The challenge
Consider the function
f: x -> sqrt(1 + x) - 1 at x = 1e-15.
We get: f(x) = 4.44089209850062616e-16
This function involves the subtraction of a pair of similar numbers when x is near 0 and the results are significantly erroneous in this region. Using pow instead of sqrt doesn’t give better results.
A “good” answer is 4.99999999999999875... * 1e-16.
Can you modify f(x) to give a good approximation of f(x) in the neighborhood of 0?
The solution in Golang
Option 1:
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package solution
import (
"math"
)
func F(x float64) float64 {
return x / (1.0 + math.Sqrt(1.0 + x))
}
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Option 2:
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package solution
func F(x float64) float64 {
return x*(0.5 - x*(0.125 - x*(0.0625 - x*0.0390625)))
}
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Option 3:
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package solution
import "math/big"
func F(x float64) float64 {
var a, b, c, d, e big.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
}
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Test cases to validate our solution
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package 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("Expected should be near: %1.12e , but got: %1.12e\n", exp ,act);
}
Expect(inrange).To(Equal(true))
}
func dotest(x float64, exp float64) {
assertFuzzyEquals(F(x), exp)
}
var _ = Describe("Test Example", func() {
It("should handle basic cases", func() {
dotest(2.6e-08, 1.29999999155e-08)
dotest(1.4e-09, 6.999999997549999e-10)
dotest(5.0e-06, 2.499996875007812e-06)
})
})
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