2sqrt (example 3.1)

Percentage Accurate: 52.9% → 99.7%

Time: 8.5s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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

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

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 53.7%

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

   1. flip-- 53.9%

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

   2. div-inv 53.9%

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

   3. add-sqr-sqrt 54.4%

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

   4. add-sqr-sqrt 54.4%

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

   5. associate--l+ 54.4%

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

3. Applied egg-rr 54.4%

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

   1. +-commutative 54.4%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 65000000:\\ \;\;\;\;\sqrt{1 + x} - {x}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 65000000.0)
   (- (sqrt (+ 1.0 x)) (pow x 0.5))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 65000000.0) {
		tmp = sqrt((1.0 + x)) - pow(x, 0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 65000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - (x ** 0.5d0)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 65000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.pow(x, 0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 65000000.0:
		tmp = math.sqrt((1.0 + x)) - math.pow(x, 0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 65000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - (x ^ 0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 65000000.0)
		tmp = sqrt((1.0 + x)) - (x ^ 0.5);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 65000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e7

   1. Initial program 99.3%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.3%

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

      2. flip3-- 99.2%

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

      3. add-cbrt-cube 99.1%

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

      4. rem-cube-cbrt 99.2%

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

      5. div-sub 99.2%

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

   5. Applied egg-rr 99.3%

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

      1. div-sub 99.3%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. metadata-eval 99.4%

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

      2. sqrt-pow2 99.2%

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

      3. metadata-eval 99.2%

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

      4. sqrt-pow2 99.2%

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

      5. add-sqr-sqrt 99.3%

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

      6. add-sqr-sqrt 99.2%

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

      7. sqrt-prod 99.2%

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

      8. *-commutative 99.2%

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

      9. flip3-- 99.3%

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

      10. add-sqr-sqrt 99.2%

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

      11. cancel-sign-sub-inv 99.2%

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

      12. pow1/2 99.3%

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

      13. sqrt-pow1 99.2%

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

      14. metadata-eval 99.2%

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

      15. pow1/2 99.3%

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

      16. sqrt-pow1 99.2%

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

      17. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. distribute-lft-neg-out 99.2%

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

       2. unsub-neg 99.2%

          \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.25} \cdot {x}^{0.25}} \]

       3. pow-sqr 99.4%

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

       4. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   if 6.5e7 < x

   1. Initial program 4.3%

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

      1. expm1-log1p-u 4.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 4.3%

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

      2. flip3-- 3.1%

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

      3. add-cbrt-cube 3.3%

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

      4. rem-cube-cbrt 2.9%

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

      5. div-sub 2.9%

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

   5. Applied egg-rr 3.1%

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

      1. div-sub 3.2%

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

      2. associate-+l+ 3.2%

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

      3. *-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 65000000:\\ \;\;\;\;\sqrt{1 + x} - {x}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 65000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 65000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 65000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 65000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 65000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 65000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 65000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 65000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 65000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e7

   1. Initial program 99.3%

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

   if 6.5e7 < x

   1. Initial program 4.3%

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

      1. expm1-log1p-u 4.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 4.3%

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

      2. flip3-- 3.1%

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

      3. add-cbrt-cube 3.3%

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

      4. rem-cube-cbrt 2.9%

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

      5. div-sub 2.9%

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

   5. Applied egg-rr 3.1%

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

      1. div-sub 3.2%

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

      2. associate-+l+ 3.2%

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

      3. *-commutative 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 65000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;1 + \left(\left(x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (+ 1.0 (- (+ (* x (* x -0.125)) (* x 0.5)) (sqrt x)))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = 1.0 + (((x * (x * -0.125)) + (x * 0.5)) - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.22d0) then
        tmp = 1.0d0 + (((x * (x * (-0.125d0))) + (x * 0.5d0)) - sqrt(x))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = 1.0 + (((x * (x * -0.125)) + (x * 0.5)) - Math.sqrt(x));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.22:
		tmp = 1.0 + (((x * (x * -0.125)) + (x * 0.5)) - math.sqrt(x))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(1.0 + Float64(Float64(Float64(x * Float64(x * -0.125)) + Float64(x * 0.5)) - sqrt(x)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.22)
		tmp = 1.0 + (((x * (x * -0.125)) + (x * 0.5)) - sqrt(x));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(1.0 + N[(N[(N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{\left(1 + \left(-0.125 \cdot {x}^{2} + 0.5 \cdot x\right)\right)} - \sqrt{x} \]
   3. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot x + -0.125 \cdot {x}^{2}\right)}\right) - \sqrt{x} \]

