2sqrt (example 3.1)

Average Error: 30.0 → 0.2

Time: 9.0s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
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 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	30.0
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.0

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

2. Applied egg-rr 29.4

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

3. Simplified 0.2

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

   [Start] 29.4	\[ \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   *-commutative [=>] 29.4	\[ \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \left(x + \left(1 - x\right)\right)} \]
   associate-*l/ [=>] 29.4	\[ \color{blue}{\frac{1 \cdot \left(x + \left(1 - x\right)\right)}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-lft-identity [=>] 29.4	\[ \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 29.4	\[ \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-+l- [=>] 0.2	\[ \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-inverses [=>] 0.2	\[ \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]
   metadata-eval [=>] 0.2	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 0.2	\[ \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 38000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 2
Error	1.9
Cost	6788

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

Alternative 3
Error	31.2
Cost	448

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

Alternative 4
Error	31.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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)))