2sqrt (example 3.1)

Percentage Accurate: 53.9% → 99.8%

Time: 7.7s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 48.7%

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

   1. flip-- 49.5%

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

   2. div-inv 49.5%

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

   3. add-sqr-sqrt 49.2%

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

   4. add-sqr-sqrt 50.4%

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

3. Applied egg-rr 50.4%

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

   1. associate-*r/ 50.4%

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

   2. *-rgt-identity 50.4%

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

   3. remove-double-neg 50.4%

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

   4. sub-neg 50.4%

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

   5. div-sub 48.8%

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

   6. rem-square-sqrt 48.5%

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

   7. sqr-neg 48.5%

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

   8. div-sub 49.2%

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

   9. sqr-neg 49.2%

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

   10. +-commutative 49.2%

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

   11. rem-square-sqrt 50.4%

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

   12. associate--l+ 99.8%

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

   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%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-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((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-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((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} 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-5:
		tmp = 0.5 * math.sqrt((1.0 / x))
	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-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((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-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[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-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)) < 5.00000000000000024e-5

   1. Initial program 5.4%

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

      1. flip-- 6.5%

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

      2. div-inv 6.5%

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

      3. add-sqr-sqrt 5.9%

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

      4. add-sqr-sqrt 7.4%

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

   3. Applied egg-rr 7.4%

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

      1. associate-*r/ 7.4%

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

      2. *-rgt-identity 7.4%

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

      3. remove-double-neg 7.4%

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

      4. sub-neg 7.4%

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

      5. div-sub 5.6%

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

      6. rem-square-sqrt 5.2%

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

      7. sqr-neg 5.2%

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

      8. div-sub 5.9%

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

      9. sqr-neg 5.9%

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

      10. +-commutative 5.9%

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

      11. rem-square-sqrt 7.4%

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

      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%

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

   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 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 68.2%

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

      5. associate-/r/ 68.1%

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

      6. sqrt-pow2 68.1%

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

      7. metadata-eval 68.1%

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

      8. sqrt-pow2 67.9%

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

      9. +-commutative 67.9%

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

      10. metadata-eval 67.9%

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

      11. add-sqr-sqrt 68.2%

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

      12. add-sqr-sqrt 67.9%

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

      13. +-commutative 67.9%

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

      14. associate--l+ 67.9%

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

   9. Applied egg-rr 47.3%

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

   10. Taylor expanded in x around inf 99.0%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 98.5%

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

4. Final simplification 98.8%

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

Alternative 3: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (1.0 - math.sqrt(x)) + (x * 0.5)
	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(1.0 - sqrt(x)) + Float64(x * 0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.8%

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

      1. associate--l+ 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

   4. Applied egg-rr 98.8%

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

   if 1 < x

   1. Initial program 8.8%

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

      1. flip-- 10.3%

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

      2. div-inv 10.3%

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

      3. add-sqr-sqrt 9.7%

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

      4. add-sqr-sqrt 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. associate-*r/ 11.8%

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

      2. *-rgt-identity 11.8%

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

      3. remove-double-neg 11.8%

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

      4. sub-neg 11.8%

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

      5. div-sub 9.1%

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

      6. rem-square-sqrt 8.6%

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

      7. sqr-neg 8.6%

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

      8. div-sub 9.7%

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

      9. sqr-neg 9.7%

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

      10. +-commutative 9.7%

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

      11. rem-square-sqrt 11.8%

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

      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%

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

   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 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 69.7%

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

      5. associate-/r/ 69.6%

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

      6. sqrt-pow2 69.6%

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

      7. metadata-eval 69.6%

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

      8. sqrt-pow2 69.4%

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

      9. +-commutative 69.4%

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

      10. metadata-eval 69.4%

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

      11. add-sqr-sqrt 69.7%

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

      12. add-sqr-sqrt 69.4%

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

      13. +-commutative 69.4%

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

      14. associate--l+ 69.4%

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

   9. Applied egg-rr 49.9%

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

   10. Taylor expanded in x around inf 96.5%

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

4. Final simplification 97.5%

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

Alternative 4: 96.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.38) (/ 1.0 (+ 1.0 (pow x 1.5))) (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.38) {
		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 <= 0.38d0) 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 <= 0.38) {
		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 <= 0.38:
		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 <= 0.38)
		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 <= 0.38)
		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, 0.38], 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 < 0.38

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      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. sqr-neg 99.9%

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

      10. +-commutative 99.9%

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

      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%

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

   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. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. sqrt-pow1 99.9%

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

      2. metadata-eval 99.9%

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

      3. inv-pow 99.9%

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

      4. flip3-+ 99.9%

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

      5. associate-/r/ 99.9%

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

      6. sqrt-pow2 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-pow2 99.9%

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

      9. +-commutative 99.9%

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

      10. metadata-eval 99.9%

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

      11. add-sqr-sqrt 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. +-commutative 99.9%

