Main:bigenough3 from C

Percentage Accurate: 53.7% → 99.7%

Time: 12.2s

Alternatives: 10

Speedup: 1.9×

Specification

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: 53.7% 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: 99.7% 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 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 48.1%

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

4. Applied egg-rr 48.1%

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

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 99.7%

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

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

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

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

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

   3. unpow1/2 N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sqrt-lowering-sqrt.f64 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0001) (* (pow x -0.5) 0.5) 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) * 0.5;
	} 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)) * 0.5d0
    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) * 0.5;
	} 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) * 0.5
	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) * 0.5);
	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) * 0.5;
	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] * 0.5), $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 5.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 99.3%

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

   7. Applied egg-rr 99.3%

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

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

   1. Initial program 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 1 < x

   1. Initial program 6.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.9%

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

   5. Simplified 97.9%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 98.2%

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 1 < x

   1. Initial program 6.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.9%

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

   5. Simplified 97.9%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 98.2%

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

   7. Applied egg-rr 98.2%

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

Alternative 5: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 99.4%

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

   5. Simplified 99.4%

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

   if 0.35999999999999999 < x

   1. Initial program 6.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.9%

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

   5. Simplified 97.9%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 98.2%

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

   7. Applied egg-rr 98.2%

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

Alternative 6: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (/ 0.5 (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.36d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = 0.5d0 / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = 0.5 / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = 0.5 / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.36)
		tmp = 1.0 - sqrt(x);
	else
		tmp = 0.5 / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 99.4%

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

   5. Simplified 99.4%

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

   if 0.35999999999999999 < x

   1. Initial program 6.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.9%

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

   5. Simplified 97.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
   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 97.8%

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

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (- 1.0 (sqrt x))
   (/ (+ 1.0 x) (* (* x (* x x)) (+ 0.0625 (/ -0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = (1.0 + x) / ((x * (x * x)) * (0.0625 + (-0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = (1.0d0 + x) / ((x * (x * x)) * (0.0625d0 + ((-0.125d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = (1.0 + x) / ((x * (x * x)) * (0.0625 + (-0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = (1.0 + x) / ((x * (x * x)) * (0.0625 + (-0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(Float64(1.0 + x) / Float64(Float64(x * Float64(x * x)) * Float64(0.0625 + Float64(-0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 1.0 - sqrt(x);
	else
		tmp = (1.0 + x) / ((x * (x * x)) * (0.0625 + (-0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.0625 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 98.5%

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

   5. Simplified 98.5%

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

   if 2 < x

   1. Initial program 6.1%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 9.7%

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

   4. Applied egg-rr 9.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

      10. *-lowering-*.f64 4.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   7. Simplified 4.1%

      \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} + \sqrt{x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{16}} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{16}} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{16} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{x}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{x}\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{\color{blue}{x}}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{-1}{8}}{x}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 4.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{-1}{8}, \color{blue}{x}\right)\right)\right)\right) \]

   10. Simplified 4.1%

       \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{1}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{-1}{8}, x\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 5.1%

       \[\leadsto \frac{x + \color{blue}{1}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 4.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{16}{x}}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 16.0 x) (* x x)))

C

double code(double x) {
	return (16.0 / x) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (16.0d0 / x) / (x * x)
end function

Java

public static double code(double x) {
	return (16.0 / x) / (x * x);
}

Python

def code(x):
	return (16.0 / x) / (x * x)

Julia

function code(x)
	return Float64(Float64(16.0 / x) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (16.0 / x) / (x * x);
end

Wolfram

code[x_] := N[(N[(16.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 48.1%

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

4. Applied egg-rr 48.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   10. *-lowering-*.f64 44.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

7. Simplified 44.6%

   \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} + \sqrt{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{16}{{x}^{3}}} \]
9. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \frac{16}{\left(x \cdot x\right) \cdot \color{blue}{x}} \]

   2. unpow2 N/A

      \[\leadsto \frac{16}{{x}^{2} \cdot x} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{16}{{x}^{2}}}{\color{blue}{x}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{16}{{x}^{2}}\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \left({x}^{2}\right)\right), x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \left(x \cdot x\right)\right), x\right) \]

   7. *-lowering-*.f64 4.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{\frac{\frac{16}{x \cdot x}}{x}} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{16}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{16}{x}}{\color{blue}{x \cdot x}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{16}{x}\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, x\right), \left(\color{blue}{x} \cdot x\right)\right) \]

    5. *-lowering-*.f64 4.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

12. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{\frac{\frac{16}{x}}{x \cdot x}} \]
13. Add Preprocessing

Alternative 9: 4.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{16}{x \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 16.0 (* x (* x x))))

C

double code(double x) {
	return 16.0 / (x * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 16.0d0 / (x * (x * x))
end function

Java

public static double code(double x) {
	return 16.0 / (x * (x * x));
}

Python

def code(x):
	return 16.0 / (x * (x * x))

Julia

function code(x)
	return Float64(16.0 / Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 16.0 / (x * (x * x));
end

Wolfram

code[x_] := N[(16.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 48.1%

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

4. Applied egg-rr 48.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   10. *-lowering-*.f64 44.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

7. Simplified 44.6%

   \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} + \sqrt{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{16}{{x}^{3}}} \]
9. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \frac{16}{\left(x \cdot x\right) \cdot \color{blue}{x}} \]

   2. unpow2 N/A

      \[\leadsto \frac{16}{{x}^{2} \cdot x} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{16}{{x}^{2}}}{\color{blue}{x}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{16}{{x}^{2}}\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \left({x}^{2}\right)\right), x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \left(x \cdot x\right)\right), x\right) \]

   7. *-lowering-*.f64 4.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(16, \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{\frac{\frac{16}{x \cdot x}}{x}} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{16}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(16, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right) \]

    4. *-lowering-*.f64 4.3%

       \[\leadsto \mathsf{/.f64}\left(16, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

12. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{\frac{16}{x \cdot \left(x \cdot x\right)}} \]
13. Add Preprocessing

Alternative 10: 2.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{-8}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -8.0 (* x x)))

C

double code(double x) {
	return -8.0 / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-8.0d0) / (x * x)
end function

Java

public static double code(double x) {
	return -8.0 / (x * x);
}

Python

def code(x):
	return -8.0 / (x * x)

Julia

function code(x)
	return Float64(-8.0 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -8.0 / (x * x);
end

Wolfram

code[x_] := N[(-8.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 48.1%

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

4. Applied egg-rr 48.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right)\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

   10. *-lowering-*.f64 44.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(x\right)\right)\right) \]

7. Simplified 44.6%

   \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} + \sqrt{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)}\right)\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{16}} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{16}} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{16} - \frac{1}{8} \cdot \frac{1}{x}\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{16} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{x}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{x}\right)\right)\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\frac{-1}{8}}{x}\right)\right)\right)\right) \]

   13. /-lowering-/.f64 2.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(1, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{/.f64}\left(\frac{-1}{8}, \color{blue}{x}\right)\right)\right)\right) \]

10. Simplified 2.7%

    \[\leadsto \frac{x + \left(1 - x\right)}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right)}} \]

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{-8}{{x}^{2}}} \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(-8, \color{blue}{\left({x}^{2}\right)}\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(-8, \left(x \cdot \color{blue}{x}\right)\right) \]

    3. *-lowering-*.f64 2.2%

       \[\leadsto \mathsf{/.f64}\left(-8, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

13. Simplified 2.2%

    \[\leadsto \color{blue}{\frac{-8}{x \cdot x}} \]
14. Add Preprocessing

Developer Target 1: 99.7% 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 2024160 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

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