Main:bigenough3 from C

Percentage Accurate: 53.3% → 99.7%

Time: 8.1s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

   1. flip-- 48.0%

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

   2. div-inv 48.0%

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

   3. add-sqr-sqrt 48.0%

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

   4. add-sqr-sqrt 48.8%

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

   5. associate--l+ 48.8%

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

4. Applied egg-rr 48.8%

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

   1. associate-*r/ 48.8%

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

   2. *-rgt-identity 48.8%

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

   3. +-commutative 48.8%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

   8. +-commutative 99.7%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-5) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 2d-5) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 2e-5:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 2e-5)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 2e-5)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-5], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 2.00000000000000016e-5

   1. Initial program 5.0%

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

      1. flip-- 6.2%

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

      2. div-inv 6.2%

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

      3. add-sqr-sqrt 6.0%

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

      4. add-sqr-sqrt 7.2%

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

      5. associate--l+ 7.2%

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

   4. Applied egg-rr 7.2%

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

      1. associate-*r/ 7.2%

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

      2. *-rgt-identity 7.2%

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

      3. +-commutative 7.2%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-cube-cbrt 97.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}} \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3}} \]

      3. cbrt-div 98.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}}^{3} \]

      4. metadata-eval 98.0%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3} \]

   8. Applied egg-rr 98.0%

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

      1. sqr-pow 98.0%

         \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}} \]

      2. inv-pow 98.0%

         \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      3. pow-pow 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      4. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      5. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      6. inv-pow 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \]

      7. pow-pow 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \]

      8. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 98.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(2 \cdot -1.5\right)}} \]

       2. metadata-eval 98.3%

          \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-3}} \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-3}} \]

   13. Taylor expanded in x around inf 99.1%

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

       1. unpow-1 99.1%

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

       2. metadata-eval 99.1%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 99.3%

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

       4. rem-sqrt-square 99.3%

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

       5. rem-square-sqrt 98.5%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

       6. fabs-sqr 98.5%

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

       7. rem-square-sqrt 99.3%

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

   15. Simplified 99.3%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = (1.0 + (x * (0.5 + (x * ((x * 0.0625) - 0.125))))) - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -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 * ((x * 0.0625d0) - 0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * (x ** (-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 * ((x * 0.0625) - 0.125))))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.0625) - 0.125))))) - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -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 * ((x * 0.0625) - 0.125))))) - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $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 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 8.6%

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

      1. flip-- 9.6%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. add-sqr-sqrt 10.9%

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

      5. associate--l+ 10.9%

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

   4. Applied egg-rr 10.9%

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

      1. associate-*r/ 10.9%

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

      2. *-rgt-identity 10.9%

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

      3. +-commutative 10.9%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}} \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3}} \]

      3. cbrt-div 98.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}}^{3} \]

      4. metadata-eval 98.0%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3} \]

   8. Applied egg-rr 98.0%

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

      1. sqr-pow 98.0%

         \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}} \]

      2. inv-pow 98.0%

         \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      3. pow-pow 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      4. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      5. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      6. inv-pow 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \]

      7. pow-pow 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \]

      8. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 98.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(2 \cdot -1.5\right)}} \]

       2. metadata-eval 98.3%

          \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-3}} \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-3}} \]

   13. Taylor expanded in x around inf 96.3%

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

       1. unpow-1 96.3%

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

       2. metadata-eval 96.3%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 96.6%

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

       4. rem-sqrt-square 96.6%

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

       5. rem-square-sqrt 95.8%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

       6. fabs-sqr 95.8%

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

       7. rem-square-sqrt 96.6%

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

   15. Simplified 96.6%

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

4. Final simplification 97.9%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = 1.0d0 + ((x * (0.5d0 + (x * (-0.125d0)))) - sqrt(x))
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 1.25 < x

   1. Initial program 8.6%

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

      1. flip-- 9.6%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. add-sqr-sqrt 10.9%

