2sqrt (example 3.1)

Percentage Accurate: 52.6% → 99.7%

Time: 7.2s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% 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 52.1%

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

   1. flip-- 52.4%

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

   2. div-inv 52.4%

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

   3. add-sqr-sqrt 52.7%

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

   4. add-sqr-sqrt 53.0%

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

   5. associate--l+ 53.0%

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

3. Applied egg-rr 53.0%

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

   1. +-commutative 53.0%

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

   2. associate-+l- 99.8%

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

   3. +-inverses 99.8%

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

   4. metadata-eval 99.8%

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

   5. associate-*r/ 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-6) (* (pow x -0.5) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 4d-6) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-6:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-6)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-6)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-6], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 4.6%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. add-sqr-sqrt 5.4%

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

      4. add-sqr-sqrt 6.1%

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

      5. associate--l+ 6.1%

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

   3. Applied egg-rr 6.1%

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

      1. +-commutative 6.1%

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

      2. associate-+l- 99.6%

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

      3. +-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. associate-*r/ 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 98.9%

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

      8. pow1/2 98.9%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 98.4%

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

       1. *-commutative 98.4%

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

       2. *-lft-identity 98.4%

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

   12. Simplified 98.4%

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

       1. *-commutative 98.4%

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

       2. unpow-prod-down 98.2%

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

       3. pow-pow 98.2%

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

       4. metadata-eval 98.2%

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

       5. sqrt-pow2 99.7%

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

       6. metadata-eval 99.7%

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

       7. metadata-eval 99.7%

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

   14. Applied egg-rr 99.7%

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

   if 3.99999999999999982e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. *-commutative 99.6%

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

   4. Simplified 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

      2. distribute-rgt-in 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 1.25 < x

   1. Initial program 6.4%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 7.6%

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

      4. add-sqr-sqrt 8.3%

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

      5. associate--l+ 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      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. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 99.0%

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

      8. pow1/2 99.0%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. *-lft-identity 97.0%

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

   12. Simplified 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow-prod-down 96.8%

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

       3. pow-pow 96.8%

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

       4. metadata-eval 96.8%

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

       5. sqrt-pow2 98.3%

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

       6. metadata-eval 98.3%

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

       7. metadata-eval 98.3%

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

   14. Applied egg-rr 98.3%

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

4. Final simplification 99.0%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. *-commutative 99.6%

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

   4. Simplified 99.6%

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

   if 1.25 < x

   1. Initial program 6.4%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 7.6%

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

      4. add-sqr-sqrt 8.3%

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

      5. associate--l+ 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      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. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 99.0%

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

      8. pow1/2 99.0%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. *-lft-identity 97.0%

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

   12. Simplified 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow-prod-down 96.8%

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

       3. pow-pow 96.8%

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

       4. metadata-eval 96.8%

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

       5. sqrt-pow2 98.3%

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

       6. metadata-eval 98.3%

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

       7. metadata-eval 98.3%

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

   14. Applied egg-rr 98.3%

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

4. Final simplification 98.9%

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

Alternative 5: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

      1. associate--l+ 99.3%

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

      2. *-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   if 1 < x

   1. Initial program 6.4%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 7.6%

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

      4. add-sqr-sqrt 8.3%

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

      5. associate--l+ 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      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. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 99.0%

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

      8. pow1/2 99.0%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. *-lft-identity 97.0%

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

   12. Simplified 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow-prod-down 96.8%

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

       3. pow-pow 96.8%

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

       4. metadata-eval 96.8%

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

       5. sqrt-pow2 98.3%

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

       6. metadata-eval 98.3%

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

       7. metadata-eval 98.3%

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

   14. Applied egg-rr 98.3%

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

4. Final simplification 98.8%

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

Alternative 6: 97.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--l+ 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. inv-pow 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. unpow-prod-down 99.8%

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

      4. +-commutative 99.8%

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

      5. add-sqr-sqrt 99.8%

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

      6. add-sqr-sqrt 99.8%

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

      7. hypot-def 99.9%

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

      8. pow1/2 99.9%

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

      9. sqrt-pow1 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. pow1/2 99.9%

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

      13. sqrt-pow1 99.9%

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

      14. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. pow-sqr 99.9%

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

      2. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 6.4%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 7.6%

