2isqrt (example 3.6)

Percentage Accurate: 37.6% → 99.9%

Time: 7.8s

Alternatives: 9

Speedup: 1.5×

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: 37.6% 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, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((x - -1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((x - -1.0) - x) / ((t_0 + sqrt(x)) * sqrt(((x - -1.0) * 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 (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((x - (-1.0d0)) - x) / ((t_0 + sqrt(x)) * sqrt(((x - (-1.0d0)) * 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 (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((x - -1.0) - x) / ((t_0 + Math.sqrt(x)) * Math.sqrt(((x - -1.0) * x)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x - -1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((x - -1.0) - x) / ((t_0 + sqrt(x)) * sqrt(((x - -1.0) * 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[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * 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 39.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 100.0%

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

   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 67.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 68.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x}}}{\sqrt{x + 1}}} \]
   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. 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. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lower--.f64 N/A

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

   6. Applied rewrites 99.4%

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

4. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x - -1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t\_0} \leq 0:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.625}{x}\right) \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\left(t\_0 + \sqrt{x}\right) \cdot \sqrt{\left(x - -1\right) \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- x -1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (/ (* (- 0.5 (/ -0.625 x)) (sqrt (/ 1.0 x))) x)
     (/ (- (- x -1.0) x) (* (+ t_0 (sqrt x)) (sqrt (* (- x -1.0) x)))))))

C

double code(double x) {
	double t_0 = sqrt((x - -1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = ((0.5 - (-0.625 / x)) * sqrt((1.0 / x))) / x;
	} else {
		tmp = ((x - -1.0) - x) / ((t_0 + sqrt(x)) * sqrt(((x - -1.0) * 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 (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 0.0d0) then
        tmp = ((0.5d0 - ((-0.625d0) / x)) * sqrt((1.0d0 / x))) / x
    else
        tmp = ((x - (-1.0d0)) - x) / ((t_0 + sqrt(x)) * sqrt(((x - (-1.0d0)) * 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 (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = ((0.5 - (-0.625 / x)) * Math.sqrt((1.0 / x))) / x;
	} else {
		tmp = ((x - -1.0) - x) / ((t_0 + Math.sqrt(x)) * Math.sqrt(((x - -1.0) * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x - -1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 0.0:
		tmp = ((0.5 - (-0.625 / x)) * math.sqrt((1.0 / x))) / x
	else:
		tmp = ((x - -1.0) - x) / ((t_0 + math.sqrt(x)) * math.sqrt(((x - -1.0) * x)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(Float64(0.5 - Float64(-0.625 / x)) * sqrt(Float64(1.0 / x))) / x);
	else
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64(Float64(t_0 + sqrt(x)) * sqrt(Float64(Float64(x - -1.0) * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x - -1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0)
		tmp = ((0.5 - (-0.625 / x)) * sqrt((1.0 / x))) / x;
	else
		tmp = ((x - -1.0) - x) / ((t_0 + sqrt(x)) * sqrt(((x - -1.0) * 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[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(0.5 - N[(-0.625 / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * 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 39.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right), 0.5, -0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\left(-\sqrt{\frac{1}{x}}\right) \cdot \left(0.5 - \frac{-0.625}{x}\right)}{\color{blue}{-x}} \]

   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 67.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 68.6%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x}}}{\sqrt{x + 1}}} \]
   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. 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. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lower--.f64 N/A

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

   6. Applied rewrites 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x - -1}} \leq 0:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.625}{x}\right) \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\left(\sqrt{x - -1} + \sqrt{x}\right) \cdot \sqrt{\left(x - -1\right) \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x - -1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t\_0} \leq 0:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.625}{x}\right) \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x - -1\right) - x}{\sqrt{\left(x - -1\right) \cdot x} + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- x -1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (/ (* (- 0.5 (/ -0.625 x)) (sqrt (/ 1.0 x))) x)
     (/ (/ (- (- x -1.0) x) (+ (sqrt (* (- x -1.0) x)) x)) t_0))))

