2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 9.1s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return 1.0 / (sqrt(x) + sqrt((1.0 + 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(x) + sqrt((1.0d0 + 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(x) + Math.sqrt((1.0 + x)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 / (sqrt(x) + sqrt((1.0 + 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[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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

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

3. Simplified 0.2

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   Proof
   (/.f64 1 (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (-.f64 1 0)) (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))): 0 points increase in error, 5 points decrease in error
   (/.f64 (-.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 x x))) (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))): 5 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 x) x)) (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))): 0 points increase in error, 5 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 1 x))) (+.f64 (sqrt.f64 (+.f64 1 x)) (sqrt.f64 x))): 0 points increase in error, 5 points decrease in error
   (/.f64 (+.f64 x (-.f64 1 x)) (+.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (sqrt.f64 x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 x (-.f64 1 x)) 1)) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))): 5 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 x (-.f64 1 x)) (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))))): 3 points increase in error, 5 points decrease in error

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.7
Cost	26308

\[\begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	6980

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

Alternative 4
Error	1.2
Cost	6852

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

Alternative 5
Error	1.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6
Error	30.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(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)))