2sqrt (example 3.1)

Percentage Accurate: 53.2% → 99.0%

Time: 7.1s

Alternatives: 7

Speedup: 2.0×

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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 1e-8) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 1e-8) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 1e-8:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 1e-8)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 1e-8)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-8], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1e-8

   1. Initial program 4.0%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. add-sqr-sqrt 4.3%

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

      4. add-sqr-sqrt 4.6%

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

      5. associate--l+ 4.6%

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

   3. Applied egg-rr 4.6%

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

      1. associate-*r/ 4.6%

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

      2. *-rgt-identity 4.6%

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

      3. +-commutative 4.6%

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

      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%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 98.9%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.0%

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

      5. add-sqr-sqrt 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. hypot-def 98.8%

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

      8. pow1/2 98.8%

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

      9. sqrt-pow1 99.0%

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

      10. metadata-eval 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}\right)\right)}^{\left(-1 + -1\right)} \]

      11. pow1/2 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. sqrt-pow1 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around inf 98.5%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot x\right)}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]
   9. Step-by-step derivation

      1. *-commutative 98.5%

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

      2. *-lft-identity 98.5%

         \[\leadsto {\left(\sqrt{2} \cdot {\color{blue}{x}}^{0.25}\right)}^{-2} \]

   10. Simplified 98.5%

       \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot {x}^{0.25}\right)}}^{-2} \]
   11. Step-by-step derivation

       1. *-commutative 98.5%

          \[\leadsto {\color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]

       2. unpow-prod-down 98.6%

          \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2} \cdot {\left(\sqrt{2}\right)}^{-2}} \]

       3. pow-pow 98.4%

          \[\leadsto \color{blue}{{x}^{\left(0.25 \cdot -2\right)}} \cdot {\left(\sqrt{2}\right)}^{-2} \]

       4. metadata-eval 98.4%

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

       5. sqrt-pow2 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto {x}^{-0.5} \cdot {2}^{\color{blue}{-1}} \]

       7. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 1e-8 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 10^{-8}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. flip-- 50.8%

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

   2. div-inv 50.8%

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

   3. add-sqr-sqrt 50.6%

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

   4. add-sqr-sqrt 50.8%

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

   5. associate--l+ 50.8%

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

3. Applied egg-rr 50.8%

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

   1. associate-*r/ 50.8%

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

   2. *-rgt-identity 50.8%

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

   3. +-commutative 50.8%

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

   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%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   4. 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%

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

   5. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 99.9%

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

      3. unpow-prod-down 99.9%

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

      4. pow-prod-up 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. hypot-def 99.9%

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

      8. pow1/2 99.9%

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

      9. sqrt-pow1 99.9%

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

      10. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}\right)\right)}^{\left(-1 + -1\right)} \]

      11. pow1/2 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. sqrt-pow1 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. metadata-eval 99.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around 0 98.1%

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

   if 1 < x

   1. Initial program 4.6%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.0%

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

      4. add-sqr-sqrt 5.3%

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

      5. associate--l+ 5.3%

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

   3. Applied egg-rr 5.3%

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

      1. associate-*r/ 5.3%

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

      2. *-rgt-identity 5.3%

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

      3. +-commutative 5.3%

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

      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%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 98.9%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.0%

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

      5. add-sqr-sqrt 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. hypot-def 98.8%

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

      8. pow1/2 98.8%

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

      9. sqrt-pow1 99.0%

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

      10. metadata-eval 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}\right)\right)}^{\left(-1 + -1\right)} \]

      11. pow1/2 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. sqrt-pow1 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around inf 98.1%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot x\right)}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]
   9. Step-by-step derivation

      1. *-commutative 98.1%

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

      2. *-lft-identity 98.1%

         \[\leadsto {\left(\sqrt{2} \cdot {\color{blue}{x}}^{0.25}\right)}^{-2} \]

   10. Simplified 98.1%

       \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot {x}^{0.25}\right)}}^{-2} \]
   11. Step-by-step derivation

       1. *-commutative 98.1%

          \[\leadsto {\color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]

       2. unpow-prod-down 98.2%

          \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2} \cdot {\left(\sqrt{2}\right)}^{-2}} \]

       3. pow-pow 98.1%

          \[\leadsto \color{blue}{{x}^{\left(0.25 \cdot -2\right)}} \cdot {\left(\sqrt{2}\right)}^{-2} \]

       4. metadata-eval 98.1%

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

       5. sqrt-pow2 99.6%

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

       6. metadata-eval 99.6%

          \[\leadsto {x}^{-0.5} \cdot {2}^{\color{blue}{-1}} \]

       7. metadata-eval 99.6%

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

   12. Applied egg-rr 99.6%

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

4. Final simplification 98.9%

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

Alternative 4: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;1 + x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.38) (+ 1.0 (* x -0.5)) (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 + (x * -0.5);
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.38d0) then
        tmp = 1.0d0 + (x * (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 + (x * -0.5);
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.38:
		tmp = 1.0 + (x * -0.5)
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.38)
		tmp = Float64(1.0 + Float64(x * -0.5));
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.38)
		tmp = 1.0 + (x * -0.5);
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.38], N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   4. 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%