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

      4. add-sqr-sqrt 8.3%

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

      5. associate--l+ 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      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. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 99.0%

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

      8. pow1/2 99.0%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. *-lft-identity 97.0%

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

   12. Simplified 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow-prod-down 96.8%

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

       3. pow-pow 96.8%

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

       4. metadata-eval 96.8%

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

       5. sqrt-pow2 98.3%

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

       6. metadata-eval 98.3%

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

       7. metadata-eval 98.3%

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

   14. Applied egg-rr 98.3%

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

4. Final simplification 98.1%

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

Alternative 7: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.7%

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

   if 0.25 < x

   1. Initial program 6.4%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. add-sqr-sqrt 7.6%

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

      4. add-sqr-sqrt 8.3%

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

      5. associate--l+ 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      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. Step-by-step derivation

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.0%

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

      3. unpow-prod-down 98.9%

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

      4. +-commutative 98.9%

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

      5. add-sqr-sqrt 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. hypot-def 99.0%

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

      8. pow1/2 99.0%

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

      9. sqrt-pow1 99.0%

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

      10. +-commutative 99.0%

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

      11. metadata-eval 99.0%

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

      12. pow1/2 99.0%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

   7. Applied egg-rr 98.8%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in x around inf 97.0%

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

       1. *-commutative 97.0%

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

       2. *-lft-identity 97.0%

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

   12. Simplified 97.0%

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

       1. *-commutative 97.0%

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

       2. unpow-prod-down 96.8%

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

       3. pow-pow 96.8%

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

       4. metadata-eval 96.8%

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

       5. sqrt-pow2 98.3%

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

       6. metadata-eval 98.3%

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

       7. metadata-eval 98.3%

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

   14. Applied egg-rr 98.3%

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

4. Final simplification 97.0%

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

Alternative 8: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5625}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.55) 1.0 (sqrt (/ 0.5625 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.55) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5625 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5625d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.55) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5625 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.55:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5625 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.55)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5625 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = 1.0;
	else
		tmp = sqrt((0.5625 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.55], 1.0, N[Sqrt[N[(0.5625 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.7%

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

   if 0.55000000000000004 < x

   1. Initial program 6.4%

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

      1. flip3-- 4.9%

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

      2. sqrt-pow2 5.2%

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

      3. metadata-eval 5.2%

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

      4. sqrt-pow2 5.0%

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

      5. metadata-eval 5.0%

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

      6. add-sqr-sqrt 5.0%

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

      7. associate-+l+ 5.0%

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

      8. add-sqr-sqrt 5.0%

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

      9. +-commutative 5.0%

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

      10. fma-def 5.0%

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

   3. Applied egg-rr 5.0%

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

   4. Taylor expanded in x around inf 20.0%

      \[\leadsto \color{blue}{0.75 \cdot \sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. *-commutative 20.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.75} \]

   6. Simplified 20.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.75} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 20.0%

         \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{1}{x}} \cdot 0.75} \cdot \sqrt{\sqrt{\frac{1}{x}} \cdot 0.75}} \]

      2. sqrt-unprod 20.0%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{\frac{1}{x}} \cdot 0.75\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.75\right)}} \]

      3. swap-sqr 20.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(0.75 \cdot 0.75\right)}} \]

      4. add-sqr-sqrt 20.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{x}} \cdot \left(0.75 \cdot 0.75\right)} \]

      5. metadata-eval 20.0%

         \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.5625}} \]

   8. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.5625}} \]
   9. Step-by-step derivation

      1. associate-*l/ 20.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.5625}{x}}} \]

      2. metadata-eval 20.0%

         \[\leadsto \sqrt{\frac{\color{blue}{0.5625}}{x}} \]

   10. Simplified 20.0%

       \[\leadsto \color{blue}{\sqrt{\frac{0.5625}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5625}{x}}\\ \end{array} \]

Alternative 9: 50.6% 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 52.1%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto 1 \]

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023325 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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