Average Error: 19.8 → 0.2
Time: 5.2s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{-1}{\left(-\sqrt{x}\right) - \sqrt{x + 1}} \cdot \left({\left(x + 1\right)}^{-0.5} \cdot {x}^{-0.5}\right) \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (/ -1.0 (- (- (sqrt x)) (sqrt (+ x 1.0))))
  (* (pow (+ x 1.0) -0.5) (pow 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 / (-sqrt(x) - sqrt((x + 1.0)))) * (pow((x + 1.0), -0.5) * pow(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) / (-sqrt(x) - sqrt((x + 1.0d0)))) * (((x + 1.0d0) ** (-0.5d0)) * (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 / (-Math.sqrt(x) - Math.sqrt((x + 1.0)))) * (Math.pow((x + 1.0), -0.5) * Math.pow(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 / (-math.sqrt(x) - math.sqrt((x + 1.0)))) * (math.pow((x + 1.0), -0.5) * math.pow(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(Float64(-sqrt(x)) - sqrt(Float64(x + 1.0)))) * Float64((Float64(x + 1.0) ^ -0.5) * (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 / (-sqrt(x) - sqrt((x + 1.0)))) * (((x + 1.0) ^ -0.5) * (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[((-N[Sqrt[x], $MachinePrecision]) - N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.8

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

  2. Simplified19.8

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

  3. Applied egg-rr19.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, -1, \sqrt{1 + x}\right) \cdot {\left(x \cdot \left(1 + x\right)\right)}^{-0.5}} \]

  4. Applied egg-rr19.4

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{x} \cdot -1\right)}^{2} - \left(x + 1\right)}{\sqrt{x} \cdot -1 - \sqrt{x + 1}}} \cdot {\left(x \cdot \left(1 + x\right)\right)}^{-0.5} \]

  5. Taylor expanded in x around 0 5.3

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

  6. Applied egg-rr0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022210 
(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)))))