Average Error: 19.9 → 0.4
Time: 9.8s
Precision: binary64
Cost: 20224
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[{\left(x \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)\right)}^{-1} \cdot \frac{1}{x + 1} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (* (pow (* x (+ (pow x -0.5) (pow (+ x 1.0) -0.5))) -1.0) (/ 1.0 (+ x 1.0))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	return pow((x * (pow(x, -0.5) + pow((x + 1.0), -0.5))), -1.0) * (1.0 / (x + 1.0));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))) ** (-1.0d0)) * (1.0d0 / (x + 1.0d0))
end function

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}
↓
public static double code(double x) {
	return Math.pow((x * (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5))), -1.0) * (1.0 / (x + 1.0));
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))
↓
def code(x):
	return math.pow((x * (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))), -1.0) * (1.0 / (x + 1.0))

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	return Float64((Float64(x * Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5))) ^ -1.0) * Float64(1.0 / Float64(x + 1.0)))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp = code(x)
	tmp = ((x * ((x ^ -0.5) + ((x + 1.0) ^ -0.5))) ^ -1.0) * (1.0 / (x + 1.0));
end

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Power[N[(x * N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{\left(x \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)\right)}^{-1} \cdot \frac{1}{x + 1}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.9

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

  2. Applied egg-rr20.0

    \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\frac{1}{x} - \frac{1}{1 + x}}}} \]

  3. Applied egg-rr5.4

    \[\leadsto \frac{1}{\frac{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}{\color{blue}{\frac{\frac{1 + \left(x - x\right)}{x}}{1 + x}}}} \]

  4. Applied egg-rr0.4

    \[\leadsto \color{blue}{{\left(\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x\right)}^{-1} \cdot \frac{1}{x + 1}} \]

  5. Final simplification0.4

    \[\leadsto {\left(x \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)\right)}^{-1} \cdot \frac{1}{x + 1} \]

Alternatives

Alternative 1
Error0.5
Cost26884
\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 10^{-17}:\\ \;\;\;\;t_0 \cdot {\left(\sqrt{x} \cdot 2\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{t_0}\\ \end{array} \]
Alternative 2
Error0.5
Cost26756
\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 10^{-17}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{t_0}\\ \end{array} \]
Alternative 3
Error0.7
Cost13696
\[\frac{-1}{\left(\sqrt{x} + x \cdot {\left(x + 1\right)}^{-0.5}\right) \cdot \left(-1 - x\right)} \]
Alternative 4
Error0.5
Cost13380
\[\begin{array}{l} \mathbf{if}\;x \leq 35000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.5\right)\\ \end{array} \]
Alternative 5
Error1.1
Cost7236
\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.5\right)\\ \end{array} \]
Alternative 6
Error1.1
Cost7044
\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{1}{x}\right)}^{1.5}\\ \end{array} \]
Alternative 7
Error1.3
Cost6916
\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{1}{x}\right)}^{1.5}\\ \end{array} \]
Alternative 8
Error21.2
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 9
Error1.7
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]
Alternative 10
Error21.6
Cost6660
\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 11
Error51.8
Cost64
\[0 \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))