Average Error: 29.2 → 0.2
Time: 2.6s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}\right)\right)} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (sqrt (expm1 (log1p (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) -2.0)))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
↓
double code(double x) {
	return sqrt(expm1(log1p(pow((sqrt((1.0 + x)) + sqrt(x)), -2.0))));
}

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.expm1(Math.log1p(Math.pow((Math.sqrt((1.0 + x)) + Math.sqrt(x)), -2.0))));
}

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)
↓
def code(x):
	return math.sqrt(math.expm1(math.log1p(math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), -2.0))))

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end
↓
function code(x)
	return sqrt(expm1(log1p((Float64(sqrt(Float64(1.0 + x)) + sqrt(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[(Exp[N[Log[1 + N[Power[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}\right)\right)}

Error

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  2. Applied egg-rr28.6

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied egg-rr0.2

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

  5. Applied egg-rr0.2

    \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}\right)\right)}} \]

  6. Final simplification0.2

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}\right)\right)} \]

Reproduce

herbie shell --seed 2022185 
(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)))