2sqrt (example 3.1)

Percentage Accurate: 53.7% → 99.7%

Time: 5.0s

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: 53.7% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\sqrt{x} + \sqrt{1 + x}\right)}^{-2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (pow (+ (sqrt x) (sqrt (+ 1.0 x))) -2.0)))

C

double code(double x) {
	return sqrt(pow((sqrt(x) + sqrt((1.0 + x))), -2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((sqrt(x) + sqrt((1.0d0 + x))) ** (-2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.pow((Math.sqrt(x) + Math.sqrt((1.0 + x))), -2.0));
}

Python

def code(x):
	return math.sqrt(math.pow((math.sqrt(x) + math.sqrt((1.0 + x))), -2.0))

Julia

function code(x)
	return sqrt((Float64(sqrt(x) + sqrt(Float64(1.0 + x))) ^ -2.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((sqrt(x) + sqrt((1.0 + x))) ^ -2.0));
end

Wolfram

code[x_] := N[Sqrt[N[Power[N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. flip-- 51.5%

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

   2. div-inv 51.5%

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

   3. add-sqr-sqrt 51.7%

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

   4. add-sqr-sqrt 52.6%

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

3. Applied egg-rr 52.6%

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

   1. *-commutative 52.6%

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

   2. associate-/r/ 52.6%

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

   3. +-commutative 52.6%

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

   4. associate--l+ 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. +-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. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.7%

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

   3. inv-pow 99.7%

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

   4. inv-pow 99.7%

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

   5. pow-prod-up 99.8%

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

   6. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-6) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-6) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-6:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-6)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 5.00000000000000041e-6

   1. Initial program 4.8%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. add-sqr-sqrt 4.9%

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

      4. add-sqr-sqrt 6.8%

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

   3. Applied egg-rr 6.8%

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

      1. *-commutative 6.8%

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

      2. associate-/r/ 6.8%

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

      3. +-commutative 6.8%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-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 99.1%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.2%

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

      5. +-commutative 99.2%

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

      6. add-sqr-sqrt 99.1%

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

      7. add-sqr-sqrt 99.0%

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

      8. hypot-def 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.1%

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

      11. metadata-eval 99.1%

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

      12. pow1/2 99.1%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 97.8%

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

      1. unpow2 97.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \sqrt{\frac{1}{x}} \]

      2. rem-square-sqrt 99.4%

         \[\leadsto \frac{1}{\color{blue}{2}} \cdot \sqrt{\frac{1}{x}} \]

      3. metadata-eval 99.4%

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

      4. unpow1/2 99.4%

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

      5. rem-exp-log 92.1%

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

      6. exp-neg 92.1%

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

      7. exp-prod 92.1%

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

      8. distribute-lft-neg-out 92.1%

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

      9. distribute-rgt-neg-in 92.1%

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

      10. metadata-eval 92.1%

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

      11. exp-to-pow 99.6%

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

   10. Simplified 99.6%

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

   if 5.00000000000000041e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

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

Alternative 3: 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 51.6%

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

   1. flip-- 51.5%

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

   2. div-inv 51.5%

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

   3. add-sqr-sqrt 51.7%

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

   4. add-sqr-sqrt 52.6%

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

3. Applied egg-rr 52.6%

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

   1. *-commutative 52.6%

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

   2. associate-/r/ 52.6%

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

   3. +-commutative 52.6%

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

   4. associate--l+ 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

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 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 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. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. +-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 99.9%

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

      3. unpow-prod-down 99.9%

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

      4. pow-prod-up 99.9%

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

      5. +-commutative 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. hypot-def 99.9%

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

      9. pow1/2 99.9%

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

      10. sqrt-pow1 99.9%

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

      11. metadata-eval 99.9%

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

      12. pow1/2 99.9%

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

      13. sqrt-pow1 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 98.9%

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

   if 1 < x

   1. Initial program 5.3%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.7%

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

      4. add-sqr-sqrt 7.5%

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

   3. Applied egg-rr 7.5%

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

      1. *-commutative 7.5%

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

      2. associate-/r/ 7.5%

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

      3. +-commutative 7.5%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-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 99.1%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.2%

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

      5. +-commutative 99.2%

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

      6. add-sqr-sqrt 99.1%

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

      7. add-sqr-sqrt 99.0%

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

      8. hypot-def 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.1%

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

      11. metadata-eval 99.1%

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

      12. pow1/2 99.1%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 97.4%

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

      1. unpow2 97.4%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \sqrt{\frac{1}{x}} \]

      2. rem-square-sqrt 99.0%

         \[\leadsto \frac{1}{\color{blue}{2}} \cdot \sqrt{\frac{1}{x}} \]

      3. metadata-eval 99.0%

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

      4. unpow1/2 99.0%

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

      5. rem-exp-log 91.7%

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

      6. exp-neg 91.7%

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

      7. exp-prod 91.7%

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

      8. distribute-lft-neg-out 91.7%

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

      9. distribute-rgt-neg-in 91.7%

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

      10. metadata-eval 91.7%

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

      11. exp-to-pow 99.2%

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

   10. Simplified 99.2%

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

4. Final simplification 99.0%

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

Alternative 5: 96.9% 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 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

   if 0.25 < x

   1. Initial program 6.0%

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

      1. flip-- 6.0%

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

      2. div-inv 6.0%

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

      3. add-sqr-sqrt 6.4%

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

      4. add-sqr-sqrt 8.2%

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

   3. Applied egg-rr 8.2%

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

      1. *-commutative 8.2%

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

      2. associate-/r/ 8.2%

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

      3. +-commutative 8.2%

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

      4. associate--l+ 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. +-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 99.1%

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

      3. unpow-prod-down 98.9%

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

      4. pow-prod-up 99.2%

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

      5. +-commutative 99.2%

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

      6. add-sqr-sqrt 99.1%

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

      7. add-sqr-sqrt 99.0%

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

      8. hypot-def 99.0%

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

      9. pow1/2 99.0%

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

      10. sqrt-pow1 99.1%

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

      11. metadata-eval 99.1%

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

      12. pow1/2 99.1%

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

      13. sqrt-pow1 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 96.8%

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

      1. unpow2 96.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \sqrt{\frac{1}{x}} \]

      2. rem-square-sqrt 98.4%

         \[\leadsto \frac{1}{\color{blue}{2}} \cdot \sqrt{\frac{1}{x}} \]

      3. metadata-eval 98.4%

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

      4. unpow1/2 98.4%

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

      5. rem-exp-log 91.2%

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

      6. exp-neg 91.2%

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

      7. exp-prod 91.2%

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

      8. distribute-lft-neg-out 91.2%

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

      9. distribute-rgt-neg-in 91.2%

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

      10. metadata-eval 91.2%

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

      11. exp-to-pow 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 98.5%

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

Alternative 6: 51.6% 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 51.6%

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

2. Taylor expanded in x around 0 51.1%

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

3. Final simplification 51.1%

   \[\leadsto 1 \]

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 2023268 
(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)))