2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 6.5s

Precision: binary64

Cost: 19584

Math

\[\sqrt{x + 1} - \sqrt{x} \]

↓

\[{\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-1} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

↓

(FPCore (x) :precision binary64 (pow (+ (sqrt x) (sqrt (+ 1.0 x))) -1.0))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

↓

double code(double x) {
	return pow((sqrt(x) + sqrt((1.0 + x))), -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt(x) + sqrt((1.0d0 + x))) ** (-1.0d0)
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

↓

public static double code(double x) {
	return Math.pow((Math.sqrt(x) + Math.sqrt((1.0 + x))), -1.0);
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

↓

def code(x):
	return math.pow((math.sqrt(x) + math.sqrt((1.0 + x))), -1.0)

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

↓

function code(x)
	return Float64(sqrt(x) + sqrt(Float64(1.0 + x))) ^ -1.0
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

↓

function tmp = code(x)
	tmp = (sqrt(x) + sqrt((1.0 + x))) ^ -1.0;
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Target

Original	29.5
Target	0.2
Herbie	0.2

\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

1. Initial program 29.5

   \[\sqrt{x + 1} - \sqrt{x} \]

2. Applied egg-rr 28.8

   \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}} \]

3. Taylor expanded in x around 0 0.2

   \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1}} \]

5. Final simplification 0.2

   \[\leadsto {\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-1} \]

Alternatives

Alternative 1
Error	0.2
Cost	13248

\[\frac{1}{\sqrt{x} + \sqrt{1 + x}} \]

Alternative 2
Error	26.6
Cost	6912

\[\frac{-1}{-\sqrt{x + \left(1 + x\right)}} \]

Alternative 3
Error	26.4
Cost	6720

\[\frac{1}{1 + \sqrt{x}} \]

Alternative 4
Error	30.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))