2sqrt (example 3.1)

Percentage Accurate: 6.9% → 99.6%

Time: 8.9s

Alternatives: 5

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: 99.6% 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 6.2%

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

   1. flip-- 6.5%

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

   2. div-inv 6.5%

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

   3. add-sqr-sqrt 7.0%

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

   4. add-sqr-sqrt 7.6%

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

   5. associate--l+ 7.6%

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

4. Applied egg-rr 7.6%

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

   1. associate-*r/ 7.6%

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

   2. *-rgt-identity 7.6%

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

   3. +-commutative 7.6%

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

   4. associate-+l- 99.6%

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

   5. +-inverses 99.6%

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

   6. metadata-eval 99.6%

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

   7. +-commutative 99.6%

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

   8. +-commutative 99.6%

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

6. Simplified 99.6%

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

Alternative 2: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

   1. flip-- 6.5%

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

   2. div-inv 6.5%

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

   3. add-sqr-sqrt 7.0%

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

   4. add-sqr-sqrt 7.6%

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

   5. associate--l+ 7.6%

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

4. Applied egg-rr 7.6%

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

   1. associate-*r/ 7.6%

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

   2. *-rgt-identity 7.6%

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

   3. +-commutative 7.6%

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

   4. associate-+l- 99.6%

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

   5. +-inverses 99.6%

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

   6. metadata-eval 99.6%

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

   7. +-commutative 99.6%

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

   8. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around inf 98.3%

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

   1. *-commutative 98.3%

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

   2. unpow-1 98.3%

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

   3. metadata-eval 98.3%

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

   4. pow-sqr 98.4%

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

   5. rem-sqrt-square 98.4%

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

   6. rem-square-sqrt 97.6%

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

   7. fabs-sqr 97.6%

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

   8. rem-square-sqrt 98.4%

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

9. Simplified 98.4%

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

Alternative 3: 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 6.2%

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

3. Taylor expanded in x around inf 98.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
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 98.3%

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

   3. *-commutative 98.3%

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

   4. *-commutative 98.3%

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

   5. swap-sqr 98.3%

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

   6. add-sqr-sqrt 98.3%

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

   7. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*l/ 98.3%

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

   2. metadata-eval 98.3%

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

7. Simplified 98.3%

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

Alternative 4: 7.0% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around inf 98.3%

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

   1. sqrt-div 98.1%

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

   2. metadata-eval 98.1%

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

   3. un-div-inv 98.1%

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

5. Applied egg-rr 98.1%

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

7. Applied egg-rr 6.9%

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

   1. unpow-1 6.9%

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

   2. *-commutative 6.9%

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

   3. associate-/r* 6.9%

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

   4. metadata-eval 6.9%

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

9. Simplified 6.9%

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

Alternative 5: 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 6.2%

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

3. Taylor expanded in x around inf 98.3%

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

   1. pow1/2 98.3%

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

   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]

Reproduce

herbie shell --seed 2024123 
(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))))

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