2sqrt (example 3.1)

Percentage Accurate: 6.9% → 99.0%

Time: 11.2s

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: 6.9% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.125}{\sqrt{x}} + \left(\frac{0.0625}{{x}^{1.5}} + \sqrt{x} \cdot 0.5\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ -0.125 (sqrt x)) (+ (/ 0.0625 (pow x 1.5)) (* (sqrt x) 0.5))) x))

C

double code(double x) {
	return ((-0.125 / sqrt(x)) + ((0.0625 / pow(x, 1.5)) + (sqrt(x) * 0.5))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.125d0) / sqrt(x)) + ((0.0625d0 / (x ** 1.5d0)) + (sqrt(x) * 0.5d0))) / x
end function

Java

public static double code(double x) {
	return ((-0.125 / Math.sqrt(x)) + ((0.0625 / Math.pow(x, 1.5)) + (Math.sqrt(x) * 0.5))) / x;
}

Python

def code(x):
	return ((-0.125 / math.sqrt(x)) + ((0.0625 / math.pow(x, 1.5)) + (math.sqrt(x) * 0.5))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-0.125 / sqrt(x)) + Float64(Float64(0.0625 / (x ^ 1.5)) + Float64(sqrt(x) * 0.5))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-0.125 / sqrt(x)) + ((0.0625 / (x ^ 1.5)) + (sqrt(x) * 0.5))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \color{blue}{\frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(0.0625 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

   1. sqrt-div 99.6%

      \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(0.0625 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{{x}^{3}}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   2. metadata-eval 99.6%

      \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(0.0625 \cdot \frac{\color{blue}{1}}{\sqrt{{x}^{3}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   3. sqrt-pow1 99.6%

      \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(0.0625 \cdot \frac{1}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   4. un-div-inv 99.6%

      \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(\color{blue}{\frac{0.0625}{{x}^{\left(\frac{3}{2}\right)}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(\frac{0.0625}{{x}^{\color{blue}{1.5}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

5. Applied egg-rr 99.6%

   \[\leadsto \frac{-0.125 \cdot \sqrt{\frac{1}{x}} + \left(\color{blue}{\frac{0.0625}{{x}^{1.5}}} + 0.5 \cdot \sqrt{x}\right)}{x} \]
6. Step-by-step derivation

   1. sqrt-div 99.6%

      \[\leadsto \frac{-0.125 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} + \left(\frac{0.0625}{{x}^{1.5}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   2. metadata-eval 99.6%

      \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{1}}{\sqrt{x}} + \left(\frac{0.0625}{{x}^{1.5}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

   3. un-div-inv 99.6%

      \[\leadsto \frac{\color{blue}{\frac{-0.125}{\sqrt{x}}} + \left(\frac{0.0625}{{x}^{1.5}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{\color{blue}{\frac{-0.125}{\sqrt{x}}} + \left(\frac{0.0625}{{x}^{1.5}} + 0.5 \cdot \sqrt{x}\right)}{x} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{-0.125}{\sqrt{x}} + \left(\frac{0.0625}{{x}^{1.5}} + \sqrt{x} \cdot 0.5\right)}{x} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $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-- 7.5%

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

   2. div-inv 7.5%

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

   3. add-sqr-sqrt 8.1%

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

   4. add-sqr-sqrt 9.1%

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

   5. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-rgt-identity 9.1%

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

   3. +-commutative 9.1%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. flip-- 7.5%

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

   2. div-inv 7.5%

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

   3. add-sqr-sqrt 8.1%

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

   4. add-sqr-sqrt 9.1%

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

   5. associate--l+ 9.1%

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

4. Applied egg-rr 9.1%

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

   1. associate-*r/ 9.1%

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

   2. *-rgt-identity 9.1%

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

   3. +-commutative 9.1%

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

   4. associate-+l- 99.5%

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

   5. +-inverses 99.5%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around inf 98.2%

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

   1. unpow-1 98.2%

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

   2. metadata-eval 98.2%

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

   3. pow-sqr 98.4%

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

   4. rem-sqrt-square 98.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-0.5}\right|} \]

   5. rem-square-sqrt 97.7%

      \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

   6. fabs-sqr 97.7%

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

   7. rem-square-sqrt 98.4%

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

9. Simplified 98.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{-0.5}} \]
10. Add Preprocessing

Alternative 4: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.25 x)))

C

double code(double x) {
	return sqrt((0.25 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.25d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.25 / x));
}

Python

def code(x):
	return math.sqrt((0.25 / x))

Julia

function code(x)
	return sqrt(Float64(0.25 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.25 / x));
end

Wolfram

code[x_] := N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 98.2%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.2%

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

   3. *-commutative 98.2%

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

   4. *-commutative 98.2%

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

   5. swap-sqr 98.2%

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

   6. add-sqr-sqrt 98.2%

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

   7. metadata-eval 98.2%

      \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.25}} \]

5. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.25}} \]
6. Step-by-step derivation

   1. associate-*l/ 98.2%

      \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.25}{x}}} \]

   2. metadata-eval 98.2%

      \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{x}} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
8. Add Preprocessing

Alternative 5: 18.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around 0 1.6%

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

4. Taylor expanded in x around inf 1.6%

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

   1. neg-mul-1 1.6%

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

6. Simplified 1.6%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 5.3%

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

   3. sqr-neg 5.3%

      \[\leadsto \sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   4. add-exp-log 5.3%

      \[\leadsto \sqrt{\color{blue}{e^{\log \left(\sqrt{x} \cdot \sqrt{x}\right)}}} \]

   5. add-sqr-sqrt 5.3%

      \[\leadsto \sqrt{e^{\log \color{blue}{x}}} \]

   6. add-sqr-sqrt 5.3%

      \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}} \]

   7. sqrt-unprod 5.3%

      \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\log x \cdot \log x}}}} \]

   8. sqr-neg 5.3%

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

   9. sqrt-unprod 0.0%

      \[\leadsto \sqrt{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}} \]

   10. add-sqr-sqrt 18.7%

       \[\leadsto \sqrt{e^{\color{blue}{-\log x}}} \]

   11. neg-log 18.7%

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

   12. inv-pow 18.7%

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

   13. add-exp-log 18.7%

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

   14. sqrt-pow1 18.7%

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

   15. metadata-eval 18.7%

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

8. Applied egg-rr 18.7%

   \[\leadsto \color{blue}{{x}^{-0.5}} \]
9. Add Preprocessing

Alternative 6: 7.0% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around 0 1.6%

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

4. Taylor expanded in x around inf 1.6%

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

   1. neg-mul-1 1.6%

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

6. Simplified 1.6%

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

8. Applied egg-rr 6.9%

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

Alternative 7: 7.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 98.2%

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

   1. pow1/2 98.2%

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

   2. pow-to-exp 91.2%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\frac{1}{x}\right) \cdot 0.5}} \]

   3. log-rec 91.2%

      \[\leadsto 0.5 \cdot e^{\color{blue}{\left(-\log x\right)} \cdot 0.5} \]

5. Applied egg-rr 91.2%

   \[\leadsto 0.5 \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]

7. Applied egg-rr 6.9%

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

   1. *-commutative 6.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot x}}{x} \]

   2. associate-/l* 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{x}} \]

   3. *-inverses 6.9%

      \[\leadsto 0.5 \cdot \color{blue}{1} \]

   4. metadata-eval 6.9%

      \[\leadsto \color{blue}{0.5} \]

9. Simplified 6.9%

   \[\leadsto \color{blue}{0.5} \]
10. Add Preprocessing

Developer Target 1: 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]

Developer Target 2: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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

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