2isqrt (example 3.6)

Percentage Accurate: 38.8% → 99.0%

Time: 7.6s

Alternatives: 10

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: 38.8% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{0.125}{x} + 0.5\right) - \frac{0.0625}{x \cdot x}}{x + 1}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (+ (/ 0.125 x) 0.5) (/ 0.0625 (* x x))) (+ x 1.0)) (sqrt x)))

C

double code(double x) {
	return ((((0.125 / x) + 0.5) - (0.0625 / (x * x))) / (x + 1.0)) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((0.125d0 / x) + 0.5d0) - (0.0625d0 / (x * x))) / (x + 1.0d0)) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((((0.125 / x) + 0.5) - (0.0625 / (x * x))) / (x + 1.0)) / Math.sqrt(x);
}

Python

def code(x):
	return ((((0.125 / x) + 0.5) - (0.0625 / (x * x))) / (x + 1.0)) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(0.125 / x) + 0.5) - Float64(0.0625 / Float64(x * x))) / Float64(x + 1.0)) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((((0.125 / x) + 0.5) - (0.0625 / (x * x))) / (x + 1.0)) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.125 / x), $MachinePrecision] + 0.5), $MachinePrecision] - N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. 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. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 39.3

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

4. Applied rewrites 39.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 6.5%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x}\right) - \frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x}\right) - \frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)} - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)} - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{8}}}{x} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{8}}{x}} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{8}}{x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{8}}{x} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{\color{blue}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]

   9. lower-*.f64 99.6

      \[\leadsto \frac{\frac{\left(\frac{0.125}{x} + 0.5\right) - \frac{0.0625}{\color{blue}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{\color{blue}{\left(\frac{0.125}{x} + 0.5\right) - \frac{0.0625}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]
10. Add Preprocessing

Alternative 2: 38.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} + x\\ \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;{t\_0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt x) x)))
   (if (<= x 4.6e+153) (pow t_0 -1.0) (/ (- (+ x 1.0) x) t_0))))

C

double code(double x) {
	double t_0 = sqrt(x) + x;
	double tmp;
	if (x <= 4.6e+153) {
		tmp = pow(t_0, -1.0);
	} else {
		tmp = ((x + 1.0) - 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) + x
    if (x <= 4.6d+153) then
        tmp = t_0 ** (-1.0d0)
    else
        tmp = ((x + 1.0d0) - x) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(x) + x;
	double tmp;
	if (x <= 4.6e+153) {
		tmp = Math.pow(t_0, -1.0);
	} else {
		tmp = ((x + 1.0) - x) / t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(x) + x
	tmp = 0
	if x <= 4.6e+153:
		tmp = math.pow(t_0, -1.0)
	else:
		tmp = ((x + 1.0) - x) / t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(x) + x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = t_0 ^ -1.0;
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(x) + x;
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = t_0 ^ -1.0;
	else
		tmp = ((x + 1.0) - x) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, 4.6e+153], N[Power[t$95$0, -1.0], $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 7.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. 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. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 7.9

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

   4. Applied rewrites 7.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      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

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lower--.f64 N/A

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

   6. Applied rewrites 14.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-+.f64 N/A

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

       5. lower-sqrt.f64 8.5

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

   11. Applied rewrites 8.5%

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

   if 4.6000000000000003e153 < x

   1. Initial program 67.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. 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. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 67.9

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

   4. Applied rewrites 67.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      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

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lower--.f64 N/A

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 67.9

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

   9. Applied rewrites 67.9%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;{\left(\sqrt{x} + x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{x} + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;{\left(\sqrt{x} + x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+153) (pow (+ (sqrt x) x) -1.0) (/ 0.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = pow((sqrt(x) + x), -1.0);
	} else {
		tmp = 0.0 / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.6d+153) then
        tmp = (sqrt(x) + x) ** (-1.0d0)
    else
        tmp = 0.0d0 / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.6e+153) {
		tmp = Math.pow((Math.sqrt(x) + x), -1.0);
	} else {
		tmp = 0.0 / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.6e+153:
		tmp = math.pow((math.sqrt(x) + x), -1.0)
	else:
		tmp = 0.0 / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+153)
		tmp = Float64(sqrt(x) + x) ^ -1.0;
	else
		tmp = Float64(0.0 / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.6e+153)
		tmp = (sqrt(x) + x) ^ -1.0;
	else
		tmp = 0.0 / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+153], N[Power[N[(N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision], -1.0], $MachinePrecision], N[(0.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000003e153

   1. Initial program 7.7%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. 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. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 7.9

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

   4. Applied rewrites 7.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      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

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lower--.f64 N/A

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

   6. Applied rewrites 14.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. rem-square-sqrt N/A

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

       4. lower-+.f64 N/A

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

       5. lower-sqrt.f64 8.5

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

   11. Applied rewrites 8.5%

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

   if 4.6000000000000003e153 < x

   1. Initial program 67.9%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. 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. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-lft-identity N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 67.9

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

   4. Applied rewrites 67.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-div N/A

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

      4. frac-sub N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 1.5%

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

   7. Taylor expanded in x around -inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. metadata-eval 67.9

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{x}} \]

   9. Applied rewrites 67.9%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;{\left(\sqrt{x} + x\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. 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. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 39.3

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

4. Applied rewrites 39.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 6.5%

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

7. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (- 0.5 (/ 0.375 x)) x) (sqrt x)))

C

double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((0.5d0 - (0.375d0 / x)) / x) / sqrt(x)
end function

