2sqrt (example 3.1)

Percentage Accurate: 7.1% → 99.6%

Time: 6.9s

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: 7.1% 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 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. div-inv 7.6%

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

   3. add-sqr-sqrt 7.2%

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

   4. add-sqr-sqrt 9.0%

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

   5. associate--l+ 9.0%

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

4. Applied egg-rr 9.0%

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

   1. associate-*r/ 9.0%

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

   2. *-rgt-identity 9.0%

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

   3. +-commutative 9.0%

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

   4. associate-+l- 99.6%

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

   5. div-sub 99.6%

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

   6. +-inverses 99.6%

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

   7. div0 99.6%

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

   8. --rgt-identity 99.6%

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

   9. +-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. Final simplification 99.6%

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

Alternative 2: 97.8% 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 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. div-inv 7.6%

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

   3. add-sqr-sqrt 7.2%

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

   4. add-sqr-sqrt 9.0%

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

   5. associate--l+ 9.0%

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

4. Applied egg-rr 9.0%

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

   1. associate-*r/ 9.0%

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

   2. *-rgt-identity 9.0%

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

   3. +-commutative 9.0%

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

   4. associate-+l- 99.6%

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

   5. div-sub 99.6%

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

   6. +-inverses 99.6%

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

   7. div0 99.6%

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

   8. --rgt-identity 99.6%

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

   9. +-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 97.9%

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

   1. *-commutative 97.9%

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

   2. unpow1/2 97.9%

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

   3. rem-exp-log 91.0%

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

   4. exp-neg 91.0%

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

   5. exp-prod 91.0%

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

   6. distribute-lft-neg-out 91.0%

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

   7. distribute-rgt-neg-in 91.0%

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

   8. metadata-eval 91.0%

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

   9. exp-to-pow 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 3: 97.6% 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 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. div-inv 7.6%

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

   3. add-sqr-sqrt 7.2%

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

   4. add-sqr-sqrt 9.0%

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

   5. associate--l+ 9.0%

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

4. Applied egg-rr 9.0%

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

   1. associate-*r/ 9.0%

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

   2. *-rgt-identity 9.0%

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

   3. +-commutative 9.0%

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

   4. associate-+l- 99.6%

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

   5. div-sub 99.6%

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

   6. +-inverses 99.6%

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

   7. div0 99.6%

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

   8. --rgt-identity 99.6%

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

   9. +-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 97.7%

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

   1. add-sqr-sqrt 97.2%

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

   2. sqrt-unprod 97.7%

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

   3. associate-/r* 97.7%

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

   4. metadata-eval 97.7%

      \[\leadsto \sqrt{\frac{\color{blue}{0.5}}{\sqrt{x}} \cdot \frac{1}{2 \cdot \sqrt{x}}} \]

   5. associate-/r* 97.7%

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{x}} \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{x}}}} \]

   6. metadata-eval 97.7%

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

   7. frac-times 97.6%

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

   8. metadata-eval 97.6%

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

   9. add-sqr-sqrt 97.9%

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

9. Applied egg-rr 97.9%

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

10. Final simplification 97.9%

    \[\leadsto \sqrt{\frac{0.25}{x}} \]
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 6.7%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 5.5%

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

   5. rem-sqrt-square 5.5%

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

   6. sqr-neg 5.5%

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

   7. rem-square-sqrt 5.5%

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

5. Simplified 5.5%

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

6. Taylor expanded in x around inf 5.5%

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

7. Final simplification 5.5%

   \[\leadsto \sqrt{x} \]
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 2024060 
(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)))