      2. unpow2 99.3%

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

      3. associate-*r* 99.3%

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

      4. distribute-rgt-out 99.3%

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

      5. *-commutative 99.3%

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

   4. Simplified 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. distribute-rgt-in 99.3%

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

      3. *-commutative 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 1.21999999999999997 < x

   1. Initial program 7.4%

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

      1. expm1-log1p-u 7.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 7.4%

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

      2. flip3-- 6.3%

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

      3. add-cbrt-cube 6.4%

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

      4. rem-cube-cbrt 6.1%

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

      5. div-sub 6.1%

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

   5. Applied egg-rr 6.3%

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

      1. div-sub 6.4%

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

      2. associate-+l+ 6.4%

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

      3. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 97.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;1 + \left(\left(x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.22d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.22:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - sqrt(x));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.22)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{\left(1 + \left(-0.125 \cdot {x}^{2} + 0.5 \cdot x\right)\right)} - \sqrt{x} \]
   3. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot x + -0.125 \cdot {x}^{2}\right)}\right) - \sqrt{x} \]

      2. unpow2 99.3%

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

      3. associate-*r* 99.3%

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

      4. distribute-rgt-out 99.3%

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

      5. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   if 1.21999999999999997 < x

   1. Initial program 7.4%

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

      1. expm1-log1p-u 7.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 7.4%

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

      2. flip3-- 6.3%

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

      3. add-cbrt-cube 6.4%

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

      4. rem-cube-cbrt 6.1%

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

      5. div-sub 6.1%

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

   5. Applied egg-rr 6.3%

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

      1. div-sub 6.4%

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

      2. associate-+l+ 6.4%

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

      3. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 97.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + ((x * 0.5) - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	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 * sqrt((1.0d0 / x))
    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.sqrt((1.0 / x));
	}
	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.sqrt((1.0 / x))
	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 * sqrt(Float64(1.0 / x)));
	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 * sqrt((1.0 / x));
	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[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.9%

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

      1. associate--l+ 98.9%

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

      2. +-commutative 98.9%

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

      3. *-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. expm1-log1p-u 7.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 7.4%

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

      2. flip3-- 6.3%

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

      3. add-cbrt-cube 6.4%

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

      4. rem-cube-cbrt 6.1%

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

      5. div-sub 6.1%

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

   5. Applied egg-rr 6.3%

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

      1. div-sub 6.4%

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

      2. associate-+l+ 6.4%

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

      3. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 97.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 7: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 95.4%

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

   if 0.25 < x

   1. Initial program 8.1%

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

      1. expm1-log1p-u 8.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 8.1%

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

      2. flip3-- 7.0%

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

      3. add-cbrt-cube 7.1%

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

      4. rem-cube-cbrt 6.8%

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

      5. div-sub 6.8%

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

   5. Applied egg-rr 7.0%

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

      1. div-sub 7.1%

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

      2. associate-+l+ 7.1%

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

      3. *-commutative 7.1%

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

   7. Simplified 7.1%

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

   8. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

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

Alternative 8: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	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 * sqrt((1.0d0 / x))
    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.sqrt((1.0 / x));
	}
	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.sqrt((1.0 / x))
	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 * sqrt(Float64(1.0 / x)));
	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 * sqrt((1.0 / x));
	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[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. 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 99.8%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. pow2 99.8%

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

      3. pow1/2 99.8%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around 0 97.2%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. expm1-log1p-u 7.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]

   3. Applied egg-rr 7.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 7.4%

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

      2. flip3-- 6.3%

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

      3. add-cbrt-cube 6.4%

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

      4. rem-cube-cbrt 6.1%

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

      5. div-sub 6.1%

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

   5. Applied egg-rr 6.3%

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

      1. div-sub 6.4%

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

      2. associate-+l+ 6.4%

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

      3. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in x around inf 97.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

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

Alternative 9: 50.9% 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 53.7%

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

2. Taylor expanded in x around 0 50.9%

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

3. Final simplification 50.9%

   \[\leadsto 1 \]

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 2023305 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

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