?

Average Error: 29.9 → 30.0
Time: 14.4s
Precision: binary64
Cost: 33472

?

\[\sqrt{x + 1} - \sqrt{x} \]
\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ t_0 \cdot 2 + \left(0 - \left(\left(t_0 \cdot \frac{1}{t_0}\right) \cdot t_0 + \sqrt{x}\right)\right) \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (+ (* t_0 2.0) (- 0.0 (+ (* (* t_0 (/ 1.0 t_0)) t_0) (sqrt x))))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return (t_0 * 2.0) + (0.0 - (((t_0 * (1.0 / t_0)) * t_0) + sqrt(x)));
}
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
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (t_0 * 2.0d0) + (0.0d0 - (((t_0 * (1.0d0 / t_0)) * t_0) + sqrt(x)))
end function
public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}
public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	return (t_0 * 2.0) + (0.0 - (((t_0 * (1.0 / t_0)) * t_0) + Math.sqrt(x)));
}
def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)
def code(x):
	t_0 = math.sqrt((x + 1.0))
	return (t_0 * 2.0) + (0.0 - (((t_0 * (1.0 / t_0)) * t_0) + math.sqrt(x)))
function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end
function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64(Float64(t_0 * 2.0) + Float64(0.0 - Float64(Float64(Float64(t_0 * Float64(1.0 / t_0)) * t_0) + sqrt(x))))
end
function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end
function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (t_0 * 2.0) + (0.0 - (((t_0 * (1.0 / t_0)) * t_0) + sqrt(x)));
end
code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]
code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * 2.0), $MachinePrecision] + N[(0.0 - N[(N[(N[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]
\sqrt{x + 1} - \sqrt{x}
\begin{array}{l}
t_0 := \sqrt{x + 1}\\
t_0 \cdot 2 + \left(0 - \left(\left(t_0 \cdot \frac{1}{t_0}\right) \cdot t_0 + \sqrt{x}\right)\right)
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie30.0
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation?

  1. Initial program 29.9

    \[\sqrt{x + 1} - \sqrt{x} \]
  2. Applied egg-rr30.0

    \[\leadsto \color{blue}{\sqrt{x + 1} \cdot 2 + \left(0 - \left(\sqrt{x + 1} + \sqrt{x}\right)\right)} \]
  3. Applied egg-rr30.0

    \[\leadsto \sqrt{x + 1} \cdot 2 + \left(0 - \left(\color{blue}{\left(\sqrt{x + 1} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \sqrt{x + 1}} + \sqrt{x}\right)\right) \]
  4. Final simplification30.0

    \[\leadsto \sqrt{x + 1} \cdot 2 + \left(0 - \left(\left(\sqrt{x + 1} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \sqrt{x + 1} + \sqrt{x}\right)\right) \]

Alternatives

Alternative 1
Error29.9
Cost13120
\[\sqrt{x + 1} - \sqrt{x} \]
Alternative 2
Error30.9
Cost6848
\[0.5 \cdot x + \left(1 - \sqrt{x}\right) \]
Alternative 3
Error30.9
Cost6848
\[\left(0.5 \cdot x - \sqrt{x}\right) + 1 \]
Alternative 4
Error30.5
Cost6724
\[\begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 5
Error31.2
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 2023094 
(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)))