2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.6%

Time: 6.5s

Alternatives: 4

Speedup: 2.0×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

   1. flip-- 5.9%

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

   2. add-sqr-sqrt 6.5%

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

   3. add-sqr-sqrt 7.2%

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

   4. div-sub 5.8%

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

4. Applied egg-rr 5.8%

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

   1. div-sub 7.2%

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

   2. +-commutative 7.2%

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

   3. associate--l+ 99.6%

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

   4. +-inverses 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-commutative 99.6%

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

6. Simplified 99.6%

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

Alternative 2: 98.1% 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 5.7%

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

   1. flip-- 5.9%

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

   2. add-sqr-sqrt 6.5%

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

   3. add-sqr-sqrt 7.2%

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

   4. div-sub 5.8%

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

4. Applied egg-rr 5.8%

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

   1. div-sub 7.2%

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

   2. +-commutative 7.2%

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

   3. associate--l+ 99.6%

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

   4. +-inverses 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 98.6%

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

   1. unpow1/2 98.6%

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

   2. rem-exp-log 91.2%

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

   3. exp-neg 91.2%

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

   4. exp-prod 91.2%

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

   5. distribute-lft-neg-out 91.2%

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

   6. distribute-rgt-neg-in 91.2%

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

   7. metadata-eval 91.2%

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

   8. exp-to-pow 98.8%

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

9. Simplified 98.8%

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

Alternative 3: 97.9% 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 5.7%

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

3. Taylor expanded in x around inf 98.9%

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

4. Taylor expanded in x around inf 98.4%

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

   1. *-commutative 98.4%

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

6. Simplified 98.4%

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

   1. add-sqr-sqrt 98.0%

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

   2. sqrt-unprod 98.3%

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

   3. associate-/l* 98.2%

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

   4. div-inv 98.3%

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

   5. metadata-eval 98.3%

      \[\leadsto \sqrt{\left(\sqrt{x} \cdot \left(0.5 \cdot \frac{\color{blue}{\sqrt[3]{1}}}{x}\right)\right) \cdot \frac{\sqrt{x} \cdot 0.5}{x}} \]

   6. rem-cbrt-cube 32.6%

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

   7. cbrt-div 32.6%

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

   8. associate-*l* 32.6%

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

   9. div-inv 32.6%

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

   10. metadata-eval 32.6%

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

   11. rem-cbrt-cube 32.6%

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

   12. cbrt-div 32.5%

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

   13. swap-sqr 32.5%

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

   14. swap-sqr 32.5%

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

   15. add-sqr-sqrt 32.5%

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

   16. metadata-eval 32.5%

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

8. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{\sqrt{\left(x \cdot 0.25\right) \cdot {x}^{-2}}} \]
9. Step-by-step derivation

   1. *-commutative 52.5%

      \[\leadsto \sqrt{\color{blue}{{x}^{-2} \cdot \left(x \cdot 0.25\right)}} \]

   2. associate-*r* 52.5%

      \[\leadsto \sqrt{\color{blue}{\left({x}^{-2} \cdot x\right) \cdot 0.25}} \]

   3. pow-plus 98.6%

      \[\leadsto \sqrt{\color{blue}{{x}^{\left(-2 + 1\right)}} \cdot 0.25} \]

   4. metadata-eval 98.6%

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

   5. unpow-1 98.6%

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

   6. associate-*l/ 98.6%

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

   7. metadata-eval 98.6%

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

10. Simplified 98.6%

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

Alternative 4: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Taylor expanded in x around 0 1.6%

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

   1. sub-neg 1.6%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.3%

      \[\leadsto 1 + \left|\color{blue}{-\sqrt{x}}\right| \]

   5. rem-sqrt-square 5.3%

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

   6. sqr-neg 5.3%

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

   7. rem-square-sqrt 5.3%

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

5. Simplified 5.3%

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

6. Taylor expanded in x around inf 5.3%

   \[\leadsto \color{blue}{\sqrt{x}} \]
7. 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 2024096 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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