Main:bigenough3 from C

Percentage Accurate: 53.2% → 99.7%

Time: 11.6s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. flip-- 53.5%

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

   2. div-inv 53.5%

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

   3. add-sqr-sqrt 53.7%

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

   4. add-sqr-sqrt 54.2%

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

   5. associate--l+ 54.2%

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

4. Applied egg-rr 54.2%

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

   1. associate-*r/ 54.2%

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

   2. *-rgt-identity 54.2%

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

   3. +-commutative 54.2%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

6. Simplified 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 1e-6) (* 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 <= 1e-6) {
		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 <= 1d-6) 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 <= 1e-6) {
		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 <= 1e-6:
		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 <= 1e-6)
		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 <= 1e-6)
		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, 1e-6], 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)) < 9.99999999999999955e-7

   1. Initial program 4.5%

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

      1. flip-- 5.0%

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

      2. div-inv 5.0%

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

      3. add-sqr-sqrt 5.2%

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

      4. add-sqr-sqrt 6.2%

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

      5. associate--l+ 6.2%

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

   4. Applied egg-rr 6.2%

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

      1. associate-*r/ 6.2%

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

      2. *-rgt-identity 6.2%

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

      3. +-commutative 6.2%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{1}{\sqrt{x} + {\left(\sqrt{\color{blue}{{\left(1 + x\right)}^{0.5}}}\right)}^{2}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\color{blue}{\left({\left(1 + x\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\left({\left(1 + x\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}} \]

   9. Taylor expanded in x around inf 99.6%

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

       1. unpow-1 99.6%

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

       2. metadata-eval 99.6%

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

       3. pow-sqr 99.8%

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

       4. rem-sqrt-square 99.8%

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

       5. rem-square-sqrt 99.1%

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

       6. fabs-sqr 99.1%

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

       7. rem-square-sqrt 99.8%

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

   11. Simplified 99.8%

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

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

   1. Initial program 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 10^{-6}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ 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 <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		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 <= 1.2:
		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 <= 1.2)
		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 <= 1.2)
		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, 1.2], 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 < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 1.19999999999999996 < x

   1. Initial program 7.1%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{1}{\sqrt{x} + {\left(\sqrt{\color{blue}{{\left(1 + x\right)}^{0.5}}}\right)}^{2}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\color{blue}{\left({\left(1 + x\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\left({\left(1 + x\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}} \]

   9. Taylor expanded in x around inf 97.6%

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

       1. unpow-1 97.6%

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

       2. metadata-eval 97.6%

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

       3. pow-sqr 97.8%

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

       4. rem-sqrt-square 97.8%

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

       5. rem-square-sqrt 97.1%

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

       6. fabs-sqr 97.1%

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

       7. rem-square-sqrt 97.8%

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

   11. Simplified 97.8%

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

Alternative 4: 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 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{1}{\sqrt{x} + {\left(\sqrt{\color{blue}{{\left(1 + x\right)}^{0.5}}}\right)}^{2}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\color{blue}{\left({\left(1 + x\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\left({\left(1 + x\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}} \]

   9. Taylor expanded in x around inf 97.6%

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

       1. unpow-1 97.6%

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

       2. metadata-eval 97.6%

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

       3. pow-sqr 97.8%

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

       4. rem-sqrt-square 97.8%

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

       5. rem-square-sqrt 97.1%

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

       6. fabs-sqr 97.1%

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

       7. rem-square-sqrt 97.8%

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

   11. Simplified 97.8%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (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 <= 1.0d0) then
        tmp = 1.0d0 / (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 <= 1.0) {
		tmp = 1.0 / (1.0 + 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 / (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 <= 1.0)
		tmp = Float64(1.0 / 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 <= 1.0)
		tmp = 1.0 / (1.0 + 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[(1.0 + 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 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l- 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.6%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{1}{\sqrt{x} + {\left(\sqrt{\color{blue}{{\left(1 + x\right)}^{0.5}}}\right)}^{2}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\color{blue}{\left({\left(1 + x\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\left({\left(1 + x\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}} \]