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

      5. associate--l+ 10.9%

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

   4. Applied egg-rr 10.9%

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

      1. associate-*r/ 10.9%

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

      2. *-rgt-identity 10.9%

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

      3. +-commutative 10.9%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}} \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3}} \]

      3. cbrt-div 98.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}}^{3} \]

      4. metadata-eval 98.0%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3} \]

   8. Applied egg-rr 98.0%

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

      1. sqr-pow 98.0%

         \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}} \]

      2. inv-pow 98.0%

         \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      3. pow-pow 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      4. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      5. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      6. inv-pow 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \]

      7. pow-pow 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \]

      8. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 98.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(2 \cdot -1.5\right)}} \]

       2. metadata-eval 98.3%

          \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-3}} \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-3}} \]

   13. Taylor expanded in x around inf 96.3%

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

       1. unpow-1 96.3%

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

       2. metadata-eval 96.3%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 96.6%

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

       4. rem-sqrt-square 96.6%

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

       5. rem-square-sqrt 95.8%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

       6. fabs-sqr 95.8%

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

       7. rem-square-sqrt 96.6%

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

   15. Simplified 96.6%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else {
		tmp = 0.5 * pow(x, -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 = 0.5d0 * (x ** (-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 = 0.5 * Math.pow(x, -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 = 0.5 * math.pow(x, -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(0.5 * (x ^ -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 = 0.5 * (x ^ -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[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $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 99.1%

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

      1. associate--l+ 99.2%

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

      2. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1 < x

   1. Initial program 8.6%

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

      1. flip-- 9.6%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. add-sqr-sqrt 10.9%

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

      5. associate--l+ 10.9%

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

   4. Applied egg-rr 10.9%

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

      1. associate-*r/ 10.9%

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

      2. *-rgt-identity 10.9%

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

      3. +-commutative 10.9%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}} \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3}} \]

      3. cbrt-div 98.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}}^{3} \]

      4. metadata-eval 98.0%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3} \]

   8. Applied egg-rr 98.0%

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

      1. sqr-pow 98.0%

         \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}} \]

      2. inv-pow 98.0%

         \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      3. pow-pow 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      4. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      5. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      6. inv-pow 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \]

      7. pow-pow 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \]

      8. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 98.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(2 \cdot -1.5\right)}} \]

       2. metadata-eval 98.3%

          \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-3}} \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-3}} \]

   13. Taylor expanded in x around inf 96.3%

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

       1. unpow-1 96.3%

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

       2. metadata-eval 96.3%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 96.6%

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

       4. rem-sqrt-square 96.6%

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

       5. rem-square-sqrt 95.8%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

       6. fabs-sqr 95.8%

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

       7. rem-square-sqrt 96.6%

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

   15. Simplified 96.6%

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

Alternative 6: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -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 = 0.5d0 * (x ** (-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 = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -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 = 0.5 * (x ^ -0.5);
	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[Power[x, -0.5], $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 97.8%

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

   if 0.35999999999999999 < x

   1. Initial program 8.6%

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

      1. flip-- 9.6%

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

      2. div-inv 9.6%

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

      3. add-sqr-sqrt 9.6%

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

      4. add-sqr-sqrt 10.9%

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

      5. associate--l+ 10.9%

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

   4. Applied egg-rr 10.9%

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

      1. associate-*r/ 10.9%

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

      2. *-rgt-identity 10.9%

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

      3. +-commutative 10.9%

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

      4. associate-+l- 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}} \]

      2. pow3 97.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3}} \]

      3. cbrt-div 98.0%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}}^{3} \]

      4. metadata-eval 98.0%

         \[\leadsto {\left(\frac{\color{blue}{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{3} \]

   8. Applied egg-rr 98.0%

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

      1. sqr-pow 98.0%

         \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)}} \]

      2. inv-pow 98.0%

         \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      3. pow-pow 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      4. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      5. metadata-eval 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \cdot {\left(\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}\right)}^{\left(\frac{3}{2}\right)} \]

