2isqrt (example 3.6)

Percentage Accurate: 39.3% → 99.9%

Time: 8.5s

Alternatives: 9

Speedup: 1.7×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 39.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {t\_0}^{-1} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(t\_0 + \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (pow (sqrt x) -1.0) (pow t_0 -1.0)) 0.0)
     (* (pow x -1.5) 0.5)
     (/ (- (+ x 1.0) x) (* (sqrt (* (+ x 1.0) x)) (+ t_0 (sqrt x)))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((pow(sqrt(x), -1.0) - pow(t_0, -1.0)) <= 0.0) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = ((x + 1.0) - x) / (sqrt(((x + 1.0) * x)) * (t_0 + sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((sqrt(x) ** (-1.0d0)) - (t_0 ** (-1.0d0))) <= 0.0d0) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = ((x + 1.0d0) - x) / (sqrt(((x + 1.0d0) * x)) * (t_0 + sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((Math.pow(Math.sqrt(x), -1.0) - Math.pow(t_0, -1.0)) <= 0.0) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = ((x + 1.0) - x) / (Math.sqrt(((x + 1.0) * x)) * (t_0 + Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (math.pow(math.sqrt(x), -1.0) - math.pow(t_0, -1.0)) <= 0.0:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = ((x + 1.0) - x) / (math.sqrt(((x + 1.0) * x)) * (t_0 + math.sqrt(x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 0.0)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64(sqrt(Float64(Float64(x + 1.0) * x)) * Float64(t_0 + sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 0.0)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = ((x + 1.0) - x) / (sqrt(((x + 1.0) * x)) * (t_0 + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision] - N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.0

   1. Initial program 39.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \frac{1}{2}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{x}^{3}}}} \cdot \frac{1}{2} \]

      5. lower-pow.f64 66.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{x}^{3}}}} \cdot 0.5 \]

   5. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 65.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {\left(\sqrt{x + 1}\right)}^{-1} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ t_1 := {\left(\sqrt{x}\right)}^{-1} - {t\_0}^{-1}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))) (t_1 (- (pow (sqrt x) -1.0) (pow t_0 -1.0))))
   (if (<= t_1 5e-13) (/ (/ 0.5 x) t_0) t_1)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double t_1 = pow(sqrt(x), -1.0) - pow(t_0, -1.0);
	double tmp;
	if (t_1 <= 5e-13) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    t_1 = (sqrt(x) ** (-1.0d0)) - (t_0 ** (-1.0d0))
    if (t_1 <= 5d-13) then
        tmp = (0.5d0 / x) / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double t_1 = Math.pow(Math.sqrt(x), -1.0) - Math.pow(t_0, -1.0);
	double tmp;
	if (t_1 <= 5e-13) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	t_1 = math.pow(math.sqrt(x), -1.0) - math.pow(t_0, -1.0)
	tmp = 0
	if t_1 <= 5e-13:
		tmp = (0.5 / x) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	t_1 = Float64((sqrt(x) ^ -1.0) - (t_0 ^ -1.0))
	tmp = 0.0
	if (t_1 <= 5e-13)
		tmp = Float64(Float64(0.5 / x) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	t_1 = (sqrt(x) ^ -1.0) - (t_0 ^ -1.0);
	tmp = 0.0;
	if (t_1 <= 5e-13)
		tmp = (0.5 / x) / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision] - N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-13], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 4.9999999999999999e-13

   1. Initial program 39.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x}}}{\sqrt{x + 1}} \]
   6. Step-by-step derivation

      1. lower-/.f64 99.0

         \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]