   9. Taylor expanded in x around inf 97.6%

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

       1. unpow-1 97.6%

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

       2. metadata-eval 97.6%

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

       3. pow-sqr 97.8%

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

       4. rem-sqrt-square 97.8%

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

       5. rem-square-sqrt 97.1%

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

       6. fabs-sqr 97.1%

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

       7. rem-square-sqrt 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 98.2%

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

Alternative 6: 98.0% 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 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   if 0.35999999999999999 < x

   1. Initial program 7.1%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{1}{\sqrt{x} + {\left(\sqrt{\color{blue}{{\left(1 + x\right)}^{0.5}}}\right)}^{2}} \]

      4. sqrt-pow1 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\color{blue}{\left({\left(1 + x\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{1}{\sqrt{x} + {\left({\left(1 + x\right)}^{\color{blue}{0.25}}\right)}^{2}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{1}{\sqrt{x} + \color{blue}{{\left({\left(1 + x\right)}^{0.25}\right)}^{2}}} \]

   9. Taylor expanded in x around inf 97.6%

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

       1. unpow-1 97.6%

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

       2. metadata-eval 97.6%

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

       3. pow-sqr 97.8%

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

       4. rem-sqrt-square 97.8%

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

       5. rem-square-sqrt 97.1%

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

       6. fabs-sqr 97.1%

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

       7. rem-square-sqrt 97.8%

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

   11. Simplified 97.8%

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

Alternative 7: 97.9% accurate, 1.9× speedup

Math

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

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 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

   if 0.35999999999999999 < x

   1. Initial program 7.1%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. add-sqr-sqrt 97.0%

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

      2. sqrt-unprod 97.6%

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

      3. swap-sqr 97.6%

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

      4. metadata-eval 97.6%

         \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right)} \]

      5. add-sqr-sqrt 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   7. Simplified 97.6%

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

Alternative 8: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = x / 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 <= 0.25d0) then
        tmp = x / x
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = x / x;
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = Float64(x / x);
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l- 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.6%

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

   8. Taylor expanded in x around inf 6.9%

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

      1. unpow-1 6.9%

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

      2. metadata-eval 6.9%

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

      3. pow-sqr 6.9%

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

      4. rem-sqrt-square 6.9%

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

      5. rem-square-sqrt 6.9%

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

      6. fabs-sqr 6.9%

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

      7. rem-square-sqrt 6.9%

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

   10. Simplified 6.9%

       \[\leadsto \color{blue}{{x}^{-0.5}} \]
   11. Step-by-step derivation

       1. metadata-eval 6.9%

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

       2. pow-div 6.9%

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

   12. Applied egg-rr 96.4%

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

   if 0.25 < x

   1. Initial program 7.1%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. add-sqr-sqrt 97.0%

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

      2. sqrt-unprod 97.6%

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

      3. swap-sqr 97.6%

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

      4. metadata-eval 97.6%

         \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right)} \]

      5. add-sqr-sqrt 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   7. Simplified 97.6%

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

Alternative 9: 57.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l- 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.6%

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

   8. Taylor expanded in x around inf 6.9%

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

      1. unpow-1 6.9%

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

      2. metadata-eval 6.9%

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

      3. pow-sqr 6.9%

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

      4. rem-sqrt-square 6.9%

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

      5. rem-square-sqrt 6.9%

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

      6. fabs-sqr 6.9%

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

      7. rem-square-sqrt 6.9%

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

   10. Simplified 6.9%

       \[\leadsto \color{blue}{{x}^{-0.5}} \]
   11. Step-by-step derivation

       1. metadata-eval 6.9%

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

       2. pow-div 6.9%

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

   12. Applied egg-rr 96.4%

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

   if 1 < x

   1. Initial program 7.1%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.1%

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

      5. associate--l+ 9.1%

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

   4. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 18.8%

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

   8. Taylor expanded in x around inf 18.7%

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

      1. unpow-1 18.7%

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

      2. metadata-eval 18.7%

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

      3. pow-sqr 18.7%

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

      4. rem-sqrt-square 18.7%

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

      5. rem-square-sqrt 18.7%

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

      6. fabs-sqr 18.7%

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

      7. rem-square-sqrt 18.7%

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

   10. Simplified 18.7%

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

Alternative 10: 51.2% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. flip-- 53.5%

