Main:bigenough3 from C

Percentage Accurate: 53.9% → 99.7%

Time: 12.3s

Alternatives: 11

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: 53.9% 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 54.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 55.6%

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

4. Applied egg-rr 55.6%

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

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. 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 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = 0.5 * sqrt((1.0 / x));
	} 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 <= 5d-5) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    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 <= 5e-5) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} 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 <= 5e-5:
		tmp = 0.5 * math.sqrt((1.0 / x))
	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 <= 5e-5)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	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 <= 5e-5)
		tmp = 0.5 * sqrt((1.0 / x));
	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, 5e-5], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $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)) < 5.00000000000000024e-5

   1. Initial program 5.4%

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

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3)
   (- (+ 1.0 (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625)))))) (sqrt x))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0)))))) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3:
		tmp = (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625)))))) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3)
		tmp = (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   5. Simplified 99.3%

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

   if 1.30000000000000004 < x

   1. Initial program 6.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.5%

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

   5. Simplified 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.2%

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

   5. Simplified 99.2%

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

   if 1.19999999999999996 < x

   1. Initial program 6.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.5%

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

   5. Simplified 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	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 = 0.5d0 * sqrt((1.0d0 / x))
    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 = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + ((x * 0.5) - math.sqrt(x))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	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(0.5 * sqrt(Float64(1.0 / x)));
	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 = 0.5 * sqrt((1.0 / x));
	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[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.0%

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

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\left(1 + x \cdot 0.5\right)} - \sqrt{x} \]
   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. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 99.0%

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

   7. Applied egg-rr 99.0%

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

   if 1 < x

   1. Initial program 6.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.5%

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

   5. Simplified 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 6: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / 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((1.0d0 / 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((1.0 / 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((1.0 / 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(Float64(1.0 / 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((1.0 / 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[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.2%

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

   5. Simplified 98.2%

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

   if 0.35999999999999999 < x

   1. Initial program 6.9%

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

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.65) (- 1.0 (sqrt x)) (pow x -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.65d0) then
        tmp = 1.0d0 - sqrt(x)
    else
        tmp = x ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.65) {
		tmp = 1.0 - Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.65:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.65)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = x ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.65)
		tmp = 1.0 - sqrt(x);
	else
		tmp = x ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.65], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Power[x, -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.650000000000000022

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 98.2%

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

   5. Simplified 98.2%

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

   if 0.650000000000000022 < x

   1. Initial program 6.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 9.2%

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

   4. Applied egg-rr 9.2%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 18.7%

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

   10. Simplified 18.7%

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

       1. inv-pow N/A

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

       2. pow1/2 N/A

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

       3. pow-pow N/A

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

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

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

       5. metadata-eval 18.7%

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

   12. Applied egg-rr 18.7%

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

Alternative 8: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{1 + x \cdot \left(x \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.82) (/ (+ x (- 1.0 x)) (+ 1.0 (* x (* x -0.125)))) (pow x -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
	} else {
		tmp = pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.82d0) then
        tmp = (x + (1.0d0 - x)) / (1.0d0 + (x * (x * (-0.125d0))))
    else
        tmp = x ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
	} else {
		tmp = Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.82:
		tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)))
	else:
		tmp = math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(1.0 + Float64(x * Float64(x * -0.125))));
	else
		tmp = x ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.82)
		tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
	else
		tmp = x ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.82], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 95.4%

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

   10. Simplified 95.4%

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

   if 0.819999999999999951 < x

   1. Initial program 6.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 18.8%

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

   7. Simplified 18.8%

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

   8. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 18.7%

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

   10. Simplified 18.7%

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

       1. inv-pow N/A

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

       2. pow1/2 N/A

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

       3. pow-pow N/A

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

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

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

       5. metadata-eval 18.7%

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

   12. Applied egg-rr 18.7%

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

Alternative 9: 50.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(1 - x\right)}{1 + x \cdot \left(x \cdot -0.125\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x (- 1.0 x)) (+ 1.0 (* x (* x -0.125)))))

C

double code(double x) {
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + (1.0d0 - x)) / (1.0d0 + (x * (x * (-0.125d0))))
end function

Java

public static double code(double x) {
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
}

Python

def code(x):
	return (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)))

Julia

function code(x)
	return Float64(Float64(x + Float64(1.0 - x)) / Float64(1.0 + Float64(x * Float64(x * -0.125))))
end

MATLAB

function tmp = code(x)
	tmp = (x + (1.0 - x)) / (1.0 + (x * (x * -0.125)));
end

Wolfram

code[x_] := N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 55.6%

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

4. Applied egg-rr 55.6%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 53.0%

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

7. Simplified 53.0%

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

8. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 51.0%

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

10. Simplified 51.0%

    \[\leadsto \frac{x + \left(1 - x\right)}{1 + \color{blue}{x \cdot \left(x \cdot -0.125\right)}} \]
11. Add Preprocessing

Alternative 10: 3.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x (* x 0.0625))))

C

double code(double x) {
	return x * (x * (x * 0.0625));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (x * 0.0625d0))
end function

Java

public static double code(double x) {
	return x * (x * (x * 0.0625));
}

Python

def code(x):
	return x * (x * (x * 0.0625))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(x * 0.0625)))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (x * 0.0625));
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 52.8%

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

5. Simplified 52.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{16} \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \frac{1}{16} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{16} \cdot \left({x}^{2} \cdot x\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\frac{1}{16} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

   4. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{16} \cdot {x}^{2}\right)} \]

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

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 3.6%

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

8. Simplified 3.6%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right)} \]
9. Add Preprocessing

Alternative 11: 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 54.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 55.6%

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

4. Applied egg-rr 55.6%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 53.0%

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

7. Simplified 53.0%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-8}{{x}^{2}}} \]
9. 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 2.1%

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

10. Simplified 2.1%

    \[\leadsto \color{blue}{\frac{-8}{x \cdot x}} \]
11. 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 2024191 
(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)))