2sqrt (example 3.1)

Average Error: 30.0 → 29.3

Time: 3.0s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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((x + 1.0d0)) + ((-1.0d0) - sqrt(x))) + 1.0d0
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((x + 1.0)) + (-1.0 - Math.sqrt(x))) + 1.0;
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	30.0
Target	0.2
Herbie	29.3

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

Derivation

1. Initial program 30.0

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

2. Applied egg-rr 29.3

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

3. Final simplification 29.3

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

Alternatives

Alternative 1
Error	30.0
Cost	13120

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

Alternative 2
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)))