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

   5. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-cube-cbrt 100.0%

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

      3. sqrt-prod 100.0%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} + \sqrt{1 + x}} \]

      4. fma-def 100.0%

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

      5. pow2 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

      2. rem-sqrt-square 100.0%

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

      3. unpow1 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

      5. pow-sqr 100.0%

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

      6. unpow1/2 100.0%

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

      7. unpow1/2 100.0%

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

      8. fabs-sqr 100.0%

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

      9. unpow1/2 100.0%

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

      10. unpow1/2 100.0%

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

      11. pow-sqr 100.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(2 \cdot 0.5\right)}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

      12. metadata-eval 100.0%

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

      13. unpow1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 97.3%

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

   if 0.38 < x

   1. Initial program 4.6%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.0%

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

      4. add-sqr-sqrt 5.3%

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

      5. associate--l+ 5.3%

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

   3. Applied egg-rr 5.3%

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

      1. associate-*r/ 5.3%

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

      2. *-rgt-identity 5.3%

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

      3. +-commutative 5.3%

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

      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%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 98.9%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.0%

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

      5. add-sqr-sqrt 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. hypot-def 98.8%

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

      8. pow1/2 98.8%

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

      9. sqrt-pow1 99.0%

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

      10. metadata-eval 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}\right)\right)}^{\left(-1 + -1\right)} \]

      11. pow1/2 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. sqrt-pow1 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around inf 98.1%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot x\right)}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]
   9. Step-by-step derivation

      1. *-commutative 98.1%

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

      2. *-lft-identity 98.1%

         \[\leadsto {\left(\sqrt{2} \cdot {\color{blue}{x}}^{0.25}\right)}^{-2} \]

   10. Simplified 98.1%

       \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot {x}^{0.25}\right)}}^{-2} \]
   11. Step-by-step derivation

       1. *-commutative 98.1%

          \[\leadsto {\color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]

       2. unpow-prod-down 98.2%

          \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2} \cdot {\left(\sqrt{2}\right)}^{-2}} \]

       3. pow-pow 98.1%

          \[\leadsto \color{blue}{{x}^{\left(0.25 \cdot -2\right)}} \cdot {\left(\sqrt{2}\right)}^{-2} \]

       4. metadata-eval 98.1%

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

       5. sqrt-pow2 99.6%

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

       6. metadata-eval 99.6%

          \[\leadsto {x}^{-0.5} \cdot {2}^{\color{blue}{-1}} \]

       7. metadata-eval 99.6%

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

   12. Applied egg-rr 99.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;1 + x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]

Alternative 5: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;1 + x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.38) (+ 1.0 (* x -0.5)) (sqrt (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 + (x * -0.5);
	} else {
		tmp = sqrt((0.25 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.38d0) then
        tmp = 1.0d0 + (x * (-0.5d0))
    else
        tmp = sqrt((0.25d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 + (x * -0.5);
	} else {
		tmp = Math.sqrt((0.25 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.38:
		tmp = 1.0 + (x * -0.5)
	else:
		tmp = math.sqrt((0.25 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.38)
		tmp = Float64(1.0 + Float64(x * -0.5));
	else
		tmp = sqrt(Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.38)
		tmp = 1.0 + (x * -0.5);
	else
		tmp = sqrt((0.25 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.38], N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.25 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   4. 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%

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

   5. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-cube-cbrt 100.0%

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

      3. sqrt-prod 100.0%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} + \sqrt{1 + x}} \]

      4. fma-def 100.0%

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

      5. pow2 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

      2. rem-sqrt-square 100.0%

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

      3. unpow1 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

      5. pow-sqr 100.0%

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

      6. unpow1/2 100.0%

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

      7. unpow1/2 100.0%

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

      8. fabs-sqr 100.0%

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

      9. unpow1/2 100.0%

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

      10. unpow1/2 100.0%

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

      11. pow-sqr 100.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(2 \cdot 0.5\right)}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

      12. metadata-eval 100.0%

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

      13. unpow1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 97.3%

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

   if 0.38 < x

   1. Initial program 4.6%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.0%

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

      4. add-sqr-sqrt 5.3%

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

      5. associate--l+ 5.3%

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

   3. Applied egg-rr 5.3%

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

      1. associate-*r/ 5.3%

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

      2. *-rgt-identity 5.3%

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

      3. +-commutative 5.3%

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

      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%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 98.9%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.0%

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

      5. add-sqr-sqrt 98.8%

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

      6. add-sqr-sqrt 98.7%

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

      7. hypot-def 98.8%

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

      8. pow1/2 98.8%

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

      9. sqrt-pow1 99.0%

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

      10. metadata-eval 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}\right)\right)}^{\left(-1 + -1\right)} \]

      11. pow1/2 99.0%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}\right)\right)}^{\left(-1 + -1\right)} \]

      12. sqrt-pow1 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}\right)\right)}^{\left(-1 + -1\right)} \]

