2sqrt (example 3.1)

Percentage Accurate: 53.0% → 99.7%

Time: 6.9s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. add-sqr-sqrt 50.1%

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

   4. add-sqr-sqrt 50.4%

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

3. Applied egg-rr 50.4%

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

   1. associate-*r/ 50.4%

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

   2. *-rgt-identity 50.4%

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

   3. remove-double-neg 50.4%

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

   4. sub-neg 50.4%

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

   5. div-sub 49.7%

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

   6. rem-square-sqrt 49.7%

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

   7. sqr-neg 49.7%

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

   8. div-sub 50.1%

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

   9. +-commutative 50.1%

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

   10. sqr-neg 50.1%

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

   11. rem-square-sqrt 50.4%

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

   12. associate--l+ 99.7%

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

   13. +-inverses 99.7%

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

   14. metadata-eval 99.7%

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

   15. sub-neg 99.7%

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

   16. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0001) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 0.0001d0) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 0.0001:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.00000000000000005e-4

   1. Initial program 4.7%

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

      1. flip3-- 3.2%

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

      2. div-inv 3.2%

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

      3. sqrt-pow2 3.5%

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

      4. metadata-eval 3.5%

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

      5. sqrt-pow2 3.1%

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

      6. metadata-eval 3.1%

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

      7. add-sqr-sqrt 3.1%

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

      8. add-sqr-sqrt 3.1%

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

      9. associate-+r+ 3.1%

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

      10. sqrt-unprod 3.1%

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

   3. Applied egg-rr 3.1%

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

      1. associate-*r/ 3.1%

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

      2. *-rgt-identity 3.1%

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

      3. +-commutative 3.1%

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

      4. associate-+l+ 3.1%

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

      5. +-commutative 3.1%

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

      6. associate-+l+ 3.1%

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

      7. distribute-rgt1-in 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. expm1-log1p-u 99.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 6.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 6.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 6.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 6.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   8. Applied egg-rr 6.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 99.5%

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

   10. Simplified 99.5%

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

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 97.5%

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

      1. associate--l+ 97.5%

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

      2. *-commutative 97.5%

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

   4. Applied egg-rr 97.5%

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

   if 1 < x

   1. Initial program 6.9%

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

      1. flip3-- 5.4%

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

      2. div-inv 5.4%

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

      3. sqrt-pow2 5.7%

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

      4. metadata-eval 5.7%

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

      5. sqrt-pow2 5.4%

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

      6. metadata-eval 5.4%

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

      7. add-sqr-sqrt 5.4%

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

      8. add-sqr-sqrt 5.4%

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

      9. associate-+r+ 5.4%

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

      10. sqrt-unprod 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-*r/ 5.4%

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

      2. *-rgt-identity 5.4%

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

      3. +-commutative 5.4%

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

      4. associate-+l+ 5.4%

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

      5. +-commutative 5.4%

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

      6. associate-+l+ 5.4%

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

      7. distribute-rgt1-in 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in x around inf 97.7%

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

      1. expm1-log1p-u 97.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 97.7%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. sqr-neg 99.9%

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

      8. div-sub 99.9%

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

      9. +-commutative 99.9%

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

      10. sqr-neg 99.9%

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

      11. rem-square-sqrt 99.9%

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

      12. associate--l+ 99.8%

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

      13. +-inverses 99.8%

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

      14. metadata-eval 99.8%

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

      15. sub-neg 99.8%

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

      16. remove-double-neg 99.8%

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

   5. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. add-cbrt-cube 99.8%

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

      3. pow3 99.8%

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

      4. pow1/2 99.8%

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

      5. pow-pow 99.8%

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

      6. +-commutative 99.8%

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

      7. metadata-eval 99.8%

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

      8. pow1/3 99.8%

         \[\leadsto \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} + \sqrt{x}} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{1}{\color{blue}{{\left({\left(1 + x\right)}^{1.5}\right)}^{0.3333333333333333}} + \sqrt{x}} \]

   8. Taylor expanded in x around 0 94.9%

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

   if 1 < x

   1. Initial program 6.9%

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

      1. flip3-- 5.4%

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

      2. div-inv 5.4%

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

      3. sqrt-pow2 5.7%

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

      4. metadata-eval 5.7%

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

      5. sqrt-pow2 5.4%

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

      6. metadata-eval 5.4%

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

      7. add-sqr-sqrt 5.4%

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

      8. add-sqr-sqrt 5.4%

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

      9. associate-+r+ 5.4%

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

      10. sqrt-unprod 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-*r/ 5.4%

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

      2. *-rgt-identity 5.4%

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

      3. +-commutative 5.4%

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

      4. associate-+l+ 5.4%

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

      5. +-commutative 5.4%

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

      6. associate-+l+ 5.4%

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

      7. distribute-rgt1-in 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in x around inf 97.7%

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

      1. expm1-log1p-u 97.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 96.5%

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

Alternative 5: 96.8% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(0.5 * N[Power[x, -0.5], $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 91.9%

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

   if 0.25 < x

   1. Initial program 6.9%

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

      1. flip3-- 5.4%

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

      2. div-inv 5.4%

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

      3. sqrt-pow2 5.7%

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

      4. metadata-eval 5.7%

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

      5. sqrt-pow2 5.4%

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

      6. metadata-eval 5.4%

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

      7. add-sqr-sqrt 5.4%

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

      8. add-sqr-sqrt 5.4%

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

      9. associate-+r+ 5.4%

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

      10. sqrt-unprod 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-*r/ 5.4%

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

      2. *-rgt-identity 5.4%

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

      3. +-commutative 5.4%

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

      4. associate-+l+ 5.4%

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

      5. +-commutative 5.4%

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

      6. associate-+l+ 5.4%

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

      7. distribute-rgt1-in 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in x around inf 97.7%

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

      1. expm1-log1p-u 97.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]

      3. inv-pow 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{x}^{-1}}}\right)} - 1\right) \]

      4. sqrt-pow1 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   8. Applied egg-rr 7.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 95.1%

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

Alternative 6: 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 49.8%

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

2. Taylor expanded in x around 0 46.0%

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

3. Final simplification 46.0%

   \[\leadsto 1 \]

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 2023276 
(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)))