Average Error: 20.2 → 0.1
Time: 9.2s
Precision: binary64
Cost: 26688
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{{x}^{-0.5}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{{x}^{-0.5}}{x + 1} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (/ (pow x -0.5) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
  (/ (pow x -0.5) (+ 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, -0.5) / (pow(x, -0.5) + pow((x + 1.0), -0.5))) * (pow(x, -0.5) / (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 ** (-0.5d0)) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))) * ((x ** (-0.5d0)) / (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, -0.5) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5))) * (Math.pow(x, -0.5) / (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, -0.5) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))) * (math.pow(x, -0.5) / (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 ^ -0.5) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5))) * Float64((x ^ -0.5) / 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 ^ -0.5) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5))) * ((x ^ -0.5) / (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[(N[Power[x, -0.5], $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.6
Herbie0.1
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 20.2

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

  2. Simplified20.2

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

  3. Applied egg-rr20.3

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

  4. Applied egg-rr5.4

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

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Alternatives

Alternative 1
Error5.4
Cost26692
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(x - x\right)}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
Alternative 2
Error0.4
Cost20096
\[\frac{\frac{{x}^{-1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{x + 1} \]
Alternative 3
Error0.5
Cost13888
\[\frac{1}{x} \cdot \frac{1}{\left(x + 1\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]
Alternative 4
Error6.0
Cost7620
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;x \cdot 0.5 + \left({x}^{-0.5} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(x - x\right)}{x}}{x + 1}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
Alternative 5
Error6.5
Cost7236
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;x \cdot 0.5 + \left({x}^{-0.5} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
Alternative 6
Error6.5
Cost7172
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;x \cdot 0.5 + \left({x}^{-0.5} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{0.5}{{x}^{-0.5}}\\ \end{array} \]
Alternative 7
Error20.6
Cost7044
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;x \cdot 0.5 + \left({x}^{-0.5} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \]
Alternative 8
Error20.7
Cost6980
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \]
Alternative 9
Error30.1
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 0.06588527500348274:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
Alternative 10
Error30.4
Cost6720
\[\frac{1}{x + \sqrt{x}} \]
Alternative 11
Error31.7
Cost6528
\[{x}^{-0.5} \]
Alternative 12
Error59.3
Cost192
\[\frac{1}{x} \]
Alternative 13
Error62.8
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022225 
(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)))))