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

   2. div-inv 53.5%

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

   3. add-sqr-sqrt 53.7%

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

   4. add-sqr-sqrt 54.2%

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

   5. associate--l+ 54.2%

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

4. Applied egg-rr 54.2%

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

   1. associate-*r/ 54.2%

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

   2. *-rgt-identity 54.2%

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

   3. +-commutative 54.2%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 58.4%

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

8. Taylor expanded in x around inf 12.9%

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

   1. unpow-1 12.9%

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

   2. metadata-eval 12.9%

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

   3. pow-sqr 12.9%

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

   4. rem-sqrt-square 12.9%

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

   5. rem-square-sqrt 12.9%

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

   6. fabs-sqr 12.9%

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

   7. rem-square-sqrt 12.9%

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

10. Simplified 12.9%

    \[\leadsto \color{blue}{{x}^{-0.5}} \]
11. Step-by-step derivation

    1. metadata-eval 12.9%

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

    2. pow-div 12.9%

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

12. Applied egg-rr 51.3%

    \[\leadsto \color{blue}{\frac{x}{x}} \]
13. Add Preprocessing

Alternative 11: 12.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around inf 52.6%

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

   1. sqrt-div 52.5%

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

   2. metadata-eval 52.5%

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

   3. un-div-inv 52.5%

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

5. Applied egg-rr 52.5%

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

7. Applied egg-rr 12.7%

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

   1. associate-*r/ 12.7%

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

   2. *-rgt-identity 12.7%

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

   3. *-commutative 12.7%

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

   4. neg-mul-1 12.7%

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

   5. times-frac 12.7%

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

   6. metadata-eval 12.7%

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

   7. *-inverses 12.7%

      \[\leadsto 0.5 \cdot \color{blue}{1} \]

   8. metadata-eval 12.7%

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

9. Simplified 12.7%

   \[\leadsto \color{blue}{0.5} \]
10. Add Preprocessing

Alternative 12: 3.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.2%

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

   1. sub-neg 53.2%

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

   2. +-commutative 53.2%

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

   3. add-cbrt-cube 52.4%

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

   4. add-sqr-sqrt 52.4%

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

   5. cbrt-prod 52.7%

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

   6. distribute-lft-neg-in 52.7%

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

   7. fma-define 52.7%

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

   8. pow1/3 53.7%

      \[\leadsto \mathsf{fma}\left(-\sqrt[3]{x}, \color{blue}{{\left(\sqrt{x}\right)}^{0.3333333333333333}}, \sqrt{x + 1}\right) \]

   9. pow1/2 53.7%

      \[\leadsto \mathsf{fma}\left(-\sqrt[3]{x}, {\color{blue}{\left({x}^{0.5}\right)}}^{0.3333333333333333}, \sqrt{x + 1}\right) \]

   10. pow-pow 53.7%

       \[\leadsto \mathsf{fma}\left(-\sqrt[3]{x}, \color{blue}{{x}^{\left(0.5 \cdot 0.3333333333333333\right)}}, \sqrt{x + 1}\right) \]

   11. metadata-eval 53.7%

       \[\leadsto \mathsf{fma}\left(-\sqrt[3]{x}, {x}^{\color{blue}{0.16666666666666666}}, \sqrt{x + 1}\right) \]

4. Applied egg-rr 53.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-\sqrt[3]{x}, {x}^{0.16666666666666666}, \sqrt{x + 1}\right)} \]

5. Taylor expanded in x around inf 3.5%

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

   1. distribute-rgt1-in 3.5%

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

   2. metadata-eval 3.5%

      \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt{\frac{1}{x}}\right) \]

   3. mul0-lft 3.5%

      \[\leadsto x \cdot \color{blue}{0} \]

   4. metadata-eval 3.5%

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

   5. distribute-rgt-out-- 3.5%

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

   6. distribute-lft-out-- 3.5%

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

   7. +-inverses 3.5%

      \[\leadsto 1 \cdot \color{blue}{0} \]

   8. metadata-eval 3.5%

      \[\leadsto \color{blue}{0} \]

7. Simplified 3.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer target: 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 2024110 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))