2sqrt (example 3.1)

Average Error: 30.1 → 0.2

Time: 6.1s

Precision: binary64

Cost: 25984

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

Julia

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

↓

function code(x)
	return sqrt((Float64(sqrt(x) + sqrt(Float64(1.0 + x))) ^ -2.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = sqrt(((sqrt(x) + sqrt((1.0 + x))) ^ -2.0));
end

Wolfram

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

↓

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

Target

Original	30.1
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.1

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

2. Applied egg-rr 29.5

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

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

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	26308

\[\begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 10^{-6}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13248

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

Alternative 3
Error	0.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10266196555053207:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10266196555053207:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5
Error	1.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10266196555053207:\\ \;\;\;\;1 - {x}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6
Error	31.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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)))