      6. inv-pow 98.1%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1}\right)}}^{\left(\frac{3}{2}\right)} \]

      7. pow-pow 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \frac{3}{2}\right)}} \]

      8. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(-1 \cdot \color{blue}{1.5}\right)} \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-1.5}} \]

   10. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5} \cdot {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-1.5}} \]
   11. Step-by-step derivation

       1. pow-sqr 98.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\left(2 \cdot -1.5\right)}} \]

       2. metadata-eval 98.3%

          \[\leadsto {\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{\color{blue}{-3}} \]

   12. Simplified 98.3%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}\right)}^{-3}} \]

   13. Taylor expanded in x around inf 96.3%

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

       1. unpow-1 96.3%

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

       2. metadata-eval 96.3%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 96.6%

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

       4. rem-sqrt-square 96.6%

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

       5. rem-square-sqrt 95.8%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}\right| \]

       6. fabs-sqr 95.8%

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

       7. rem-square-sqrt 96.6%

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

   15. Simplified 96.6%

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

Alternative 7: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.36:\\ \;\;\;\;1 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.36) (- 1.0 (sqrt x)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.36) {
		tmp = 1.0 - sqrt(x);
	} else {
		tmp = sqrt((0.25 / 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 = sqrt((0.25d0 / 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 = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.36:
		tmp = 1.0 - math.sqrt(x)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.36)
		tmp = Float64(1.0 - sqrt(x));
	else
		tmp = sqrt(Float64(0.25 / 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 = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.36], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / 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 97.8%

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

   if 0.35999999999999999 < x

   1. Initial program 8.6%

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

   3. Taylor expanded in x around inf 96.3%

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

      1. add-sqr-sqrt 95.7%

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

      2. sqrt-unprod 96.3%

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

      3. *-commutative 96.3%

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

      4. *-commutative 96.3%

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

      5. swap-sqr 96.3%

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

      6. add-sqr-sqrt 96.3%

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

      7. metadata-eval 96.3%

         \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.25}} \]
   6. Step-by-step derivation

      1. associate-*l/ 96.3%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.25}{x}}} \]

      2. metadata-eval 96.3%

         \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{x}} \]

   7. Simplified 96.3%

      \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ 0.25 x)))

C

double code(double x) {
	return sqrt((0.25 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((0.25d0 / x))
end function

Java

public static double code(double x) {
	return Math.sqrt((0.25 / x));
}

Python

def code(x):
	return math.sqrt((0.25 / x))

Julia

function code(x)
	return sqrt(Float64(0.25 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((0.25 / x));
end

Wolfram

code[x_] := N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.5%

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

3. Taylor expanded in x around inf 58.4%

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

   1. add-sqr-sqrt 58.0%

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

   2. sqrt-unprod 58.4%

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

   3. *-commutative 58.4%

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

   4. *-commutative 58.4%

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

   5. swap-sqr 58.4%

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

   6. add-sqr-sqrt 58.4%

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

   7. metadata-eval 58.4%

      \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.25}} \]

5. Applied egg-rr 58.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.25}} \]
6. Step-by-step derivation

   1. associate-*l/ 58.4%

      \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.25}{x}}} \]

   2. metadata-eval 58.4%

      \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{x}} \]

7. Simplified 58.4%

   \[\leadsto \color{blue}{\sqrt{\frac{0.25}{x}}} \]
8. Add Preprocessing

Alternative 9: 6.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 47.5%

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

3. Taylor expanded in x around 0 42.5%

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

4. Taylor expanded in x around inf 1.7%

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

   1. neg-mul-1 1.7%

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

6. Simplified 1.7%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 6.4%

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

   3. sqr-neg 6.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   4. add-sqr-sqrt 6.4%

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

   5. pow1/2 6.4%

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

8. Applied egg-rr 6.4%

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

9. Taylor expanded in x around 0 6.4%

   \[\leadsto \color{blue}{\sqrt{x}} \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024182 
(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)))