2sqrt (example 3.1)

Percentage Accurate: 51.4% → 99.7%

Time: 8.3s

Alternatives: 7

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: 51.4% 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

\[\begin{array}{l} \\ {\left({\left(\sqrt{x + 1} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ x 1.0)) (sqrt x)) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt((x + 1.0)) + sqrt(x)), 2.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt((x + 1.0d0)) + sqrt(x)) ** 2.0d0) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt((x + 1.0)) + Math.sqrt(x)), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt((x + 1.0)) + math.sqrt(x)), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(Float64(x + 1.0)) + sqrt(x)) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((x + 1.0)) + sqrt(x)) ^ 2.0) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 55.0%

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

   1. flip-- 55.1%

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

   2. div-inv 55.1%

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

   3. add-sqr-sqrt 55.8%

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

   4. add-sqr-sqrt 56.4%

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

3. Applied egg-rr 56.4%

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

   1. associate-*r/ 56.4%

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

   2. *-rgt-identity 56.4%

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

   3. remove-double-neg 56.4%

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

   4. sub-neg 56.4%

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

   5. div-sub 55.1%

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

   6. rem-square-sqrt 54.9%

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

   7. sqr-neg 54.9%

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

   8. div-sub 55.8%

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

   9. +-commutative 55.8%

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

   10. sqr-neg 55.8%

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

   11. rem-square-sqrt 56.4%

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

   12. associate--l+ 99.8%

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

   13. +-inverses 99.8%

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

   14. metadata-eval 99.8%

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

   15. sub-neg 99.8%

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

   16. remove-double-neg 99.8%

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

5. Simplified 99.8%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.8%

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

   3. inv-pow 99.8%

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

   4. +-commutative 99.8%

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

   5. inv-pow 99.8%

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

   6. +-commutative 99.8%

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

   7. pow-prod-up 99.7%

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

   8. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sqrt-pow1 99.8%

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. pow-sqr 99.6%

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

   5. pow-prod-down 99.8%

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

   6. pow2 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 2e-5) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = 0.5 * sqrt((1.0 / x));
	} 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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 2d-5) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 2e-5:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 2e-5)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 2e-5)
		tmp = 0.5 * sqrt((1.0 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-5], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 2.00000000000000016e-5

   1. Initial program 4.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 6.7%

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

      4. add-sqr-sqrt 7.8%

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

   3. Applied egg-rr 7.8%

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

      1. associate-*r/ 7.8%

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

      2. *-rgt-identity 7.8%

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

      3. remove-double-neg 7.8%

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

      4. sub-neg 7.8%

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

      5. div-sub 5.1%

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

      6. rem-square-sqrt 4.9%

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

      7. sqr-neg 4.9%

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

      8. div-sub 6.7%

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

      9. +-commutative 6.7%

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

      10. sqr-neg 6.7%

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

      11. rem-square-sqrt 7.8%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

      16. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. +-commutative 99.6%

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

      5. inv-pow 99.6%

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

      6. +-commutative 99.6%

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

      7. pow-prod-up 99.5%

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

      8. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 71.2%

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

      5. associate-/r/ 71.2%

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

      6. sqrt-pow2 71.1%

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

      7. metadata-eval 71.1%

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

      8. sqrt-pow2 70.9%

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

      9. metadata-eval 70.9%

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

      10. add-sqr-sqrt 71.2%

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

      11. add-sqr-sqrt 70.8%

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

      12. associate-+r- 70.8%

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

   9. Applied egg-rr 49.5%

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

   10. Taylor expanded in x around inf 99.3%

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

   if 2.00000000000000016e-5 < (-.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 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 3: 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]

Derivation

1. Initial program 55.0%

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

   1. flip-- 55.1%

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

   2. div-inv 55.1%

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

   3. add-sqr-sqrt 55.8%

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

   4. add-sqr-sqrt 56.4%

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

3. Applied egg-rr 56.4%

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

   1. associate-*r/ 56.4%

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

   2. *-rgt-identity 56.4%

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

   3. remove-double-neg 56.4%

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

   4. sub-neg 56.4%

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

   5. div-sub 55.1%

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

   6. rem-square-sqrt 54.9%

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

   7. sqr-neg 54.9%

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

   8. div-sub 55.8%

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

   9. +-commutative 55.8%

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

   10. sqr-neg 55.8%

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

   11. rem-square-sqrt 56.4%

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

   12. associate--l+ 99.8%

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

   13. +-inverses 99.8%

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

   14. metadata-eval 99.8%

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

   15. sub-neg 99.8%

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

   16. remove-double-neg 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. associate--l+ 99.0%

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

      3. *-commutative 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 1 < x

   1. Initial program 6.3%

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

      1. flip-- 6.6%

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

      2. div-inv 6.6%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.3%

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

   3. Applied egg-rr 9.3%

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

      1. associate-*r/ 9.3%

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

      2. *-rgt-identity 9.3%

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

      3. remove-double-neg 9.3%

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

      4. sub-neg 9.3%

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

      5. div-sub 6.5%

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

      6. rem-square-sqrt 6.2%

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

      7. sqr-neg 6.2%

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

      8. div-sub 8.1%

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

      9. +-commutative 8.1%

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

      10. sqr-neg 8.1%

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

      11. rem-square-sqrt 9.3%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

      16. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. +-commutative 99.6%

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

      5. inv-pow 99.6%

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

      6. +-commutative 99.6%

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

      7. pow-prod-up 99.5%

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

      8. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 71.6%

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

      5. associate-/r/ 71.6%

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

      6. sqrt-pow2 71.5%

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

      7. metadata-eval 71.5%

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

      8. sqrt-pow2 71.4%

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

      9. metadata-eval 71.4%

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

      10. add-sqr-sqrt 71.6%

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

      11. add-sqr-sqrt 71.3%

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

      12. associate-+r- 71.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in x around inf 98.1%

