2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 8.7s

Alternatives: 5

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: 6.7% 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 8.4%

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

   1. flip-- 9.5%

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

   2. div-inv 9.5%

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

   3. add-sqr-sqrt 9.7%

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

   4. add-sqr-sqrt 10.6%

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

   5. associate--l+ 10.6%

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

4. Applied egg-rr 10.6%

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

   1. +-commutative 10.6%

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

   2. associate-+l- 99.6%

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

   3. +-inverses 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate-*r/ 99.6%

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

   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. Final simplification 99.6%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 32500000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 32500000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 32500000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 32500000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 32500000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 32500000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 32500000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25e7

   1. Initial program 90.0%

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

   if 3.25e7 < x

   1. Initial program 5.5%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. inv-pow 99.0%

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

      3. sqrt-pow1 99.2%

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

      4. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. *-lft-identity 99.2%

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

   7. Simplified 99.2%

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 32500000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 8.4%

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

3. Taylor expanded in x around inf 96.6%

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

   1. *-un-lft-identity 96.6%

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

   2. inv-pow 96.6%

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

   3. sqrt-pow1 96.8%

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

   4. metadata-eval 96.8%

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

5. Applied egg-rr 96.8%

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

   1. *-lft-identity 96.8%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 4: 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 8.4%

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

   1. flip-- 9.5%

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

   2. div-inv 9.5%

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

   3. add-sqr-sqrt 9.7%

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

   4. add-sqr-sqrt 10.6%

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

   5. associate--l+ 10.6%

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

4. Applied egg-rr 10.6%

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

   1. +-commutative 10.6%

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

   2. associate-+l- 99.6%

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

   3. +-inverses 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate-*r/ 99.6%

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

   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 96.4%

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

   1. *-commutative 96.4%

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

9. Simplified 96.4%

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

    1. add-sqr-sqrt 96.0%

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

    2. sqrt-unprod 96.4%

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

    3. frac-times 96.0%

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

    4. metadata-eval 96.0%

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

    5. swap-sqr 96.0%

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

    6. add-sqr-sqrt 96.2%

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

    7. metadata-eval 96.2%

       \[\leadsto \sqrt{\frac{1}{x \cdot \color{blue}{4}}} \]

11. Applied egg-rr 96.2%

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

    1. *-commutative 96.2%

       \[\leadsto \sqrt{\frac{1}{\color{blue}{4 \cdot x}}} \]

    2. associate-/r* 96.6%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4}}{x}}} \]

    3. metadata-eval 96.6%

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

13. Simplified 96.6%

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

14. Final simplification 96.6%

    \[\leadsto \sqrt{\frac{0.25}{x}} \]
15. Add Preprocessing

Alternative 5: 1.6% 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 Float64(-sqrt(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. Final simplification 1.6%

   \[\leadsto -\sqrt{x} \]
8. 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 2024077 
(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)))