2sqrt (example 3.1)

Percentage Accurate: 6.6% → 99.5%

Time: 8.8s

Alternatives: 6

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: 6.6% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt((1.0 + x)) + sqrt(x)), 2.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt((1.0d0 + x)) + sqrt(x)) ** 2.0d0) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt((1.0 + x)) + Math.sqrt(x)), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.7%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. +-commutative 8.4%

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

   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%

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

6. Simplified 99.5%

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

   1. inv-pow 99.5%

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

   2. sqr-pow 99.1%

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

   3. pow-prod-down 99.6%

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

   4. pow2 99.6%

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

   5. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

   \[\leadsto {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) -1.0))

C

double code(double x) {
	return pow((sqrt((1.0 + x)) + sqrt(x)), -1.0);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), -1.0)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ -1.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.7%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. +-commutative 8.4%

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

   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%

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

6. Simplified 99.5%

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

   1. inv-pow 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1} \]
10. Add Preprocessing

Alternative 3: 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 6.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.7%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. +-commutative 8.4%

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

   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%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (sqrt (/ 1.0 x))))

C

double code(double x) {
	return 0.5 * sqrt((1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * sqrt((1.0d0 / x))
end function

Java

public static double code(double x) {
	return 0.5 * Math.sqrt((1.0 / x));
}

Python

def code(x):
	return 0.5 * math.sqrt((1.0 / x))

Julia

function code(x)
	return Float64(0.5 * sqrt(Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * sqrt((1.0 / x));
end

Wolfram

code[x_] := N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.7%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. +-commutative 8.4%

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

   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%

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

6. Simplified 99.5%

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

   1. inv-pow 99.5%

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

   2. sqr-pow 99.1%

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

   3. pow-prod-down 99.6%

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

   4. pow2 99.6%

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

   5. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5}} \]
9. Step-by-step derivation

   1. pow-pow 99.5%

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

   2. metadata-eval 99.5%

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

   3. inv-pow 99.5%

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

   4. flip3-+ 66.4%

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

   5. associate-/r/ 66.4%

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

   6. sqrt-pow2 66.3%

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

   7. metadata-eval 66.3%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   8. sqrt-pow2 66.1%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   9. metadata-eval 66.1%

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

   10. add-sqr-sqrt 66.3%

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

   11. add-sqr-sqrt 66.2%

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

   12. associate-+r- 66.2%

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

10. Applied egg-rr 47.9%

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

11. Taylor expanded in x around inf 98.0%

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

12. Final simplification 98.0%

    \[\leadsto 0.5 \cdot \sqrt{\frac{1}{x}} \]
13. Add Preprocessing

Alternative 5: 7.0% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 x))

C

double code(double x) {
	return 2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / x
end function

Java

public static double code(double x) {
	return 2.0 / x;
}

Python

def code(x):
	return 2.0 / x

Julia

function code(x)
	return Float64(2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / x;
end

Wolfram

code[x_] := N[(2.0 / x), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.7%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. +-commutative 8.4%

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

   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%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 7.0%

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

8. Taylor expanded in x around inf 7.0%

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

9. Final simplification 7.0%

   \[\leadsto \frac{2}{x} \]
10. Add Preprocessing

Alternative 6: 6.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 6.6%

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

3. Taylor expanded in x around 0 6.9%

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

4. Final simplification 6.9%

   \[\leadsto 1 \]
5. 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 2024040 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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