2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.5%

Time: 10.1s

Alternatives: 7

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt{x} + \sqrt{1 + x}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt x) (sqrt (+ 1.0 x))) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt(x) + sqrt((1.0 + x))), 2.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt(x) + sqrt((1.0d0 + x))) ** 2.0d0) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt(x) + Math.sqrt((1.0 + x))), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt(x) + math.sqrt((1.0 + x))), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(x) + sqrt(Float64(1.0 + x))) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt(x) + sqrt((1.0 + x))) ^ 2.0) ^ -0.5;
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. inv-pow 99.6%

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

   4. inv-pow 99.6%

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

   5. pow-prod-up 99.5%

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

   6. +-commutative 99.5%

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

   7. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. sqrt-pow1 99.6%

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

   2. metadata-eval 99.6%

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

   3. metadata-eval 99.6%

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

   4. pow-prod-up 99.2%

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

   5. pow-prod-down 99.6%

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

   6. pow2 99.6%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

    \[\leadsto {\left({\left(\sqrt{x} + \sqrt{1 + x}\right)}^{2}\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 2: 22.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.35 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.35e+15)
   (- (sqrt (+ 1.0 x)) (sqrt x))
   (/ 1.0 (sqrt (+ 1.0 (* x 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.35e+15) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 1.0 / sqrt((1.0 + (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.35d+15) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 1.0d0 / sqrt((1.0d0 + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.35e+15) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / Math.sqrt((1.0 + (x * 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.35e+15:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 1.0 / math.sqrt((1.0 + (x * 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.35e+15)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(1.0 / sqrt(Float64(1.0 + Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.35e+15)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 1.0 / sqrt((1.0 + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.35e+15], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.35e15

   1. Initial program 51.4%

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

   if 4.35e15 < x

   1. Initial program 3.9%

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

      1. flip-- 3.9%

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

      2. add-sqr-sqrt 3.8%

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

      3. add-sqr-sqrt 3.9%

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

      4. associate--l+ 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. frac-2neg 3.9%

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

      2. div-inv 3.9%

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

   6. Applied egg-rr 3.9%

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

      1. associate--r+ 20.3%

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

      2. +-inverses 20.3%

         \[\leadsto \left(\color{blue}{0} - 1\right) \cdot \frac{1}{-\sqrt{x + \left(x + 1\right)}} \]

      3. metadata-eval 20.3%

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

      4. associate-*r/ 20.3%

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

      5. metadata-eval 20.3%

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

      6. neg-mul-1 20.3%

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

      7. associate-/r* 20.3%

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

      8. metadata-eval 20.3%

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

      9. +-commutative 20.3%

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

      10. +-commutative 20.3%

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

      11. associate-+l+ 20.3%

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

      12. count-2 20.3%

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

   8. Simplified 20.3%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.35 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \frac{1}{\sqrt{x} + \sqrt{1 + x}} \]
7. Add Preprocessing

Alternative 4: 20.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x + \left(1 + x\right)}{\sqrt{x \cdot 2}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ (+ x (+ 1.0 x)) (sqrt (* x 2.0)))))

C

double code(double x) {
	return 1.0 / ((x + (1.0 + x)) / sqrt((x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x + (1.0d0 + x)) / sqrt((x * 2.0d0)))
end function

Java

public static double code(double x) {
	return 1.0 / ((x + (1.0 + x)) / Math.sqrt((x * 2.0)));
}

Python

def code(x):
	return 1.0 / ((x + (1.0 + x)) / math.sqrt((x * 2.0)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x + Float64(1.0 + x)) / sqrt(Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x + (1.0 + x)) / sqrt((x * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

   1. flip-+ 3.0%

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

   2. add-sqr-sqrt 4.7%

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

   3. add-sqr-sqrt 2.9%

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

   4. flip-- 4.0%

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

   5. add-sqr-sqrt 3.6%

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

   6. add-sqr-sqrt 5.5%

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

   7. associate-+r- 5.5%

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

   8. associate-+r- 5.5%

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

6. Applied egg-rr 4.8%

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

7. Taylor expanded in x around inf 4.8%

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

   1. *-commutative 4.8%

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

9. Simplified 4.8%

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

10. Taylor expanded in x around 0 20.3%

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

11. Final simplification 20.3%

    \[\leadsto \frac{1}{\frac{x + \left(1 + x\right)}{\sqrt{x \cdot 2}}} \]
12. Add Preprocessing

Alternative 5: 20.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{1 + x \cdot 2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (sqrt (+ 1.0 (* x 2.0)))))

C

double code(double x) {
	return 1.0 / sqrt((1.0 + (x * 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / sqrt((1.0d0 + (x * 2.0d0)))
end function

Java

public static double code(double x) {
	return 1.0 / Math.sqrt((1.0 + (x * 2.0)));
}

Python

def code(x):
	return 1.0 / math.sqrt((1.0 + (x * 2.0)))

Julia

function code(x)
	return Float64(1.0 / sqrt(Float64(1.0 + Float64(x * 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / sqrt((1.0 + (x * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

   1. frac-2neg 9.1%

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

   2. div-inv 9.1%

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

6. Applied egg-rr 4.8%

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

   1. associate--r+ 20.3%

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

   2. +-inverses 20.3%

      \[\leadsto \left(\color{blue}{0} - 1\right) \cdot \frac{1}{-\sqrt{x + \left(x + 1\right)}} \]

   3. metadata-eval 20.3%

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

   4. associate-*r/ 20.3%

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

   5. metadata-eval 20.3%

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

   6. neg-mul-1 20.3%

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

   7. associate-/r* 20.3%

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

   8. metadata-eval 20.3%

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

   9. +-commutative 20.3%

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

   10. +-commutative 20.3%

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

   11. associate-+l+ 20.3%

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

   12. count-2 20.3%

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

8. Simplified 20.3%

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

9. Final simplification 20.3%

   \[\leadsto \frac{1}{\sqrt{1 + x \cdot 2}} \]
10. Add Preprocessing

Alternative 6: 7.0% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 x))

C

double code(double x) {
	return 2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / x
end function

Java

public static double code(double x) {
	return 2.0 / x;
}

Python

def code(x):
	return 2.0 / x

Julia

function code(x)
	return Float64(2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / x;
end

Wolfram

code[x_] := N[(2.0 / x), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.6%

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

   2. add-sqr-sqrt 7.1%

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

   3. add-sqr-sqrt 9.1%

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

   4. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around 0 4.4%

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

6. Taylor expanded in x around inf 7.1%

   \[\leadsto \color{blue}{\frac{2}{x}} \]

7. Final simplification 7.1%

   \[\leadsto \frac{2}{x} \]
8. Add Preprocessing

Alternative 7: 6.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around 0 7.0%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 7.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024036 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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