2sqrt (example 3.1)

Average Error: 29.8 → 0.2

Time: 7.8s

Precision: binary64

Cost: 26048

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ x 1.0)) (sqrt 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 + 1.0)) + sqrt(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 + 1.0d0)) + sqrt(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 + 1.0)) + Math.sqrt(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 + 1.0)) + math.sqrt(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(Float64(x + 1.0)) + sqrt(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 + 1.0)) + sqrt(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[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Target

Original	29.8
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.8

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

2. Applied egg-rr 29.2

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

3. Taylor expanded in x around 0 0.2

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

4. Applied egg-rr 0.3

   \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\sqrt[3]{x + 1}\right)}^{2}} \cdot \sqrt{\sqrt[3]{x + 1}}} + \sqrt{x}} \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9528963858121105 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x}}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 23.744287494643725:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x}}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	13248

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

Alternative 4
Error	26.6
Cost	6720

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

Alternative 5
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022317 
(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)))