2sqrt (example 3.1)

Percentage Accurate: 6.7% → 99.6%

Time: 11.6s

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 7.9%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right), \color{blue}{\left(\sqrt{x + 1} + \sqrt{x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - \sqrt{x} \cdot \sqrt{x}\right), \left(\sqrt{\color{blue}{x + 1}} + \sqrt{x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 - x\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\sqrt{x + 1}\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 9.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 9.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 99.5%

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

   1. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\sqrt{x + 1}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\left(x + 1\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   4. +-lowering-+.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 0.0001:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0001) (/ (pow x -0.5) 2.0) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = pow(x, -0.5) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 0.0001d0) then
        tmp = (x ** (-0.5d0)) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = Math.pow(x, -0.5) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 0.0001:
		tmp = math.pow(x, -0.5) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64((x ^ -0.5) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = (x ^ -0.5) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(N[Power[x, -0.5], $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   5. Simplified 99.2%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{{x}^{\frac{-1}{2}}}{2} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({x}^{\frac{-1}{2}}\right), \color{blue}{2}\right) \]

      12. pow-lowering-pow.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), 2\right) \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{2}} \]

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 82.6%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;\frac{{x}^{-0.5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.5}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x -0.5) 2.0))

C

double code(double x) {
	return pow(x, -0.5) / 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, -0.5) / 2.0

Julia

function code(x)
	return Float64((x ^ -0.5) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 96.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 96.9%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. inv-pow N/A

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

   9. sqrt-pow2 N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{{x}^{\frac{-1}{2}}}{2} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left({x}^{\frac{-1}{2}}\right), \color{blue}{2}\right) \]

   12. pow-lowering-pow.f64 97.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), 2\right) \]

7. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{2}} \]
8. Add Preprocessing

Alternative 4: 97.7% 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 7.9%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

   3. /-lowering-/.f64 96.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

5. Simplified 96.9%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(x\right)\right) \]

7. Applied egg-rr 96.7%

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

Alternative 5: 3.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.9%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right), \color{blue}{\left(\sqrt{x + 1} + \sqrt{x}\right)}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - \sqrt{x} \cdot \sqrt{x}\right), \left(\sqrt{\color{blue}{x + 1}} + \sqrt{x}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x + 1\right) - x\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 - x\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x + \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 \cdot 1 - x \cdot 1\right)\right), \left(\color{blue}{\sqrt{x + 1}} + \sqrt{x}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x \cdot 1\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \left(1 - x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \left(\sqrt{x + 1} + \sqrt{x}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\sqrt{x + 1}\right), \color{blue}{\left(\sqrt{x}\right)}\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left({\left(x + 1\right)}^{\frac{1}{2}}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

   14. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{2}\right), \left(\sqrt{\color{blue}{x}}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 9.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 9.8%

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

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\sqrt{\frac{x}{1}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\sqrt{\frac{1}{\frac{1}{x}}}\right)\right)\right) \]

   3. sqrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{1}{x}}}}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \left(\frac{1}{\sqrt{\color{blue}{\frac{1}{x}}}}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right)\right)\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \left(\sqrt{{x}^{-1}}\right)\right)\right)\right) \]

   7. sqrt-pow1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \left({x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \left({x}^{\frac{-1}{2}}\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \left({x}^{\left(\frac{1}{2} \cdot \color{blue}{-1}\right)}\right)\right)\right)\right) \]

   10. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot -1\right)}\right)\right)\right)\right) \]

   11. metadata-eval 9.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

6. Applied egg-rr 9.8%

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

7. Taylor expanded in x around -inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

9. Simplified 2.6%

   \[\leadsto \color{blue}{\frac{1}{x \cdot 0}} \]
10. Step-by-step derivation

    1. inv-pow N/A

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

    2. mul0-rgt N/A

       \[\leadsto {0}^{-1} \]

    3. pow-base-0 3.8%

       \[\leadsto 0 \]

11. Applied egg-rr 3.8%

    \[\leadsto \color{blue}{0} \]
12. 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 2024158 
(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)))