Main:bigenough3 from C

Percentage Accurate: 53.1% → 99.9%

Time: 10.1s

Alternatives: 14

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: 53.1% 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

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

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) (+ t_0 (sqrt x))))))

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) / (t_0 + sqrt(x));
	}
	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) / (t_0 + sqrt(x))
    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) / (t_0 + Math.sqrt(x));
	}
	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) / (t_0 + math.sqrt(x))
	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(Float64(x + 1.0) - x) / Float64(t_0 + sqrt(x)));
	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) / (t_0 + sqrt(x));
	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[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $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. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 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 97.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

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

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

      10. +-lowering-+.f64 96.8%

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

   4. Applied egg-rr 96.8%

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

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

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. metadata-eval 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. flip-- N/A

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

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

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

      3. rem-square-sqrt N/A

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

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

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

      5. pow2 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. +-commutative N/A

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

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

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

      13. +-commutative N/A

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

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

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

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

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

      16. pow-flip N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. unpow1 N/A

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

      20. metadata-eval N/A

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

      21. pow-pow N/A

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.002:\\ \;\;\;\;\frac{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1}{x + 1}}} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.002)
   (/ (+ (/ -0.125 (sqrt x)) (* (sqrt x) 0.5)) x)
   (- (/ 1.0 (sqrt (/ 1.0 (+ x 1.0)))) (sqrt x))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.002) {
		tmp = ((-0.125 / sqrt(x)) + (sqrt(x) * 0.5)) / x;
	} else {
		tmp = (1.0 / sqrt((1.0 / (x + 1.0)))) - sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((sqrt((x + 1.0d0)) - sqrt(x)) <= 0.002d0) then
        tmp = (((-0.125d0) / sqrt(x)) + (sqrt(x) * 0.5d0)) / x
    else
        tmp = (1.0d0 / sqrt((1.0d0 / (x + 1.0d0)))) - sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) <= 0.002) {
		tmp = ((-0.125 / Math.sqrt(x)) + (Math.sqrt(x) * 0.5)) / x;
	} else {
		tmp = (1.0 / Math.sqrt((1.0 / (x + 1.0)))) - Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.sqrt((x + 1.0)) - math.sqrt(x)) <= 0.002:
		tmp = ((-0.125 / math.sqrt(x)) + (math.sqrt(x) * 0.5)) / x
	else:
		tmp = (1.0 / math.sqrt((1.0 / (x + 1.0)))) - math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.002)
		tmp = Float64(Float64(Float64(-0.125 / sqrt(x)) + Float64(sqrt(x) * 0.5)) / x);
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(1.0 / Float64(x + 1.0)))) - sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.002)
		tmp = ((-0.125 / sqrt(x)) + (sqrt(x) * 0.5)) / x;
	else
		tmp = (1.0 / sqrt((1.0 / (x + 1.0)))) - sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(-0.125 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 2e-3

   1. Initial program 5.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

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

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

      10. +-lowering-+.f64 5.5%

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{x}} \cdot -0.125 + 0.5 \cdot \sqrt{x}}{x}} \]
   8. Step-by-step derivation

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

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

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

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. sqrt-lowering-sqrt.f64 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 2e-3 < (-.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. Step-by-step derivation

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

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

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

      10. +-lowering-+.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.002:\\ \;\;\;\;\frac{\frac{-0.125}{\sqrt{x}} + \sqrt{x} \cdot 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1}{x + 1}}} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 1e-5) (* (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-5) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 1d-5) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-5) {
		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-5:
		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-5)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 1.00000000000000008e-5

   1. Initial program 5.1%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

      3. /-lowering-/.f64 99.2%

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

   5. Simplified 99.2%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1.00000000000000008e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.4%

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

Alternative 4: 98.9% 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

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

      1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. sqrt-lowering-sqrt.f64 99.5%

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

   5. Simplified 99.5%

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

   if 1.30000000000000004 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 97.7%

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

   7. Applied egg-rr 97.7%

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

Alternative 5: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = (1.0 - sqrt(x)) + (x * (0.5 + (x * -0.125)));
	} 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 <= 1.2d0) then
        tmp = (1.0d0 - sqrt(x)) + (x * (0.5d0 + (x * (-0.125d0))))
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.3%

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

   5. Simplified 99.3%

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

   if 1.19999999999999996 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 97.7%

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

   7. Applied egg-rr 97.7%

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

Alternative 6: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 98.9%

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

   7. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 97.7%

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

   7. Applied egg-rr 97.7%

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

4. Final simplification 98.2%

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

Alternative 7: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

   if 1 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 97.7%

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

   7. Applied egg-rr 97.7%

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

Alternative 8: 98.1% 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 97.0%

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

   5. Simplified 97.0%

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

   if 0.35999999999999999 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 97.7%

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

   7. Applied egg-rr 97.7%

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

Alternative 9: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.35999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 97.0%

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

   5. Simplified 97.0%

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

   if 0.35999999999999999 < x

   1. Initial program 7.4%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 97.5%

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

   5. Simplified 97.5%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 97.3%

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

   7. Applied egg-rr 97.3%

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

Alternative 10: 49.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. sqrt-lowering-sqrt.f64 44.4%

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

5. Simplified 44.4%

   \[\leadsto \color{blue}{1 - \sqrt{x}} \]
6. Add Preprocessing

Alternative 11: 49.5% accurate, 15.8× speedup

Math

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

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 * x) * Float64(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 * x), $MachinePrecision] * N[(x * N[(0.0625 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 46.5%

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

5. Simplified 46.5%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. sub-neg N/A

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

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

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. /-lowering-/.f64 43.9%

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

8. Simplified 43.9%

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

Alternative 12: 49.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 46.5%

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

5. Simplified 46.5%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 43.9%

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

8. Simplified 43.9%

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

9. Final simplification 43.9%

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

Alternative 13: 3.6% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 46.5%

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

5. Simplified 46.5%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 3.6%

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

8. Simplified 3.6%

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

9. Final simplification 3.6%

   \[\leadsto x \cdot \left(0.0625 \cdot \left(x \cdot x\right)\right) \]
10. Add Preprocessing

Alternative 14: 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 49.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 45.2%

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

5. Simplified 45.2%

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

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: 99.8% 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 2024145 
(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)))