2sqrt (example 3.1)

Average Error: 29.4 → 0.2

Time: 6.0s

Precision: binary64

Cost: 26048

Math

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

↓

\[{\left({\left(\sqrt{x} + \sqrt{1 + x}\right)}^{2}\right)}^{-0.5} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt x) (sqrt (+ 1.0 x))) 2.0) -0.5))

C

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

↓

double code(double x) {
	return pow(pow((sqrt(x) + sqrt((1.0 + x))), 2.0), -0.5);
}

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))) ** 2.0d0) ** (-0.5d0)
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.pow((Math.sqrt(x) + Math.sqrt((1.0 + x))), 2.0), -0.5);
}

Python

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

↓

def code(x):
	return math.pow(math.pow((math.sqrt(x) + math.sqrt((1.0 + x))), 2.0), -0.5)

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))) ^ 2.0) ^ -0.5
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))) ^ 2.0) ^ -0.5;
end

Wolfram

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

↓

code[x_] := N[Power[N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Target

Original	29.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.4

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

2. Applied egg-rr 28.8

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

3. Applied egg-rr 28.9

   \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{{\left({\left(x + 1\right)}^{1.5}\right)}^{0.3333333333333333}} + \sqrt{x}} \]

4. Taylor expanded in x around 0 11.5

   \[\leadsto \frac{\color{blue}{1}}{{\left({\left(x + 1\right)}^{1.5}\right)}^{0.3333333333333333} + \sqrt{x}} \]

5. Applied egg-rr 0.2

   \[\leadsto \color{blue}{{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5}} \]

6. Final simplification 0.2

   \[\leadsto {\left({\left(\sqrt{x} + \sqrt{1 + x}\right)}^{2}\right)}^{-0.5} \]

Alternatives

Alternative 1
Error	0.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 14707591.402461676:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	1.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.0029421815227809534:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.0029421815227809534:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 5
Error	26.3
Cost	6720

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

Alternative 6
Error	30.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022325 
(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)))