2sqrt (example 3.1)

Percentage Accurate: 9.2% → 99.6%

Time: 17.3s

Alternatives: 6

Speedup: 2.0×

• could not determine a ground truth

  1. x = 3.373155319120939e+218

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: 9.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p (expm1 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))))

C

double code(double x) {
	return log1p(expm1((1.0 / (sqrt((1.0 + x)) + sqrt(x)))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x)))));
}

Python

def code(x):
	return math.log1p(math.expm1((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x)))))

Julia

function code(x)
	return log1p(expm1(Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 10.1%

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

   1. flip-- 12.2%

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

   2. div-inv 12.2%

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

   3. add-sqr-sqrt 13.2%

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

   4. add-sqr-sqrt 15.8%

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

   5. associate--l+ 15.8%

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

4. Applied egg-rr 15.8%

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

   1. associate-*r/ 15.8%

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

   2. *-rgt-identity 15.8%

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

   3. +-commutative 15.8%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. log1p-expm1-u 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]

9. Final simplification 99.7%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right) \]
10. Add Preprocessing

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 68000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 68000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 68000000.0) {
		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 <= 68000000.0d0) 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 <= 68000000.0) {
		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 <= 68000000.0:
		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 <= 68000000.0)
		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 <= 68000000.0)
		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, 68000000.0], 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 < 6.8e7

   1. Initial program 76.7%

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

   if 6.8e7 < x

   1. Initial program 6.0%

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

      1. flip-- 7.7%

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

      2. div-inv 7.7%

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

      3. add-sqr-sqrt 8.6%

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

      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. associate-*r/ 10.6%

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

      2. *-rgt-identity 10.6%

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

      3. +-commutative 10.6%

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

      4. associate-+l- 99.7%

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

      5. div-sub 99.7%

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

      6. +-inverses 99.7%

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

      7. div0 99.7%

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

      8. --rgt-identity 99.7%

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

      9. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 98.7%

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

      1. rem-exp-log 93.0%

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

      2. exp-neg 93.0%

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

      3. unpow1/2 93.0%

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

      4. exp-prod 93.0%

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

      5. distribute-lft-neg-out 93.0%

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

      6. distribute-rgt-neg-in 93.0%

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

      7. metadata-eval 93.0%

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

      8. exp-to-pow 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 68000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. 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 10.1%

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

   1. flip-- 12.2%

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

   2. div-inv 12.2%

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

   3. add-sqr-sqrt 13.2%

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

   4. add-sqr-sqrt 15.8%

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

   5. associate--l+ 15.8%

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

4. Applied egg-rr 15.8%

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

   1. associate-*r/ 15.8%

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

   2. *-rgt-identity 15.8%

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

   3. +-commutative 15.8%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 96.2% 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 10.1%

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

   1. flip-- 12.2%

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

   2. div-inv 12.2%

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

   3. add-sqr-sqrt 13.2%

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

   4. add-sqr-sqrt 15.8%

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

   5. associate--l+ 15.8%

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

4. Applied egg-rr 15.8%

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

   1. associate-*r/ 15.8%

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

   2. *-rgt-identity 15.8%

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

   3. +-commutative 15.8%

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

   4. associate-+l- 99.7%

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

   5. div-sub 99.7%

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

   6. +-inverses 99.7%

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

   7. div0 99.7%

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

   8. --rgt-identity 99.7%

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

   9. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 95.6%

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

   1. rem-exp-log 90.2%

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

   2. exp-neg 90.2%

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

   3. unpow1/2 90.2%

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

   4. exp-prod 90.2%

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

   5. distribute-lft-neg-out 90.2%

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

   6. distribute-rgt-neg-in 90.2%

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

   7. metadata-eval 90.2%

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

   8. exp-to-pow 95.8%

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

9. Simplified 95.8%

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

10. Final simplification 95.8%

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

Alternative 5: 95.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.1%

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

3. Taylor expanded in x around inf 98.0%

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

4. Taylor expanded in x around inf 95.4%

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

   1. *-commutative 95.4%

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

6. Simplified 95.4%

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

   1. add-sqr-sqrt 95.3%

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

   2. times-frac 95.5%

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

8. Applied egg-rr 95.5%

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

   1. *-inverses 95.5%

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

   2. *-lft-identity 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

    \[\leadsto \frac{0.5}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 6: 6.9% 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 10.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 7.0%

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

   5. rem-sqrt-square 7.0%

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

   6. sqr-neg 7.0%

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

   7. rem-square-sqrt 7.0%

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

5. Simplified 7.0%

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

6. Taylor expanded in x around inf 7.0%

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

7. Final simplification 7.0%

   \[\leadsto \sqrt{x} \]
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 2024084 
(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)))