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

      14. associate--l+ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 96.1%

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

   if 0.38 < x

   1. Initial program 8.8%

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

      1. flip-- 10.3%

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

      2. div-inv 10.3%

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

      3. add-sqr-sqrt 9.7%

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

      4. add-sqr-sqrt 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. associate-*r/ 11.8%

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

      2. *-rgt-identity 11.8%

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

      3. remove-double-neg 11.8%

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

      4. sub-neg 11.8%

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

      5. div-sub 9.1%

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

      6. rem-square-sqrt 8.6%

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

      7. sqr-neg 8.6%

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

      8. div-sub 9.7%

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

      9. sqr-neg 9.7%

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

      10. +-commutative 9.7%

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

      11. rem-square-sqrt 11.8%

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

      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%

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

   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 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 69.7%

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

      5. associate-/r/ 69.6%

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

      6. sqrt-pow2 69.6%

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

      7. metadata-eval 69.6%

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

      8. sqrt-pow2 69.4%

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

      9. +-commutative 69.4%

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

      10. metadata-eval 69.4%

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

      11. add-sqr-sqrt 69.7%

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

      12. add-sqr-sqrt 69.4%

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

      13. +-commutative 69.4%

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

      14. associate--l+ 69.4%

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

   9. Applied egg-rr 49.9%

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

   10. Taylor expanded in x around inf 96.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]

Alternative 5: 96.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.1%

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

   if 0.25 < x

   1. Initial program 8.8%

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

      1. flip-- 10.3%

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

      2. div-inv 10.3%

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

      3. add-sqr-sqrt 9.7%

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

      4. add-sqr-sqrt 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. associate-*r/ 11.8%

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

      2. *-rgt-identity 11.8%

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

      3. remove-double-neg 11.8%

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

      4. sub-neg 11.8%

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

      5. div-sub 9.1%

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

      6. rem-square-sqrt 8.6%

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

      7. sqr-neg 8.6%

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

      8. div-sub 9.7%

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

      9. sqr-neg 9.7%

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

      10. +-commutative 9.7%

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

      11. rem-square-sqrt 11.8%

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

      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%

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

   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 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. sqrt-pow1 99.6%

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

      2. metadata-eval 99.6%

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

      3. inv-pow 99.6%

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

      4. flip3-+ 69.7%

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

      5. associate-/r/ 69.6%

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

      6. sqrt-pow2 69.6%

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

      7. metadata-eval 69.6%

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

      8. sqrt-pow2 69.4%

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

      9. +-commutative 69.4%

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

      10. metadata-eval 69.4%

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

      11. add-sqr-sqrt 69.7%

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

      12. add-sqr-sqrt 69.4%

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

      13. +-commutative 69.4%

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

      14. associate--l+ 69.4%

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

   9. Applied egg-rr 49.9%

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

   10. Taylor expanded in x around inf 96.5%

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

4. Final simplification 96.3%

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

Alternative 6: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 / sqrt(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(x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.1%

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

   if 0.25 < x

   1. Initial program 8.8%

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

      1. flip-- 10.3%

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

      2. div-inv 10.3%

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

      3. add-sqr-sqrt 9.7%

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

      4. add-sqr-sqrt 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. associate-*r/ 11.8%

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

      2. *-rgt-identity 11.8%

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

      3. remove-double-neg 11.8%

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

      4. sub-neg 11.8%

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

      5. div-sub 9.1%

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

      6. rem-square-sqrt 8.6%

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

      7. sqr-neg 8.6%

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

      8. div-sub 9.7%

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

      9. sqr-neg 9.7%

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

      10. +-commutative 9.7%

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

      11. rem-square-sqrt 11.8%

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

      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%

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

   5. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. add-exp-log 94.0%

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

      3. pow1/2 94.0%

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

      4. log-pow 94.0%

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

      5. +-commutative 94.0%

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

      6. log1p-udef 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. *-commutative 94.0%

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

      2. exp-prod 94.0%

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

      3. unpow1/2 94.0%

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

   9. Simplified 94.0%

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

   10. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. exp-diff 0.0%

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

       5. rem-exp-log 0.0%

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

       6. rem-exp-log 96.3%

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

   12. Simplified 96.3%

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

       1. expm1-log1p-u 96.3%

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

       2. expm1-udef 9.7%

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

       3. associate-/r/ 9.7%

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

       4. metadata-eval 9.7%

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

       5. *-un-lft-identity 9.7%

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

   14. Applied egg-rr 9.7%

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

       1. expm1-def 96.3%

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

       2. expm1-log1p 96.3%

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

       3. count-2 96.3%

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

       4. associate-/r* 96.3%

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

       5. metadata-eval 96.3%

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

   16. Simplified 96.3%

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

4. Final simplification 96.2%

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

Alternative 7: 51.5% 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 48.7%

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

2. Taylor expanded in x around 0 46.0%

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

3. Final simplification 46.0%

   \[\leadsto 1 \]

Developer target: 99.8% 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 2023187 
(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)))