Average Error: 19.6 → 0.4
Time: 10.7s
Precision: binary64
Cost: 13760
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{\frac{1}{1 + x}}{x \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ 1.0 x)) (* x (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	return (1.0 / (1.0 + x)) / (x * (pow(x, -0.5) + pow((1.0 + x), -0.5)));
}

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 = (1.0d0 / (1.0d0 + x)) / (x * ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0))))
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 (1.0 / (1.0 + x)) / (x * (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5)));
}

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

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(1.0 / Float64(1.0 + x)) / Float64(x * Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp = code(x)
	tmp = (1.0 / (1.0 + x)) / (x * ((x ^ -0.5) + ((1.0 + x) ^ -0.5)));
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[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.6

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

  2. Applied egg-rr19.7

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

  3. Applied egg-rr5.2

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

  4. Applied egg-rr0.5

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost13380
\[\begin{array}{l} \mathbf{if}\;x \leq 11273832.584030142:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
Alternative 2
Error1.3
Cost7236
\[\begin{array}{l} \mathbf{if}\;x \leq 7.401378024311287 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
Alternative 3
Error1.3
Cost7108
\[\begin{array}{l} \mathbf{if}\;x \leq 7.401378024311287 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
Alternative 4
Error2.2
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 7.401378024311287 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
Alternative 5
Error1.3
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 7.401378024311287 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]
Alternative 6
Error31.3
Cost6528
\[{x}^{-0.5} \]
Alternative 7
Error62.8
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022298 
(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)))))