The problem
Think about the operate
f: x -> sqrt(1 + x) - 1 at x = 1e-15.
We get: f(x) = 4.44089209850062616e-16
This operate entails the subtraction of a pair of comparable numbers when x is close to 0 and the outcomes are considerably faulty on this area. Utilizing pow as a substitute of sqrt doesn’t give higher outcomes.
A “good” reply is 4.99999999999999875... * 1e-16.
Are you able to modify f(x) to present a very good approximation of f(x) within the neighborhood of 0?
The answer in Golang
Possibility 1:
bundle answer
import (
"math"
)
func F(x float64) float64 {
return x / (1.0 + math.Sqrt(1.0 + x))
}
Possibility 2:
bundle answer
func F(x float64) float64 {
return x*(0.5 - x*(0.125 - x*(0.0625 - x*0.0390625)))
}
Possibility 3:
bundle answer
import "math/huge"
func F(x float64) float64 {
var a, b, c, d, e huge.Float
a.SetInt64(1)
b.SetFloat64(x)
c.SetPrec(106)
c.Add(&a, &b)
d.Sqrt(&c)
e.Sub(&d, &a)
r, _ := e.Float64()
return r
}
Take a look at instances to validate our answer
bundle solution_test
import (
. "github.com/onsi/ginkgo"
. "github.com/onsi/gomega"
"math"
"fmt"
)
func assertFuzzyEquals(act float64, exp float64) {
var inrange bool
var merr float64 = 1e-12
var e float64
if (exp == 0.0) {
e = math.Abs(act)
} else {
e = math.Abs((act - exp) / exp)
}
inrange = (e <= merr)
if (inrange == false) {
fmt.Printf("Anticipated ought to be close to: %1.12e , however bought: %1.12en", exp ,act);
}
Count on(inrange).To(Equal(true))
}
func dotest(x float64, exp float64) {
assertFuzzyEquals(F(x), exp)
}
var _ = Describe("Take a look at Instance", func() {
It("ought to deal with fundamental instances", func() {
dotest(2.6e-08, 1.29999999155e-08)
dotest(1.4e-09, 6.999999997549999e-10)
dotest(5.0e-06, 2.499996875007812e-06)
})
})