   if 4.9999999999999999e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 86.2%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {\left(\sqrt{x + 1}\right)}^{-1} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1} - {\left(\sqrt{x + 1}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {t\_0}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{0.125}{x}}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(t\_0 + \sqrt{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (pow (sqrt x) -1.0) (pow t_0 -1.0)) 5e-12)
     (/ (/ (- 0.5 (/ 0.125 x)) x) t_0)
     (/ (- (+ x 1.0) x) (* (sqrt (* (+ x 1.0) x)) (+ t_0 (sqrt x)))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((pow(sqrt(x), -1.0) - pow(t_0, -1.0)) <= 5e-12) {
		tmp = ((0.5 - (0.125 / x)) / x) / t_0;
	} else {
		tmp = ((x + 1.0) - x) / (sqrt(((x + 1.0) * x)) * (t_0 + sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((sqrt(x) ** (-1.0d0)) - (t_0 ** (-1.0d0))) <= 5d-12) then
        tmp = ((0.5d0 - (0.125d0 / x)) / x) / t_0
    else
        tmp = ((x + 1.0d0) - x) / (sqrt(((x + 1.0d0) * x)) * (t_0 + sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((Math.pow(Math.sqrt(x), -1.0) - Math.pow(t_0, -1.0)) <= 5e-12) {
		tmp = ((0.5 - (0.125 / x)) / x) / t_0;
	} else {
		tmp = ((x + 1.0) - x) / (Math.sqrt(((x + 1.0) * x)) * (t_0 + Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (math.pow(math.sqrt(x), -1.0) - math.pow(t_0, -1.0)) <= 5e-12:
		tmp = ((0.5 - (0.125 / x)) / x) / t_0
	else:
		tmp = ((x + 1.0) - x) / (math.sqrt(((x + 1.0) * x)) * (t_0 + math.sqrt(x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 5e-12)
		tmp = Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) / t_0);
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64(sqrt(Float64(Float64(x + 1.0) * x)) * Float64(t_0 + sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 5e-12)
		tmp = ((0.5 - (0.125 / x)) / x) / t_0;
	else
		tmp = ((x + 1.0) - x) / (sqrt(((x + 1.0) * x)) * (t_0 + sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision] - N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 4.9999999999999997e-12

   1. Initial program 40.0%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x + 1}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x + 1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} \cdot 1}{x}}}{x}}{\sqrt{x + 1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{1}{8}}}{x}}{x}}{\sqrt{x + 1}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

      5. lower-/.f64 99.6

         \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]

   if 4.9999999999999997e-12 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 89.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {\left(\sqrt{x + 1}\right)}^{-1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{\left(x + 1\right) \cdot x} \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {t\_0}^{-1} \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \sqrt{\frac{x}{1 + x}}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (pow (sqrt x) -1.0) (pow t_0 -1.0)) 4e-13)
     (/ (/ 0.5 x) t_0)
     (/ (- 1.0 (sqrt (/ x (+ 1.0 x)))) (sqrt x)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((pow(sqrt(x), -1.0) - pow(t_0, -1.0)) <= 4e-13) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = (1.0 - sqrt((x / (1.0 + x)))) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((sqrt(x) ** (-1.0d0)) - (t_0 ** (-1.0d0))) <= 4d-13) then
        tmp = (0.5d0 / x) / t_0
    else
        tmp = (1.0d0 - sqrt((x / (1.0d0 + x)))) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((Math.pow(Math.sqrt(x), -1.0) - Math.pow(t_0, -1.0)) <= 4e-13) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = (1.0 - Math.sqrt((x / (1.0 + x)))) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (math.pow(math.sqrt(x), -1.0) - math.pow(t_0, -1.0)) <= 4e-13:
		tmp = (0.5 / x) / t_0
	else:
		tmp = (1.0 - math.sqrt((x / (1.0 + x)))) / math.sqrt(x)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 4e-13)
		tmp = Float64(Float64(0.5 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((sqrt(x) ^ -1.0) - (t_0 ^ -1.0)) <= 4e-13)
		tmp = (0.5 / x) / t_0;
	else
		tmp = (1.0 - sqrt((x / (1.0 + x)))) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision] - N[Power[t$95$0, -1.0], $MachinePrecision]), $MachinePrecision], 4e-13], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 4.0000000000000001e-13

   1. Initial program 39.8%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x}}}{\sqrt{x + 1}} \]
   6. Step-by-step derivation

      1. lower-/.f64 99.2

         \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]

   if 4.0000000000000001e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 82.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 84.6

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

   6. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-/r* N/A

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

   8. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{1 - \sqrt{\frac{x}{1 + x}}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{x}\right)}^{-1} - {\left(\sqrt{x + 1}\right)}^{-1} \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \sqrt{\frac{x}{1 + x}}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 5.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{{x}^{-1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (pow x -1.0)))

C

double code(double x) {
	return sqrt(pow(x, -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x ** (-1.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.pow(x, -1.0));
}

Python

def code(x):
	return math.sqrt(math.pow(x, -1.0))

