sqrt(x+1) - sqrt(x)

?

Percentage Accurate: 99.2% → 99.9%
Time: 4.5s
Precision: binary64
Cost: 13248

?

\[-1000000000 \leq x \land x \leq 1000000000\]
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
↓
double code(double x) {
	return 1.0 / (sqrt((1.0 + x)) + 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
    code = 1.0d0 / (sqrt((1.0d0 + x)) + 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) {
	return 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)
↓
def code(x):
	return 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))

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

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

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

\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{1 + x} + \sqrt{x}}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.3%

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

  2. Applied egg-rr99.9%

    \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
    Step-by-step derivation

    [Start]99.3

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

    flip-- [=>]99.8

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

    div-inv [=>]99.8

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

    add-sqr-sqrt [<=]99.8

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

    add-sqr-sqrt [<=]99.9

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

  3. Simplified99.9%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
    Step-by-step derivation

    [Start]99.9

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

    associate-*r/ [=>]99.9

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

    *-rgt-identity [=>]99.9

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

    remove-double-neg [<=]99.9

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

    sub-neg [<=]99.9

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

    div-sub [=>]99.3

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

    rem-square-sqrt [<=]99.3

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

    sqr-neg [<=]99.3

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

    div-sub [<=]99.8

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

    sqr-neg [=>]99.8

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

    +-commutative [=>]99.8

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

    rem-square-sqrt [=>]99.9

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

    associate--l+ [=>]99.9

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

    +-inverses [=>]99.9

    \[ \frac{1 + \color{blue}{0}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]

    metadata-eval [=>]99.9

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

    sub-neg [=>]99.9

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

  4. Final simplification99.9%

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

Alternatives

Alternative 1
Accuracy99.2%
Cost13120
\[\sqrt{1 + x} - \sqrt{x} \]
Alternative 2
Accuracy96.4%
Cost6848
\[x \cdot 0.5 + \left(1 - \sqrt{x}\right) \]
Alternative 3
Accuracy93.2%
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x+1) - sqrt(x)"
  :precision binary64
  :pre (and (<= -1000000000.0 x) (<= x 1000000000.0))
  (- (sqrt (+ x 1.0)) (sqrt x)))