      13. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{\color{blue}{0.25}}\right)\right)}^{\left(-1 + -1\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   8. Taylor expanded in x around inf 98.1%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot x\right)}^{0.25} \cdot \sqrt{2}\right)}}^{-2} \]
   9. Step-by-step derivation

      1. *-commutative 98.1%

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

      2. *-lft-identity 98.1%

         \[\leadsto {\left(\sqrt{2} \cdot {\color{blue}{x}}^{0.25}\right)}^{-2} \]

   10. Simplified 98.1%

       \[\leadsto {\color{blue}{\left(\sqrt{2} \cdot {x}^{0.25}\right)}}^{-2} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 98.1%

          \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \cdot \sqrt{{\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}}} \]

       2. sqrt-unprod 98.1%

          \[\leadsto \color{blue}{\sqrt{{\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2} \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}}} \]

       3. unpow-prod-down 98.2%

          \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt{2}\right)}^{-2} \cdot {\left({x}^{0.25}\right)}^{-2}\right)} \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       4. sqrt-pow2 98.5%

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

       5. metadata-eval 98.5%

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

       6. metadata-eval 98.5%

          \[\leadsto \sqrt{\left(\color{blue}{0.5} \cdot {\left({x}^{0.25}\right)}^{-2}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       7. pow-pow 98.8%

          \[\leadsto \sqrt{\left(0.5 \cdot \color{blue}{{x}^{\left(0.25 \cdot -2\right)}}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       8. metadata-eval 98.8%

          \[\leadsto \sqrt{\left(0.5 \cdot {x}^{\color{blue}{-0.5}}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       9. metadata-eval 98.8%

          \[\leadsto \sqrt{\left(0.5 \cdot {x}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       10. sqrt-pow1 98.8%

           \[\leadsto \sqrt{\left(0.5 \cdot \color{blue}{\sqrt{{x}^{-1}}}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       11. inv-pow 98.8%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \cdot {\left(\sqrt{2} \cdot {x}^{0.25}\right)}^{-2}} \]

       12. unpow-prod-down 98.8%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{-2} \cdot {\left({x}^{0.25}\right)}^{-2}\right)}} \]

       13. sqrt-pow2 99.1%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\color{blue}{{2}^{\left(\frac{-2}{2}\right)}} \cdot {\left({x}^{0.25}\right)}^{-2}\right)} \]

       14. metadata-eval 99.1%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \left({2}^{\color{blue}{-1}} \cdot {\left({x}^{0.25}\right)}^{-2}\right)} \]

       15. metadata-eval 99.1%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(\color{blue}{0.5} \cdot {\left({x}^{0.25}\right)}^{-2}\right)} \]

       16. pow-pow 99.4%

           \[\leadsto \sqrt{\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(0.5 \cdot \color{blue}{{x}^{\left(0.25 \cdot -2\right)}}\right)} \]

       17. metadata-eval 99.4%

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

       18. metadata-eval 99.4%

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

       19. sqrt-pow1 99.4%

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

       20. inv-pow 99.4%

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

       21. swap-sqr 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;1 + x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.25}{x}}\\ \end{array} \]

Alternative 6: 51.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 1.0 (* x 0.5))))

C

double code(double x) {
	return 1.0 / (1.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + (x * 0.5));
}

Python

def code(x):
	return 1.0 / (1.0 + (x * 0.5))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

   1. flip-- 50.8%

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

   2. div-inv 50.8%

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

   3. add-sqr-sqrt 50.6%

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

   4. add-sqr-sqrt 50.8%

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

   5. associate--l+ 50.8%

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

3. Applied egg-rr 50.8%

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

   1. associate-*r/ 50.8%

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

   2. *-rgt-identity 50.8%

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

   3. +-commutative 50.8%

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

   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%

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

5. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. add-cube-cbrt 99.6%

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

   3. sqrt-prod 99.5%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} + \sqrt{1 + x}} \]

   4. fma-def 99.6%

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

   5. pow2 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)}} \]
8. Step-by-step derivation

   1. unpow2 99.6%

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

   2. rem-sqrt-square 99.6%

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

   3. unpow1 99.6%

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

   4. metadata-eval 99.6%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\left|{\left(\sqrt[3]{x}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

   5. pow-sqr 99.4%

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

   6. unpow1/2 99.4%

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

   7. unpow1/2 99.4%

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

   8. fabs-sqr 99.4%

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

   9. unpow1/2 99.4%

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

   10. unpow1/2 99.4%

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

   11. pow-sqr 99.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(2 \cdot 0.5\right)}}, \sqrt{\sqrt[3]{x}}, \sqrt{1 + x}\right)} \]

   12. metadata-eval 99.6%

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

   13. unpow1 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 50.3%

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

    1. *-commutative 50.3%

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

12. Simplified 50.3%

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

13. Final simplification 50.3%

    \[\leadsto \frac{1}{1 + x \cdot 0.5} \]

Alternative 7: 51.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 66000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 66000000.0)
   (- (sqrt (+ 1.0 x)) (sqrt x))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 66000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 66000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 66000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 66000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 66000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023339 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (if (<= x 66000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))

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