C

double code(double x) {
	double t_0 = sqrt((x - -1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = ((0.5 - (-0.625 / x)) * sqrt((1.0 / x))) / x;
	} else {
		tmp = (((x - -1.0) - x) / (sqrt(((x - -1.0) * x)) + x)) / t_0;
	}
	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 (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 0.0d0) then
        tmp = ((0.5d0 - ((-0.625d0) / x)) * sqrt((1.0d0 / x))) / x
    else
        tmp = (((x - (-1.0d0)) - x) / (sqrt(((x - (-1.0d0)) * x)) + x)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x - -1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = ((0.5 - (-0.625 / x)) * Math.sqrt((1.0 / x))) / x;
	} else {
		tmp = (((x - -1.0) - x) / (Math.sqrt(((x - -1.0) * x)) + x)) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x - -1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 0.0:
		tmp = ((0.5 - (-0.625 / x)) * math.sqrt((1.0 / x))) / x
	else:
		tmp = (((x - -1.0) - x) / (math.sqrt(((x - -1.0) * x)) + x)) / t_0
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(Float64(0.5 - Float64(-0.625 / x)) * sqrt(Float64(1.0 / x))) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x - -1.0) - x) / Float64(sqrt(Float64(Float64(x - -1.0) * x)) + x)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x - -1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0)
		tmp = ((0.5 - (-0.625 / x)) * sqrt((1.0 / x))) / x;
	else
		tmp = (((x - -1.0) - x) / (sqrt(((x - -1.0) * x)) + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(0.5 - N[(-0.625 / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 39.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right), 0.5, -0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{x}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\left(-\sqrt{\frac{1}{x}}\right) \cdot \left(0.5 - \frac{-0.625}{x}\right)}{\color{blue}{-x}} \]

   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 67.5%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 99.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x - -1}} \leq 0:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.625}{x}\right) \cdot \sqrt{\frac{1}{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x - -1\right) - x}{\sqrt{\left(x - -1\right) \cdot x} + x}}{\sqrt{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x - -1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t\_0} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(x - -1\right) \cdot x} - x}{x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- x -1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-10)
     (/ (/ 1.0 (fma 2.0 x 0.5)) t_0)
     (/ (/ (- (sqrt (* (- x -1.0) x)) x) x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((x - -1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-10) {
		tmp = (1.0 / fma(2.0, x, 0.5)) / t_0;
	} else {
		tmp = ((sqrt(((x - -1.0) * x)) - x) / x) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x - -1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-10)
		tmp = Float64(Float64(1.0 / fma(2.0, x, 0.5)) / t_0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(x - -1.0) * x)) - x) / x) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-10], N[(N[(1.0 / N[(2.0 * x + 0.5), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $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))))) < 5.00000000000000031e-10

   1. Initial program 39.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 39.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 41.5%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. lft-mult-inverse N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 41.5

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

   9. Applied rewrites 41.5%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{x + 1}} \]

   if 5.00000000000000031e-10 < (-.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 87.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-div N/A

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

      4. frac-sub N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-*.f64 90.4

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

   6. Applied rewrites 90.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x - -1}} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(x - -1\right) \cdot x} - x}{x}}{\sqrt{x - -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{x - -1}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (fma 2.0 x 0.5)) (sqrt (- x -1.0))))

C

double code(double x) {
	return (1.0 / fma(2.0, x, 0.5)) / sqrt((x - -1.0));
}

Julia

function code(x)
	return Float64(Float64(1.0 / fma(2.0, x, 0.5)) / sqrt(Float64(x - -1.0)))
end

Wolfram

code[x_] := N[(N[(1.0 / N[(2.0 * x + 0.5), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 41.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 43.8%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. lft-mult-inverse N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 41.8

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

9. Applied rewrites 41.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 97.7%

    \[\leadsto \frac{\frac{\color{blue}{1}}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{x + 1}} \]

13. Final simplification 97.7%

    \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(2, x, 0.5\right)}}{\sqrt{x - -1}} \]
14. Add Preprocessing

Alternative 6: 97.7% 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 41.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 41.5%

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

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 96.5

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

   \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{x - -1}} \]
9. Add Preprocessing

Alternative 7: 97.6% accurate, 1.4× speedup

Math

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

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(Float64(-0.5 / x) / Float64(-sqrt(x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r/ N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 41.5%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.3

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

7. Applied rewrites 96.3%

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

Alternative 8: 80.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 83.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right), 0.5, -0.5 \cdot \sqrt{\frac{1}{x}}\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 82.3%

   \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
9. Add Preprocessing

Alternative 9: 5.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5.8

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

5. Applied rewrites 5.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
6. Add Preprocessing

Developer Target 1: 37.6% 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 2024295 
(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)))))