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x ^ -1.0));
end

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

6. Final simplification 5.6%

   \[\leadsto \sqrt{{x}^{-1}} \]
7. Add Preprocessing

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.125}{x}}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.125 x)) x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.125d0 / x)) / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return ((0.5 - (0.125 / x)) / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return ((0.5 - (0.125 / x)) / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.125 / x)) / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.125 / x)) / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 41.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{x}}{x}}}{\sqrt{x + 1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} \cdot 1}{x}}}{x}}{\sqrt{x + 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{1}{8}}}{x}}{x}}{\sqrt{x + 1}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8}}{x}}}{x}}{\sqrt{x + 1}} \]

   5. lower-/.f64 98.0

      \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.125}{x}}}{x}}{\sqrt{x + 1}} \]

7. Applied rewrites 98.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.125}{x}}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 7: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x}}{\sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 x) (sqrt (+ x 1.0))))

C

double code(double x) {
	return (0.5 / x) / sqrt((x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 / x) / sqrt((x + 1.0d0))
end function

Java

public static double code(double x) {
	return (0.5 / x) / Math.sqrt((x + 1.0));
}

Python

def code(x):
	return (0.5 / x) / math.sqrt((x + 1.0))

Julia

function code(x)
	return Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) / sqrt((x + 1.0));
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 41.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{x}}}{\sqrt{x + 1}} \]
6. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]

7. Applied rewrites 97.0%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x + 1}} \]
8. Add Preprocessing

Alternative 8: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\left(-x\right) \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -0.5 (* (- x) (sqrt x))))

C

double code(double x) {
	return -0.5 / (-x * sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.5d0) / (-x * sqrt(x))
end function

Java

public static double code(double x) {
	return -0.5 / (-x * Math.sqrt(x));
}

Python

def code(x):
	return -0.5 / (-x * math.sqrt(x))

Julia

function code(x)
	return Float64(-0.5 / Float64(Float64(-x) * sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = -0.5 / (-x * sqrt(x));
end

Wolfram

code[x_] := N[(-0.5 / N[((-x) * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{x}} - \frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} - \frac{\frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{-1}{2}}}{{x}^{2}} - \frac{\frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{\frac{-1}{2}}{{x}^{2}}} - \frac{\frac{-1}{2} \cdot \sqrt{x}}{{x}^{2}} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{\frac{-1}{2}}{{x}^{2}} - \frac{\color{blue}{\sqrt{x} \cdot \frac{-1}{2}}}{{x}^{2}} \]

   5. associate-/l* N/A

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{\frac{-1}{2}}{{x}^{2}} - \color{blue}{\sqrt{x} \cdot \frac{\frac{-1}{2}}{{x}^{2}}} \]

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 80.7

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

5. Applied rewrites 80.7%

   \[\leadsto \color{blue}{\frac{\frac{-0.5}{x}}{x} \cdot \left(\sqrt{\frac{1}{x}} - \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 80.6%

   \[\leadsto \frac{\frac{-0.5}{x}}{x} \cdot \color{blue}{\frac{1 - x}{\sqrt{x}}} \]
8. Step-by-step derivation

9. Applied rewrites 95.9%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{\frac{-1}{2}}{\color{blue}{\left(-x\right)} \cdot \sqrt{x}} \]
11. Step-by-step derivation

12. Applied rewrites 95.8%

    \[\leadsto \frac{-0.5}{\color{blue}{\left(-x\right)} \cdot \sqrt{x}} \]
13. Add Preprocessing

Alternative 9: 37.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{x}{x \cdot x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ x (* x x))))

C

double code(double x) {
	return sqrt((x / (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x / (x * x)))
end function

Java

public static double code(double x) {
	return Math.sqrt((x / (x * x)));
}

Python

def code(x):
	return math.sqrt((x / (x * x)))

Julia

function code(x)
	return sqrt(Float64(x / Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x / (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.6

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

5. Applied rewrites 5.6%

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

7. Applied rewrites 5.6%

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

9. Applied rewrites 39.0%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
10. Add Preprocessing

Developer Target 1: 39.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} - {\left(x + 1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (pow x -0.5) (pow (+ x 1.0) -0.5)))

C

double code(double x) {
	return pow(x, -0.5) - pow((x + 1.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
}

Python

def code(x):
	return math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)

Julia

function code(x)
	return Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024314 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))