Main:bigenough3 from C

Percentage Accurate: 53.2% → 99.7%

Time: 10.6s

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: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% 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 58.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 60.1%

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

4. Applied egg-rr 60.1%

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

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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (* (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 <= 5e-5) {
		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 <= 5d-5) 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 <= 5e-5) {
		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 <= 5e-5:
		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 <= 5e-5)
		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 <= 5e-5)
		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, 5e-5], 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)) < 5.00000000000000024e-5

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 99.3%

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

   10. Applied egg-rr 99.3%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.6%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35)
   (+ 1.0 (- (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))) (sqrt x)))
   (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = 1.0 + ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) - 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 <= 1.35d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0))))) - 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 <= 1.35) {
		tmp = 1.0 + ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) - Math.sqrt(x));
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35:
		tmp = 1.0 + ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) - math.sqrt(x))
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))) - 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 <= 1.35)
		tmp = 1.0 + ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) - sqrt(x));
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 1.3500000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
   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 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} - \sqrt{x} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 1.3500000000000001 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.2%

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

   8. Simplified 98.2%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (+ 1.0 (- (* x (+ 0.5 (* x -0.125))) (sqrt x)))
   (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - 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 <= 1.25d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - 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 <= 1.25) {
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - Math.sqrt(x));
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - math.sqrt(x))
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) - 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 <= 1.25)
		tmp = 1.0 + ((x * (0.5 + (x * -0.125))) - sqrt(x));
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 + N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $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 < 1.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.8%

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

   5. Simplified 99.8%

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

   if 1.25 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.2%

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

   8. Simplified 98.2%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

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(1.0 - Float64(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[(1.0 - N[(N[Sqrt[x], $MachinePrecision] + N[(x * -0.5), $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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
   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 99.3%

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

   5. Simplified 99.3%

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

      1. associate-+l- N/A

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

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

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

      3. *-commutative N/A

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

      4. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

      8. metadata-eval 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 1 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.2%

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

   8. Simplified 98.2%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.1%

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

Alternative 6: 98.6% 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 99.3%

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

   5. Simplified 99.3%

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

   if 1 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.2%

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

   8. Simplified 98.2%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

Alternative 7: 98.0% 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 97.2%

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

   5. Simplified 97.2%

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

   if 0.35999999999999999 < x

   1. Initial program 6.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 98.2%

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

   8. Simplified 98.2%

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

      1. /-rgt-identity N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. times-frac N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-pow N/A

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

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

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

      18. metadata-eval 98.7%

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

   10. Applied egg-rr 98.7%

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

Alternative 8: 58.3% 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 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 97.2%

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

   5. Simplified 97.2%

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

   if 0.650000000000000022 < 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 9.0%

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

   4. Applied egg-rr 9.0%

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

   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

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

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

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

      2. /-lowering-/.f64 18.7%

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

   10. Simplified 18.7%

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

       1. pow1/2 N/A

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

       2. inv-pow N/A

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

       3. pow-pow N/A

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

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

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

       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: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;1 + -0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6) (+ 1.0 (* -0.125 (* x x))) (pow x -0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = 1.0 + (-0.125 * (x * 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 <= 2.6d0) then
        tmp = 1.0d0 + ((-0.125d0) * (x * x))
    else
        tmp = x ** (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = 1.0 + (-0.125 * (x * x));
	} else {
		tmp = Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = 1.0 + (-0.125 * (x * x))
	else:
		tmp = math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(1.0 + Float64(-0.125 * Float64(x * x)));
	else
		tmp = x ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.6)
		tmp = 1.0 + (-0.125 * (x * x));
	else
		tmp = x ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(1.0 + N[(-0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 94.1%

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

   8. Simplified 94.1%

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

   if 2.60000000000000009 < 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 9.0%

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

   4. Applied egg-rr 9.0%

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

   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

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

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

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

      2. /-lowering-/.f64 18.7%

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

   10. Simplified 18.7%

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

       1. pow1/2 N/A

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

       2. inv-pow N/A

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

       3. pow-pow N/A

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

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

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

       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 10: 51.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. sqrt-lowering-sqrt.f64 56.6%

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

5. Simplified 56.6%

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

6. Taylor expanded in x around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 53.4%

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

8. Simplified 53.4%

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

9. Taylor expanded in x around 0

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

11. Simplified 55.9%

    \[\leadsto \color{blue}{1} \]
12. 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 2024155 
(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)))