.0, IntervalFloat(4.7, 0.09), IntervalFloat(4.7, 0.09)))
println("\nSmartFloat: Kahan")
println(triangleKahan(9.0, SmartFloat(4.8, 0.09), SmartFloat(4.8, 0.09)))
println(triangleKahan(9.0, SmartFloat(4.7, 0.09), SmartFloat(4.7, 0.09)))
println("IntervalFloat: Kahan")
println(triangleKahan(9.0, IntervalFloat(4.8, 0.09), IntervalFloat(4.8, 0.09)))
println(triangleKahan(9.0, IntervalFloat(4.7, 0.09), IntervalFloat(4.7, 0.09)))
println("\nAffineFloat for these values: a = 9.0, b = 4.53, c=4.53")
println("according to Mathematica A=2.34216246234115877566387422739")
println(triangleKahan(AffineFloat(9.0), AffineFloat(4.61), AffineFloat(4.61)))
}
def triangleTextbook(a: IntervalFloat, b: IntervalFloat, c: IntervalFloat) = {
import IntervalFloat._
val s = (a + b + c)/2.0
sqrt(s * (s - a) * (s - b) * (s - c))
}
def triangleTextbook(a: SmartFloat, b: SmartFloat, c: SmartFloat): SmartFloat = {
import SmartFloat._
val s = (a + b + c)/2.0
sqrt(s * (s - a) * (s - b) * (s - c))
}
def triangleTextbook(a: AffineFloat, b: AffineFloat, c: AffineFloat): AffineFloat = {
import AffineFloat._
val s = (a + b + c)/2.0
sqrt(s * (s - a) * (s - b) * (s - c))
}
/**
* Simulates a undamped, unforced spring.
*/
def springSimulation(h:SmartFloat, tmax: Double) = {
import SmartFloat._
println("\nSpring simulation for h = " + h)
//setup
val k: SmartFloat = 1.0
val m: SmartFloat = 1.0
val xmax: SmartFloat = 5.0
var x: SmartFloat = xmax //current horizontal position
var vx: SmartFloat = 0.0 //current velocity
var t: SmartFloat = 0.0 //current 'time'
var methodError = k*m*xmax * (h*h)/2.0
while(t < tmax) { //global flag fails if 1.0
val x_next = x + h * vx
val vx_next = vx - h * k/m * x
x = x_next.addError(methodError)
vx = vx_next
t = t + h
}
println("t: " + t.toStringWithAbsErrors +
",\n x: " + x.toStringWithAbsErrors)
}
/**
* Computes the roots of a quadratic equation in two different ways.
*/
def quadraticEquation {
import AffineFloat._
var a = AffineFloat(2.999)
var b = AffineFloat(56.0001)
var c = AffineFloat(1.00074)
val discr = b*b - a * c * 4.0
//classical way
var r2 = (-b + sqrt(discr))/(a * 2.0)
var r1 = (-b - sqrt(discr))/(a * 2.0)
println("classic r1 = " + r1.toStringWithErrors +
" , r2 = " + r2.toStringWithErrors)
//smarter way
val (rk1: AffineFloat, rk2: AffineFloat) =
if(b*b - a*c > 10.0) {
if(b > 0.0) ((-b - sqrt(discr))/(a * 2.0), c * 2.0 /(-b - sqrt(discr)))
else if(b < 0.0) (c * 2.0 /(-b + sqrt(discr)), (-b + sqrt(discr))/(a * 2.0))
else ((-b - sqrt(discr))/(a * 2.0), (-b + sqrt(discr))/(a * 2.0))
}
else {
((-b - sqrt(discr))/(a * 2.0), (-b + sqrt(discr))/(a * 2.0))
}
println("smarter r1 = " + rk1.toStringWithErrors + " , r2 = " + rk2.toStringWithErrors)
}
/**
* Computes the frequency shift from doppler effect.
* Parameters need to be tweaked a bit to get the best result:
* maxNoiseCount = 200, smartQueueLimit = 23, packingFactor = 0.0, packingAvrgScale = 0.50,
* smartPackingFactor = 0.0, smartPackingAvrgScale = 1.0
*/
def dopplerEffect = {
import SmartFloat._
val T = SmartFloat(10.0, 40.0)
val v = SmartFloat(10010.0, 9990.0)
val u = SmartFloat(0.0, 100.0)
val q1 = SmartFloat(331.4) + T*0.6
val q2 = q1 * v
val q3 = q1 + u
val q4 = q3 * q3
val z = q2/q4
println("q1: " + q1.toStringWithAbsErrors)
println("q2: " + q2.toStringWithAbsErrors)
println("q3: " + q3.toStringWithAbsErrors)
println("q4: " + q4.toStringWithAbsErrors)
println("z: " + z.toStringWithAbsErrors)
}
/**
* Analyzer the roundoff from the triangle example.
* For this to work, you need to change the second import statement in SmartFloat.scala from
* import tools.{SmartAForm => CurrentAForm}
* to import tools.{ParamAForm => CurrentAForm}
*/
def analyzeTriangles = {
triangleTextbook(SmartFloat(9.0), SmartFloat(4.7, 0.19), SmartFloat(4.7, 0.19))
println("\n********* affine float **************")
triangleTextbook(AffineFloat(9.0), 4.51, 4.51) //-1, -1
triangleTextbook(AffineFloat(9.0), AffineFloat(4.89), AffineFloat(4.89)) //1, 1
triangleTextbook(AffineFloat(9.0), AffineFloat(4.51), AffineFloat(4.89)) //-1, 1
triangleTextbook(AffineFloat(9.0), AffineFloat(4.600001), AffineFloat(4.59999999)) //0, 0
}
/**
* This is not in the paper, but fun. Just try...
*/
def rumpsFunction = {
import AffineFloat._
val x = AffineFloat(77617.0)
val y = AffineFloat(33096.0)
val f = (333.75 - x*x) * (y*y*y*y*y*y)
+ x*x *(11.0*x*x*y*y - 121.0*y*y*y*y - 2.0)
+ 5.5 * y*y*y*y*y*y*y*y + x/(2.0*y)
println("f : " + f.interval)
}
}
