Main:bigenough3 from C

Percentage Accurate: 53.3% → 99.9%

Time: 11.4s

Alternatives: 13

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: 53.3% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (* (pow x -0.5) 0.5)
     (/ (+ x (- 1.0 x)) (+ (sqrt x) t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (sqrt(x) + 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((x + 1.0d0))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = (x + (1.0d0 - x)) / (sqrt(x) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = (x + (1.0 - x)) / (Math.sqrt(x) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = (x + (1.0 - x)) / (math.sqrt(x) + t_0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. /-lowering-/.f64 99.8%

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

   5. Simplified 99.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.2%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. rem-square-sqrt N/A

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

      3. rem-square-sqrt N/A

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

      4. flip-- N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.1%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow1/2 N/A

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

      7. unpow1/2 N/A

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

      8. rem-square-sqrt N/A

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

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

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 1e-6) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-6) {
		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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 1d-6) 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((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-6) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 1e-6:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 1e-6)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 1e-6)
		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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-6], 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)) < 9.99999999999999955e-7

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. /-lowering-/.f64 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 9.99999999999999955e-7 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.6%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3)
   (+ 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 <= 1.3) {
		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 <= 1.3d0) 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 <= 1.3) {
		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 <= 1.3:
		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 <= 1.3)
		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 <= 1.3)
		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, 1.3], 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 < 1.30000000000000004

   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 1.30000000000000004 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 99.4%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

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(Float64(1.0 + 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[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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 < 1.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
   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 99.6%

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

   5. Simplified 99.6%

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

   if 1.25 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

Alternative 5: 98.5% 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 99.2%

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

   5. Simplified 99.2%

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

   if 1 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
   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 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 99.2%

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

   8. Simplified 99.2%

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

   if 1 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

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

Alternative 7: 97.9% 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 97.6%

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

   5. Simplified 97.6%

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

   if 0.35999999999999999 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. pow-flip N/A

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

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

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

Alternative 8: 97.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 97.6%

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

   5. Simplified 97.6%

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

   if 0.35999999999999999 < x

   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 98.8%

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

   5. Simplified 98.8%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 98.7%

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

   7. Applied egg-rr 98.7%

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

Alternative 9: 49.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* (* x (* x x)) (+ 0.0625 (/ -0.125 x)))))

C

double code(double x) {
	return 1.0 + ((x * (x * x)) * (0.0625 + (-0.125 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * (x * x)) * (0.0625d0 + ((-0.125d0) / x)))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * (x * x)) * (0.0625 + (-0.125 / x)));
}

Python

def code(x):
	return 1.0 + ((x * (x * x)) * (0.0625 + (-0.125 / x)))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * Float64(x * x)) * Float64(0.0625 + Float64(-0.125 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * (x * x)) * (0.0625 + (-0.125 / x)));
end

Wolfram

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

Derivation

1. Initial program 50.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 49.2%

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

5. Simplified 49.2%

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} - \sqrt{x} \]
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 49.2%

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

7. Applied egg-rr 49.2%

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

8. Taylor expanded in x around inf

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

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sub-neg N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 47.4%

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

10. Simplified 47.4%

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

11. Final simplification 47.4%

    \[\leadsto 1 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0625 + \frac{-0.125}{x}\right) \]
12. Add Preprocessing

Alternative 10: 49.4% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 1 + 0.0625 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* 0.0625 (* x (* x x)))))

C

double code(double x) {
	return 1.0 + (0.0625 * (x * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (0.0625d0 * (x * (x * x)))
end function

Java

public static double code(double x) {
	return 1.0 + (0.0625 * (x * (x * x)));
}

Python

def code(x):
	return 1.0 + (0.0625 * (x * (x * x)))

Julia

function code(x)
	return Float64(1.0 + Float64(0.0625 * Float64(x * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (0.0625 * (x * (x * x)));
end

Wolfram

code[x_] := N[(1.0 + N[(0.0625 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 49.2%

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

5. Simplified 49.2%

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)} - \sqrt{x} \]
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 49.2%

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

7. Applied egg-rr 49.2%

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

8. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{16}, \left({x}^{3}\right)\right), 1\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 47.4%

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

10. Simplified 47.4%

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

11. Final simplification 47.4%

    \[\leadsto 1 + 0.0625 \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
12. Add Preprocessing

Alternative 11: 4.3% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. rem-square-sqrt N/A

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

   3. rem-square-sqrt N/A

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

   4. flip-- N/A

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

   5. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 49.2%

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

5. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 48.5%

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

7. Simplified 48.5%

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

8. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{/.f64}\left(16, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 4.1%

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

10. Simplified 4.1%

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

Alternative 12: 3.6% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

   \[\sqrt{x + 1} - \sqrt{x} \]
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 49.2%

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

5. Simplified 49.2%

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

6. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{16}, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 3.4%

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

8. Simplified 3.4%

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

Alternative 13: 1.9% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (* x -0.125)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * (x * -0.125)

Julia

function code(x)
	return Float64(x * Float64(x * -0.125))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right)}, \mathsf{sqrt.f64}\left(x\right)\right) \]
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 48.1%

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

5. Simplified 48.1%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 1.8%

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

8. Simplified 1.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.125\right)} \]
9. 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 2024158 
(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)))