Java

public static double code(double x) {
	return ((0.5 - (0.375 / x)) / x) / Math.sqrt(x);
}

Python

def code(x):
	return ((0.5 - (0.375 / x)) / x) / math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(Float64(0.5 - Float64(0.375 / x)) / x) / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = ((0.5 - (0.375 / x)) / x) / sqrt(x);
end

Wolfram

code[x_] := N[(N[(N[(0.5 - N[(0.375 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. 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. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 39.3

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}}{\sqrt{x}} \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{1} \cdot \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}{x}}{\sqrt{x}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{8} \cdot x + \frac{1}{4} \cdot x}{{x}^{2}}}}{x}}{\sqrt{x}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{x \cdot \left(\frac{1}{8} + \frac{1}{4}\right)}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{x \cdot \color{blue}{\frac{3}{8}}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{3}{8} \cdot x}}{{x}^{2}}}{x}}{\sqrt{x}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\frac{3}{8} \cdot x}{\color{blue}{x \cdot x}}}{x}}{\sqrt{x}} \]

   10. times-frac N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{3}{8}}{x} \cdot \frac{x}{x}}}{x}}{\sqrt{x}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{1}{8} + \frac{1}{4}}}{x} \cdot \frac{x}{x}}{x}}{\sqrt{x}} \]

   12. div-add-rev N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right)} \cdot \frac{x}{x}}{x}}{\sqrt{x}} \]

   13. *-rgt-identity N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \frac{\color{blue}{x \cdot 1}}{x}}{x}}{\sqrt{x}} \]

   14. associate-*r/ N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{x}\right)}}{x}}{\sqrt{x}} \]

   15. rgt-mult-inverse N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot \color{blue}{1}}{x}}{\sqrt{x}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\left(\frac{\frac{1}{8}}{x} + \frac{\frac{1}{4}}{x}\right) \cdot 1}}{x}}{\sqrt{x}} \]

   17. div-add-rev N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \color{blue}{\frac{\frac{1}{8} + \frac{1}{4}}{x}} \cdot 1}{x}}{\sqrt{x}} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{1}{2} - \frac{\color{blue}{\frac{3}{8}}}{x} \cdot 1}{x}}{\sqrt{x}} \]

   19. lower-/.f64 99.3

       \[\leadsto \frac{\frac{0.5 - \color{blue}{\frac{0.375}{x}} \cdot 1}{x}}{\sqrt{x}} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\frac{0.5 - \frac{0.375}{x} \cdot 1}{x}}}{\sqrt{x}} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\frac{0.5 - \frac{0.375}{x}}{x}}{\sqrt{x}}} \]
10. Add Preprocessing

Alternative 7: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. 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. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 39.3

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

4. Applied rewrites 39.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 6.5%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x}\right) - \frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{x}\right) - \frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)} - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)} - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{\frac{1}{8}}}{x} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{1}{8}}{x}} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{{x}^{2}}}{x + 1}}{\sqrt{x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{8}}{x} + \frac{1}{2}\right) - \color{blue}{\frac{\frac{1}{16}}{{x}^{2}}}}{x + 1}}{\sqrt{x}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\frac{\frac{1}{8}}{x} + \frac{1}{2}\right) - \frac{\frac{1}{16}}{\color{blue}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]

   9. lower-*.f64 99.6

      \[\leadsto \frac{\frac{\left(\frac{0.125}{x} + 0.5\right) - \frac{0.0625}{\color{blue}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{\color{blue}{\left(\frac{0.125}{x} + 0.5\right) - \frac{0.0625}{x \cdot x}}}{x + 1}}{\sqrt{x}} \]

10. Taylor expanded in x around inf

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

12. Applied rewrites 98.8%

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

Alternative 8: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Applied rewrites 82.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\sqrt{\frac{1}{{x}^{5}}} \cdot \mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, -\sqrt{\frac{1}{x}}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.4%

   \[\leadsto \frac{-0.5 \cdot \left(\frac{0.75 \cdot \sqrt{\frac{1}{x}}}{x} - \sqrt{\frac{1}{x}}\right)}{\color{blue}{x}} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 98.7%

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

13. Applied rewrites 98.7%

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

Alternative 9: 80.9% 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 39.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{5}}} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right)}{{x}^{2}}} \]

5. Applied rewrites 82.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\sqrt{\frac{1}{{x}^{5}}} \cdot \mathsf{fma}\left(0.5, x, 1\right) - \sqrt{x}\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, -\sqrt{\frac{1}{x}}\right)\right)}{x \cdot x}} \]

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 82.1%

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

Alternative 10: 36.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.0 (sqrt x)))

C

double code(double x) {
	return 0.0 / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 / sqrt(x)
end function

Java

public static double code(double x) {
	return 0.0 / Math.sqrt(x);
}

Python

def code(x):
	return 0.0 / math.sqrt(x)

Julia

function code(x)
	return Float64(0.0 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 / sqrt(x);
end

Wolfram

code[x_] := N[(0.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. 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. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-lft-identity N/A

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

   10. *-rgt-identity N/A

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

   11. lower--.f64 39.3

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

4. Applied rewrites 39.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-div N/A

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

   4. frac-sub N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 6.5%

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

7. Taylor expanded in x around -inf

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

   1. unpow2 N/A

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

   2. rem-square-sqrt N/A

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

   3. metadata-eval 37.6

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{x}} \]

9. Applied rewrites 37.6%

   \[\leadsto \frac{\color{blue}{0}}{\sqrt{x}} \]
10. Add Preprocessing

Developer Target 1: 38.8% 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 2024339 
(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)))))