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

4. Final simplification 98.6%

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

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8) (/ 1.0 (+ 1.0 (pow x 1.5))) (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 + pow(x, 1.5));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = 1.0d0 / (1.0d0 + (x ** 1.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 + Math.pow(x, 1.5));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = 1.0 / (1.0 + math.pow(x, 1.5))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(1.0 / Float64(1.0 + (x ^ 1.5)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = 1.0 / (1.0 + (x ^ 1.5));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(1.0 / N[(1.0 + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   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. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. sqr-neg 99.9%

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

      8. div-sub 99.9%

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

      9. +-commutative 99.9%

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

      10. sqr-neg 99.9%

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

      11. rem-square-sqrt 99.9%

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

      12. associate--l+ 99.9%

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

      13. +-inverses 99.9%

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

      14. metadata-eval 99.9%

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

      15. sub-neg 99.9%

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

      16. remove-double-neg 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. inv-pow 99.9%

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

      4. +-commutative 99.9%

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

      5. inv-pow 99.9%

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

      6. +-commutative 99.9%

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

      7. pow-prod-up 99.9%

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

      8. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. sqrt-pow1 99.9%

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

      2. metadata-eval 99.9%

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

      3. inv-pow 99.9%

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

      4. flip3-+ 99.9%

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

      5. associate-/r/ 99.9%

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

      6. sqrt-pow2 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-pow2 99.9%

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

      9. metadata-eval 99.9%

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

      10. add-sqr-sqrt 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. associate-+r- 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 95.6%

       \[\leadsto \color{blue}{\frac{1}{1 + {x}^{1.5}}} \]

   if 1.80000000000000004 < x

   1. Initial program 6.3%

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

      1. flip-- 6.6%

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

      2. div-inv 6.6%

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

      3. add-sqr-sqrt 8.1%

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

      4. add-sqr-sqrt 9.3%

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

   3. Applied egg-rr 9.3%

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

      1. associate-*r/ 9.3%

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

      2. *-rgt-identity 9.3%

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

      3. remove-double-neg 9.3%

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

      4. sub-neg 9.3%

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

      5. div-sub 6.5%

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

      6. rem-square-sqrt 6.2%

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

      7. sqr-neg 6.2%

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

      8. div-sub 8.1%

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

      9. +-commutative 8.1%

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

      10. sqr-neg 8.1%

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

      11. rem-square-sqrt 9.3%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

      16. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. +-commutative 99.6%

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

      5. inv-pow 99.6%

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

      6. +-commutative 99.6%

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

      7. pow-prod-up 99.5%

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

      8. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 71.6%

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

      5. associate-/r/ 71.6%

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

      6. sqrt-pow2 71.5%

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

      7. metadata-eval 71.5%

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

      8. sqrt-pow2 71.4%

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

      9. metadata-eval 71.4%

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

      10. add-sqr-sqrt 71.6%

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

      11. add-sqr-sqrt 71.3%

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

      12. associate-+r- 71.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in x around inf 98.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.2%

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

   if 0.25 < x

   1. Initial program 7.1%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. add-sqr-sqrt 8.8%

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

      4. add-sqr-sqrt 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. associate-*r/ 10.0%

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

      2. *-rgt-identity 10.0%

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

      3. remove-double-neg 10.0%

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

      4. sub-neg 10.0%

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

      5. div-sub 7.3%

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

      6. rem-square-sqrt 7.0%

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

      7. sqr-neg 7.0%

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

      8. div-sub 8.8%

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

      9. +-commutative 8.8%

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

      10. sqr-neg 8.8%

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

      11. rem-square-sqrt 10.0%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

      16. remove-double-neg 99.6%

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

   5. Simplified 99.6%

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

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.6%

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

      3. inv-pow 99.6%

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

      4. +-commutative 99.6%

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

      5. inv-pow 99.6%

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

      6. +-commutative 99.6%

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

      7. pow-prod-up 99.5%

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

      8. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 71.9%

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

      5. associate-/r/ 71.8%

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

      6. sqrt-pow2 71.7%

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

      7. metadata-eval 71.7%

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

      8. sqrt-pow2 71.6%

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

      9. metadata-eval 71.6%

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

      10. add-sqr-sqrt 71.9%

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

      11. add-sqr-sqrt 71.5%

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

      12. associate-+r- 71.5%

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

   9. Applied egg-rr 50.7%

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

   10. Taylor expanded in x around inf 97.5%

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

4. Final simplification 96.8%

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

Alternative 7: 49.8% 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 55.0%

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

2. Taylor expanded in x around 0 53.0%

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

3. Final simplification 53.0%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023278 
(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)))