Average Error: 29.8 → 0.2
Time: 8.5s
Precision: binary64
Cost: 25984
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\sqrt{{\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-2}} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (sqrt (pow (+ (sqrt x) (sqrt (+ 1.0 x))) -2.0)))
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));
}

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

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));
}

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))

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

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

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]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 29.8

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

  2. Applied egg-rr29.1

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

  3. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
    Proof
    (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= --rgt-identity_binary64 (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) 0)): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) (Rewrite<= div0_binary64 (/.f64 0 (neg.f64 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) (/.f64 0 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 0 -1) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) (/.f64 (Rewrite=> metadata-eval 0) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)))): 0 points increase in error, 0 points decrease in error
    (-.f64 (/.f64 1 (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))) (/.f64 (Rewrite<= +-inverses_binary64 (-.f64 x x)) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= div-sub_binary64 (/.f64 (-.f64 1 (-.f64 x x)) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 x) x)) (+.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))): 115 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (-.f64 1 x))) (+.f64 (sqrt.f64 (+.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))): 0 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))))): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost13248
\[\frac{1}{\sqrt{x} + \sqrt{1 + x}} \]
Alternative 2
Error29.8
Cost13120
\[\sqrt{1 + x} - \sqrt{x} \]

Error

Reproduce

herbie shell --seed 2022334 
(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)))