Average Error: 19.4 → 0.1
Time: 4.2s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{\left(-{x}^{-0.5}\right) \cdot \frac{{x}^{-0.5}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{-1 - x} \]
(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 -0.5) (pow (+ x 1.0) -0.5))))
  (- -1.0 x)))
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, -0.5) + pow((x + 1.0), -0.5)))) / (-1.0 - x);
}

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

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, -0.5) + math.pow((x + 1.0), -0.5)))) / (-1.0 - x)

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(Float64(-(x ^ -0.5)) * Float64((x ^ -0.5) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)))) / Float64(-1.0 - x))
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 ^ -0.5) + ((x + 1.0) ^ -0.5)))) / (-1.0 - x);
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[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.4

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

  2. Applied egg-rr19.4

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

  3. Applied egg-rr18.8

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

  4. Applied egg-rr0.1

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

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022134 
(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)))))