Floating-point Approximation in Golang

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:

package solution import ( "math" ) func F(x float64) float64 { return x / (1.0 + math.Sqrt(1.0 + x)) }
Code language: Go (go)

Option 2:

package solution func F(x float64) float64 { return x*(0.5 - x*(0.125 - x*(0.0625 - x*0.0390625))) }
Code language: Go (go)

Option 3:

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 }
Code language: Go (go)

Test cases to validate our solution

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) }) })
Code language: Go (go)
Tags:
Subscribe
Notify of
guest
0 Comments
Inline Feedbacks
View all comments